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= gk (αp ) = 0 for k ∈ I ∗ (p). Therefore, according to (8), gk (αp(l) ) = (gk (αp ))(kl) = 0, (z)) = g0 (z) and, according to (6), (8) and (11), g0 (αp ) = g0 (αp ) = (l) 1(ν1 ) for l ∈ I(p). The values z = αp , l ∈ I(p), are thus simple roots of g1 (z). Another simple root of g1 (z) is z = 0. From (7) and (12), we also have, for n ∈ N, n > 1, g1 (αp n(l) ) = 1(ν1 ) g1 [αp(l) (n − 1)] = (1(ν1 ) )n−1 g1 (αp(l) ) = 0. All n(l) , for n ∈ N and l ∈ I(p), are thus simple roots of g1 (αp z). From (7) and (12), it is easy to show that any sum of these n(l) is also a simple root of g1 (αp z). Therefore, (l) all roots of g1 (αp z) in Ep are simple and can be expressed as n = l∈I(p) nl , where nl ∈ N for l ∈ I(p). Let −p 1 = k∈I ∗ (p) 1(k) . For every j ∈ I(p), we set (αp z), we first note that n∈Np (1 −p z/n) defines a function having a simple root at each n ∈ Np . The infinite product (αp z) is p-addiholomorphic on all Ep and has the same roots, with the same multiplicities, as the infinite product (14) multiplied by z. A reasoning similar to the one done for usual holomorphic functions allows us here to conclude that there exits an p-addiholomorphic function ψ defined on Ep and a constant K ∈ Ep such that g1 (αp z) = zK exp(ψ(z)) (αp z) 1 g (z) = αp 0 . (19) = ψ (z) + + z n∈N• z p −p np g1 (αp z) p (z) = g (z) −p h (z), where h (z) = that we shall use below. Lemma 4.1. The function G has a power series representation on Ep . Proof. It is clear that the function g is p-addiholomorphic on Ep , except at the roots of g1 (αp z), that is at z = 0 and z = n for n ∈ Np . For the function h , one can see that the series (20) converges uniformly as the series n∈N•p 1/np , whose convergence follows from the one of n∈N•p 1/np p proved above. This series will be p-addiderivable, except at z = 0 and z = n for n ∈ Np . Thus, the functions g and h have simple (z) = (n, z) , z −p n (z) = (n, z) h , z −p n are functions which have power series on Ep for each n ∈ Np . Thus where g˜ and h g˜ (n, z) = (n, z) = h (z) = and (z) = lim z→n (z −p n)αp2 1(1) gp−1 (αp z) + αp g0 (αp z) (αp z) (z) = lim ⎣ z→n0 z→n0 z n(z −p n) n∈Np ⎡ ⎤ ⎡ is periodic of period 1 with respect to each copy of the m monoids M in Ap . (z + 1(k) ) = g (z), k ∈ I(p), being direct, we shall only prove that h (z + 1(k) ) = h (z) for k ∈ I(p). In what follows we shall use the sets (l) (l) " p,k,j = { # N l∈I(p) nl ∈ $Np,j : nk% = 1} and Np,k,j = { l∈I(p) nl ∈ Np,j : nk = 0} for j ∈ I(p), j = k and j = p−1 + k (mod p). We first have 2 (z). is uniformly bounded with respect to the absolute pprevalue on Ep . Proof. Let xk ∈ Ap , k ∈ I(p), and z = k∈I(p) xk skp ∈ Ep . Due to the result of Lemma 4.2, it will be sufficient to show that G is uniformly bounded with respect to the absolute p-prevalue on each of the regions {z ∈ Ep : x0 = y (k) , where y ∈ R+ , y ≤ 1 and k ∈ I(p)}. But in these regions the analytic function G is bounded with respect to the absolute p-prevalue for every xj p ≤ a such that a ∈ R+ and j ∈ I ∗ (p), because every continuous function defined on a compact set is bounded on this set. It thus remains to show that G is bounded with respect to the absolute p-prevalues for xj p > a. To this end, we shall successively show that g (z) and h (z) tends toward finite values when xj tends toward ∞(k) for j ∈ I ∗ (p) and k ∈ I(p), where ∞(k) designates 1(k) ∞. We begin with the function g . Using (10) and (19), we obtain lim (z) = (x), one directly proves that αp cannot be an antiprimary p-addisym number. This implies that *& 'xp−1 +αp exp(1(k) ) lim = 0. xp−1 →∞ exp 1 ∗ k∈I (p) (z) = αp sp . (22) xp−1 →∞(p−1) (z) = αp(i) sp , (z) when xl , l = 1, 2, . . . , p − 2, tends toward ∞(p−1−i) , i ∈ I(p), (p−l) can be computed by writting g0 (z) and g1 (z) in terms of exp(z (j) sp ), j ∈ I(p), instead of exp(z (j) sp ), j ∈ I(p). We find lim (z) = αp(i) sp−l p is uniformly bounded with respect to the absolute p-prevalue on Ep . To prove that the function h is also uniformly bounded with respect to the absolute p-prevalue on Ep , we use its expression (20). We have that z p−1 1 (z) p ≤ z + p z p −p np . p p n∈N• p lim h (z) p ≤ p lim p−1 p−1 x1 →∞ x1 →∞(k) np0 + xp1 n0 ∈N∗ pp−2 −1 njp xpp−1 −jp−1 0 1 aj . (24) = p lim pp−1 pp−1 x1 →∞ n + x ∗ 0 1 j=0 n0 ∈N Each of the terms between parentheses in (24) is uniformly bounded with respect to the absolute p-prevalue on Ep . For j = 0, 1, . . . , pp−2 − 1, we have indeed njp xpp−1 −jp−1 0 1 n0 ∈N∗ (z) is bounded with respect to the absolute p-prevalue when x1 → ∞(k) and k ∈ I(p). Repeating the above reasoning for each j ∈ I ∗ (p) shows that h (z) is bounded with respect to the absolute p-prevalue when xj → ∞(k) and k ∈ I(p). This shows that h is uniformly bounded with respect to absolute p-prevalue on Ep . The same conclusion can then be applied to the function G . Using the distance resulting from the absolute p-prevalue, let Eδp be the subset of Ep = Ep,1 whose points are at a distance larger then or equal to a given δ ∈ R∗+ form any point of E◦p = E◦p,1 . We call p-addisym surface of Ep every subset S of Ep such that at every z ∈ S one can define a patch having two real dimensions. By defining the curvilinear integral on Ep in a manner similar to the one this integral is defined on C, one directly shows that this new integral has the same properties as the later. We then have the following generalization of the Cauchy integral formula. Lemma 4.4. Let f be an p-addiholomorphic function inside and on the boundary C of a simply connected subset D of an p-addisym surface of Eδp . If z0 is an interior point of (z) = is p-addiholomorphic and uniformly bounded with respect to the absolute p-prevalue on Ep . From Lemma 4.5, it follows that G is constant on Eδp . The fact that G is periodic with respect to each copies of M in Ap and that E◦p is made of (p − 1)p/2 subspaces of codimension 2 of the vectorial space E2;p not containing any of the p2 monoids M of Ep implies that G is constant on all Ep . To determine the value of this constant, we shall compute G (0) g 1 (αp z) 1 ⎠⎦ G (0) = lim ⎣αp z 0 −p ⎝1 + pz p p p z→0 z z − n g1 (αp z) p n∈N•p ⎤ ⎡ 1(1) (αp z)p+1 1(2) (αp z)2p+1 + + ... 1 αp z + p! (2p)! = lim ⎣ −p 1⎦ (1) (2) p+1 2p+1 (αp z) (αp z) z→0 z αp z + 1 (p+1)! + 1 (2p+1)! + ... ) / .( 1(1) pαpp z p 1(2) p{2[(p + 1)!]2 −p (2p + 1)!}αp2p z 2p 1 + . . . −p 1 = lim 1+ + z→0 z (p + 1)! (2p + 1)![(p + 1)!]2 ( ) 1(1) pαpp z p−1 1(2) p{2[(p + 1)!]2 −p (2p + 1)!}αp2p z 2p−1 = lim + . . . = 0. + z→0 (p + 1)! (2p + 1)![(p + 1)!]2 Thus G (z) ≡ 0 and g (z) = h (z) for z ∈ Ep , from which the result follows. Theorem 4.6 shows that ψ (z) ≡ 0 in (19), so that ψ(z) = c for every z ∈ Ep , where c ∈ Ep is a constant. From (18), we then deduce that g1 (αp z) (z). From (5), it follows that K0 = lim K0 z→0 (αp z) = lim = αp . z→0 z (αp z) (αp z) is given by its Maclaurin’s series g1 (αp z) (αp z)/αp z. This means that −p (z + pαp ) = gk (z), for z ∈ Ap and k ∈ I(p). It follows that the functions generalizing the trigonometric functions to the set Ap have no non-zero root in Ap if p ≥ 3. These functions are thus non-periodic on Ap . This completes the proof of Theorem 4.9. If p ≥ 3, none of the functions gk , k ∈ I(p), is periodic on Ap .
l ∈ I(p),
k ∈ I ∗ (p).
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C. Gauthier / Central European Journal of Mathematics 4(3) 2006 395–412 (l)
However (g1
Np,j =
⎧ ⎨ ⎩
(( p−1 +j+1)( mod p))
(l)
p−1 + j + 2)(modp) 2 and n( p−1 +j+2)( mod p) ∈ N∗ ,
nl −p n( p−12 +j+1)( mod p) : nl ∈ N, l ∈ I(p), l = ( 2
l∈I(p)
2
and Np = {
(l)
nl : nl ∈ N, l ∈ I(p) and
l∈I(p)
(l)
nl = 0}.
(13)
l∈I(p)
It is easy to see that the Np,j form a partition of Np . Writing the reduced expressions of n ∈ Np , it is also easy to show that the set of n ∈ Np is identical to the set of n(k) for n ∈ Np, p−3 = N•p and k ∈ I(p). 2 In order to obtain an infinite product to represent g1
1 −p
n∈N•p
k∈I(p)
z p z = 1 − p n(k) n n∈N•
(14)
p
is then a function whose all roots are simple and are given by n ∈ Np . (0) (1) (p−2) For every n0 +n1 +· · ·+np−2 ∈ N•p , let ni0 = max{n0 , n1 , . . . , np−2 }. We can replace (0) (1) (p−2) (0) (1) (p−2) (0) n(k) = (n0 + n1 + · · · + np−2 )(k) of (14) by [(ni0 + ni1 + · · · + nip−2 )(i0 ) ](k) = (ni0 + (1)
(p−2)
ni1 + · · · + nip−2 )((i0 +k)( mod p)) , where ij ∈ {0, 1, . . . , p − 2} and j = 0, 1, . . . , p − 2. For (0)
(1)
(p−2)
n0 +n1 +· · ·+np−2 ∈ N•p , one can thus always assume that n0 = max{n0 , n1 , . . . , np−2 }. Let us now show that the infinite product (14) converges with respect to the absolute (0) (1) (p−2) (0) (1) p-prevalue on Ep . If n0 + n1 + · · · + np−2 = n ∈ N•p , then for every m0 + m1 + · · · +
C. Gauthier / Central European Journal of Mathematics 4(3) 2006 395–412
401
(p−2)
mp−2 ∈ N•p we have that m0 m1 mp−2 mp−2 m0 m1 z p p z ≤ 1 + −p ··· ··· 1 −p n n p n0 =1 n1 =0 np−2 =0 n0 =1 n1 =0 np−2 =0 p ⎛ ⎞ mp−2 mp−2 m0 m0 m1 m1 p p z z ⎠ ≤ ··· exp −p ··· = exp ⎝ −p n n p p n0 =1 n1 =0 np−2 =0 n0 =1 n1 =0 np−2 =0 ⎞ ⎛ mp−2 m0 m1 1 p ⎠ ⎝ (15) ≤ exp z p (p − 1) ··· np . n0 =1 n1 =0
np−2 =0 (0)
p
(1)
(p−2)
Consider the last absolute p-prevalue of (15). We know that n0 + n1 + · · · + np−2 = (0) (0) (p−3) (0) (1) (p−2) n0 + 1(1) (n1 + · · · + np−2 ). We can thus treat n0 + n1 + · · · + np−2 as a p-addisym number whose coordinates are (p − 2)-addisym numbers. Multiplying this number by (0) (p−3) the p − 2 numbers which are its p-addisym conjugates and by n0 + (n1 + · · · + np−2 ), (0) (p−3) we find np0 + (n1 + · · · + np−2 )p . For some Mi ∈ R+ , i ∈ I(p − 2), it follows that (0) (1) (p−3) (i) (n1 + n2 + · · · + np−2 )p = M0 + 1(1) i∈I(p−3) Mi+1 . If Ni = Mi+1 for i ∈ I(p − 3), then (0) (p−3) (i) Ni . (16) np0 + (n1 + · · · + np−2 )p = np0 + M0 + 1(1) i∈I(p−3)
Considering (16) as a p-addisym number having (p − 3)-addisym coordinates, it follows that the multiplication of this number by the p − 2 numbers which are its p-addisym (i) (i) conjugates and by np0 +M0 + i∈I(p−3) Ni gives (np0 +M0 )p +( i∈I(p−3) Ni )p . Repeating this reasoning the appropriate number of times to render primary the final product of the p−2 p-addisym conjugates, we arrive at an expression which is larger then np0 . If we do the same multiplications of p-addisym conjugates as above for the numerator of the last absolute p-prevalue of (15), we find a sum of terms that are all products of powers of ni , i = 0, 1, . . . , p − 2, the total power of which being always equal to pp−2 − 1. Consequently ep−2 p ne0o ne11 . . . np−2 1 . (0) ≤ (p−2) p pp−2 (n0 + n(1) n 0 e ,e ,...,e ∈N 1 + · · · + np−2 ) 0 1 p−2 p e0 +e1 +···+ep−2 =pp−2 −1
Knowing that we can set n0 ≥ ni for i = 0, 1, . . . , p − 2, it follows that 1 c0 (0) ≤ p, (1) (p−2) (n0 + n1 + · · · + np−2 )p n0 p where c0 ∈ R+ is a constant. We then obtain ⎞ ⎛ mp−2 mp−2 m0 m1 m1 z p c0 m0 ≤ exp ⎝ z pp (p − 1) ⎠ ··· ··· 1 −p p n n n0 =1 n1 =0 np−2 =0 n0 =1 n1 =0 np−2 =0 0 p m0 1 p = exp c z p , np n =1 0 0
(17)
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C. Gauthier / Central European Journal of Mathematics 4(3) 2006 395–412
where c ∈ R+ is a constant. Therefore ∞ z p ≤ exp c z pp 1 −p n−p . n n∈N•p n=1 p
−p Since ∞ converges for every p > 1, the infinite product (14) will converge with n=1 n respect to the absolute p-prevalue on Ep . This result legitimizes the rearrangement of its factors we have already done. The function g1
1 −p
n∈N•p
z p n
.
(18)
To show that the function ψ is constant on Ep , let us define the function g(u) = log(1+u), u ∈ Ep , as the reciprocal series of the one representing the function f (u) = exp(u) −p 1 ([1, chapter 1]). One can then see ([8, p. 350]) that the series n∈N•p log (1 −p (z/n)p ) converges uniformly on every compact subset of Ep which does not include n ∈ N•p . We can thus apply the logarithmic p-addiderivation to the two members of (18). This implies that pz p−1 g
We shall now show that ψ (z) ≡ 0 on Ep . Let us consider the function G
pz p−1 1 + z n∈N• z p −p np
(20)
p
1 1 1 1 z = + = + + z n∈N n (z −p n) z n∈N z −p n n
p
p
for every z ∈ Ep such that z = n et n ∈ Np . The three following lemmas will give properties of the function G
C. Gauthier / Central European Journal of Mathematics 4(3) 2006 395–412
403
poles at every n ∈ Np and n = 0. This implies that g
g˜
h
˜
˜
∞
aj (n)(z −p n)j ,
j=0
if z −p n ≤ r(n) and ˜
∞
bj (n)(z −p n)j ,
j=0
if z−p n ≤ R(n), where aj (n), bj (n) ∈ Ep for j ∈ N and r(n), R(n) ∈ R+ . Consequently a0 (n) aj+1 (n)(z −p n)j + z −p n j=0 ∞
g
b0 (n) (z) = bj+1 (n)(z −p n)j . + z −p n j=0 ∞
h
Applying l’Hˆopital’s rule, which directly generalizes to Ep , we find that a0 (n) = lim (z −p n)g
αp g0
z→n
= 1.
For every n0 ∈ Np , we also have ⎤ z(z −p n0 ) ⎦ z −p n0 + b0 (n0 ) = lim (z −p n0 )h
z(z −p n0 ) ⎥ ⎢z ⎥ = 1. + = lim ⎢ ⎦ z→n0 ⎣ n0 n(z − n) p n∈N p
n =n0
Similarly, we show that b0 (0) = 1. Thus G
(z) = g
(z) −p h
(z) =
∞
[aj+1 (n) −p bj+1 (n)] (z −p n)j .
j=0
Lemma 4.2. The function G
404
C. Gauthier / Central European Journal of Mathematics 4(3) 2006 395–412
Proof. The verification of g
h
1 1 1 (z + 1 ) = + + z + 1(k) n∈N z + 1(k) −p n n p,k 1 1 1 1 + + + 1(p−k) + = (k) (k) z + 1 −p n n z+1 z n∈N (k)
p,l
l∈I(p),l =k
+
n∈Np,k n =1(k)
1 1 + (k) z −p (n −p 1 ) n
+
n∈Np,l
1 1 + (k) z −p (n −p 1 ) n
.
l∈I(p),l =k
Setting m = n −p 1(k) , it follows that
h
1 1 1 1 (p−k) (z + 1 ) = + +1 + + z + 1(k) z z − m + 1(k) pm m∈Np,k 1 1 + = + z −p m m + 1(k) (k) (k)
m∈Np,l ,m=−p 1
l∈I(p),l =k
1 1 + + 1(p−k) + (k) z+1 z
&
1 1 + z −p m m
m∈Np,j ,j∈I(p)
j=k,k+1,...,( p−1 +k−1)( mod p) 2
&
+
m∈N p−1 p, 2 +k ( mod p) m =−p 1(k)
(
)
1 1 + z −p m m &
+
m∈Np,j ,j∈I(p)
(k) ,j∈I(p) " m∈N p,k,j ,m=1 p−1 j=k+1,..., 2 +k−1 ( mod
(
+
1 + −p 1(k)
)
(k) ,j∈I(p) " m∈N p,k,j ,m=1 p−1 j=k+1,..., 2 +k−1 ( mod
(
1 1 −p m m + 1(k)
1 1 −p (k) m+1 m
−p
'
1 1 −p m m + 1(k)
m∈Np,j ,j∈I(p)
)
1 + m m∈N
( p−1 2 +k)( mod p)
'
1 1 + z −p m m
j=k,k+1,...,( p−1 +k−1)( mod p) 2
−p
+
−p
1 1 + + 1(p−k) + = (k) z+1 z −p 1
1 1 + z −p m m
j=( p−1 +k+1)( mod p),...,(p+k−1)( mod p) 2
(p−k)
1 1 + z −p m m
p,
p)
m =−p 1(k)
1 + m p)
j=(
p−1 +k+1 2
# m∈N p,k,j ,j∈I(p)
)( mod p),...,(p+k−1)( mod p)
1 + m
'
C. Gauthier / Central European Journal of Mathematics 4(3) 2006 395–412
m∈Np,j ,j∈I(p)
1 1 + z −p m m
1 + z m∈N
=
p
1 1 + z −p m m
−p
j=( p−1 +k+1)( mod p),...,(p+k−1)( mod p) 2
j=(
405
p−1 +k+1 2
1 m
# m∈N p,k,j ,j∈I(p)
)( mod p),...,(p+k−1)( mod p)
= h
Lemma 4.3. The function G
xp−1
→∞(p−1)
g
1 + limxp−1 →∞(p−1) = αp sp
(
p−1 sp x0 +···+sp xp−2 1+ k∈I ∗ (p) exp (1) xp−1 ) ( p−1 sp x0 +···+sp xp−2 (1) exp αp xp−1 1+ (1) xp−1
(
p−1 sp x0 +···+sp xp−2 (1) xp−1 ) ( p−1 sp x0 +···+sp xp−2 (1) exp αp xp−1 1+ (1) xp−1
k∈I ∗ (p)
1 + limxp−1 →∞(p−1)
)
(k) (1) αp xp−1
)
.
(21)
(k) (1)
1(p−k) exp αp xp−1 1+
But
' & sp x0 +···+sp−1 xp−2 (k) (1) p (1) k∈I ∗ (p) exp αp xp−1 1 + xp−1 ' & lim xp−1 →∞(p−1) sp x0 +···+sp−1 xp−2 (1) p exp αp xp−1 1 + (1) xp−1 (k) (1) (k) (1) exp α x ∗ p p−1 exp(αp xp−1 ) k∈I (p) = lim lim = (1) (1) xp−1 →∞(p−1) xp−1 →∞(p−1) exp(αp x exp(αp xp−1 ) ∗ p−1 ) k∈I (p) *& + ' α x p p−1 exp(1(k) ) = lim . xp−1 →∞ exp 1 ∗
k∈I (p)
(p+k−1) Using (4) with x = (sp−1 and the continuity on Ep of the function defined by p )
406
C. Gauthier / Central European Journal of Mathematics 4(3) 2006 395–412
the absolute p-prevalue allow us to show that & 'xp−1 exp(1(k) ) = 0. lim xp−1 →∞ exp 1 From expression (5) for g1
Therefore, the numerator of (21) equals 1. The same reasoning shows that its denominator also equals 1. Thus lim g
The method used to obtain (22) also gives lim
xp−1 →∞(p−1−i)
g
i ∈ I(p).
Finally, the limits of g
xl →∞(p−1−i)
g
.
This shows that the function g
h
But, for k ∈ I(p),
1 1 = lim = 0. p−1 limx1 →∞(k) (x0 + sp x1 + · · · + sp xp−1 ) x1 →∞(k) z p p
Thus lim h
x1 →∞(k)
z p−1 (z) p ≤ p lim z p −p np x1 →∞(k) p n∈N•p 1 p−1 ≤ p lim
z p z p −p np x1 →∞(k) p n∈N•p p−1 1 . = p lim x1 p (p−1) np x1 →∞ x − p1 1 • p n∈N p
(23)
C. Gauthier / Central European Journal of Mathematics 4(3) 2006 395–412
407
The method used to obtain a lower bound for the denominator of the last absolute pp−1 p−1 prevalue of (15) here shows that the denominator of (23) has np0 +xp1 as lower bound. Applying to the numerator of (23) the same multiplications of p-addisym conjugates which have leaded to the lower bound of its denominator, and then multiplying the result p by xp−1 1 , one finds a sum of terms which are the products of power of ni and of x1 multiplied by xp−1 and whose total power is always equal to pp−1 − 1. For some aj ∈ R+ , 1 j = 0, 1, . . . , pp−2 − 1, one can thus write pp−2 −1 pp−1 −jp−1 aj njp 0 x1 j=0
np0
p−1
+ xp1
p−1
But
=
njp xpp−1 −jp−1 0 1 n0 ≤x1
np0
p−1
+ xp1
p−1
njp xpp−1 −jp−1 0 1 n0 ≤x1
np0
p−1
+ xp1
p−1
+
njp xpp−1 −jp−1 0 1 n0 >x1
np0
p−1
+ xp1
p−1
.
≤1
and, if N is the smallest integer such that 2N > x1 njp xpp−1 −jp−1 0 1 n0 >x1
p−1 np0
+
p−1 xp1
< 2N − 1 + 2(p
p−1 −jp−1)N
ζ(pp−1 − jp).
It follows that h
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C. Gauthier / Central European Journal of Mathematics 4(3) 2006 395–412
D, w −p z0 ∈ Ep for w ∈ D\{z0 } and αp = 0, then for every n ∈ N
n! f (z0 ) = pαp sp [n]
, C
f (w) dw. (w −p z0 )n+1
Proof. It is similar to the one of the corresponding result on C ([9, p. 369]).
(25)
The multiplication of z = k∈I(p) ak skp ∈ Ep by its conjugates in Ep gives a number ckk ∈ Ap [4]. If p = 2, this number is a20 + a21 ∈ R+ . If p > 2, the multiplication k∈I(p) of k∈I(p) ckk by its p-addisym conjugates gives a primary or an antiprimary p-addisym number. From these observations, we obtain a new function z → |z|p of Ep on R+ which has properties close to those of an absolute value. This function, called absolute p-quasivalue, is defined by
|z|p =
where R =
j∈I ∗ (p)
⎧ 1 ⎪ ⎨ R (p−1)p
,
1 ⎪ ⎩ (−p R) (p−1)p ,
(jk)
k∈I(p) ck
if R is primary if R is antiprimary,
. We have shown in [4] that for any z, w ∈ Ep
|zw|p = |z|p |w|p
(26)
and for every δ > 0, there exist ap , bp ∈ R∗+ such that for every z ∈ Eδp
ap z p ≤ |z|p ≤ bp z p .
(27)
We have then the following generalization of Liouville’s theorem. Lemma 4.5. If f is p-addiholomorphic and uniformly bounded with respect to the absolute p-prevalue on Eδp and αp = 0 then f is constant on this set. Proof. Since f is p-addiholomorphic on Eδp , it will be p-addiholomorphic on every paddisym surface of Eδp . One can thus apply Lemma 4.4 to every p-addisym surface S passing through any fixed point z of Eδp , by using the simple closed curve C(z, r) = {w ∈ S : w −p z p = r}, where r ∈ R∗+ . If M is an upper bound of f (w) p on Eδp then,
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according to (25), (26) and (27)
f (z) p = ≤ ≤ ≤ = ≤
But, if w =
, 1 f (w) dw pαp sp 2 (w − z) p C(z,r) p , 1 1 f (w) dw p p αp sp p C(z,r) (w −p z)2 p , 1 1 dw p 1 1
f (w) p 2 p αp p sp p C(z,r) (w −p z) p - - - - , M -- 1 -- -- 1 -1 - dw p pa3p - αp -p - sp -p C(z,r) - (w −p z)2 -p , M
dw p 3 pap |αp |p |sp |p C(z,r) |w −p z|2p , ,
dw p M M = 5
dw p . pa5p |αp |p C(z,r) w −p z 2p pap |αp |p r2 C(z,r)
k k∈I(p) vk sp , from (3) we find that dw p =
,
dw p = C(z,r)
⎛ ⎝
k∈I(p)
k∈I(p)
(dv) . Thus k,j j∈I(p)
⎞
, j∈I(p)
(dv)k,j ⎠ ≤ p2 r.
C(z,r)
Consequently f (z) p ≤ M p/a5p |αp |p r. Taking the limit when r tends toward infinity, we find that f (z) p tends toward zero. Therefore f (z) = 0 and the function f is constant on the considered p-addisym surface of Eδp . Since this surface was arbitrary, the result applies to every p-addisym surfaces of Eδp , that is on all Eδp . Theorem 4.6. For every z ∈ Ep , one has
g
pz p−1 1 . + z n∈N• z p −p np
(28)
p
Proof. The Lemmas 4.1 and 4.3 show that the function G
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by using the expression (20) of h(z): ⎛ ⎞⎤ ⎡
= zK0
1 −p
z p n
n∈N•p
,
(29)
where K0 = K exp c. From (29), one can determine the value of K0 by using the series which represents g1
1 −p
z p
n∈N•p
n
g1
We have thus proved Theorem 4.7. For every z ∈ Ep , one has g1
= αp z
1 −p
z p
n∈N•p
n
.
(30)
A second expression of g1
= αp z
∞ 1(n) (αp z)np n=0
(np + 1)!
.
(31)
The uniqueness of this series makes it possible to identify the coefficients of z p in the expressions (30) and (31) of g1
1 1(1) αpp = . p n (p + 1)! • n∈N p
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A generalization of Riemann’s zeta function is ζp (z) =
1 , nz n∈N• p
for z ∈ Ep . Consequently ζp (p) = −p
1(1) αpp , (p + 1)!
and, in particular, ζ2 (2) = ζ(2) =
α22 . 3!
(32)
Comparing (32) with (1), we obtain that α2 = π. We shall say that a number of Ap belongs to its primary sector if its reduced expression has the form of the numbers in N•p , that is if this expression has a non-zero first coordinate and its (p − 1)th coordinate is zero. We designate by αp the pth root of αpp belonging to the primary sector of Ap . The proof of the following theorem is completed. Theorem 4.8. For every prime number p ∈ N, we have ζp (p) =
−p 1(1) αpp , (p + 1)!
(33)
where αp is a number of the primary sector of Ap . The expression (33) will allow us to compute the value of αp , p ≥ 3, if we determine the value of ζp (p). To this end, we recall the remark following the definition of Np given by (13) which says that the set of n ∈ Np coincides with the set of n(k) for n ∈ N•p and k ∈ I(p). Therefore 1 1 1 = . p p n p n • n∈N n∈N p
p
But for every n ∈ Np we also have −p n ∈ Np . We can thus decompose Np into two subsets Mp and Np \Mp such that the additive inverse of every n ∈ Mp is in Np \Mp . Therefore 1 1 1 = 0. = + p p p n n (− p n) n∈N n∈M p
p
Consequently ζp (p) = 0 and αp = 0 if p ≥ 3. This result contradicts our assumption about the existence of a αp ∈ Ap such that exp(pαp sp ) = 1 and which has a minimal absolute p-prevalue without being zero. Thus, if p ≥ 3, then there exists no αp ∈ A∗p such that gk
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Acknowledgment The author thanks Paul Deguire and Pierre Gravel for fruitful discussions about some results contained in this paper.
References [1] H. Cartan: Th´eorie ´el´ementaire des fonctions analytiques d’une ou plusieurs variables complexes, Hermann, Paris, 1961. [2] P. Deguire and C. Gauthier: “Sur la d´erivation dans certains anneaux quotients de R[x]”, Ann. Sci. Math. Qu´ebec, Vol. 24, (2000), pp. 19–31. [3] L. Euler: “De summis serierum reciprocarum”, Comment. Acad. Sci. Petropolit., Vol. 7(1734/35), (1740), pp. 123–134; Opera omnia, Ser. 1, Vol. 14, Leipzig-Berlin, 1924, pp. 73–86. [4] C. Gauthier: “Quelques propri´et´es alg´ebriques des ensembles de nombres `a inverse additif compos´e”, Ann. Sci. Math. Qu´ebec, Vol. 26, (2002), pp. 47–59. [5] I.J. Good: “A simple generalization of analytic function theory”, Expo. Math., Vol. 6 (1988), pp. 289–311. [6] E. Grosswald: Topics from the Theory of Numbers, Birkh¨auser, Boston, 1984. [7] M.E. Muldoon and A.A. Ungar: “Beyond sin and cos”, Math. Mag., Vol. 69 (1996), pp. 2–14. [8] H. Silverman: Complex Variables, Houghton Mifflin, Boston, 1975. [9] G. Valiron: Th´eorie des fonctions, Masson, Paris, 1948.
DOI: 10.2478/s11533-006-0017-6 Research article CEJM 4(3) 2006 413–434
On presentations of Brauer-type monoids Ganna Kudryavtseva1∗ , Volodymyr Mazorchuk2† 1
Department of Mathematics and Mechanics, Kyiv Taras Shevchenko University, 01033 Kyiv, Ukraine 2
Department of Mathematics, Uppsala University, SE-75106, Uppsala, Sweden
Received 30 November 2005; accepted 17 March 2006 Abstract: We obtain presentations for the Brauer monoid, the partial analogue of the Brauer monoid, and for the greatest factorizable inverse submonoid of the dual symmetric inverse monoid. In all three cases we apply the same approach, based on the realization of all these monoids as Brauer-type monoids. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Brauer semigroup, monoid presentation, braid relations MSC (2000): 20M05, 20M20
1
Introduction and preliminaries
The classical Coxeter presentation of the symmetric group Sn plays an important role in many branches of modern mathematics and physics. In semigroup theory there are several “natural” analogues of the symmetric group. For example the symmetric inverse semigroup IS n or the full transformation semigroup Tn . Perhaps a “less natural” generalization of Sn is the so-called Brauer semigroup Bn , which arose in the context of centralizer algebras in representation theory in [6]. The basis of this algebra can be described in a nice combinatorial way using special diagrams (see Section 2). This combinatorial description motivated a generalization of the Brauer algebra, the so-called partition algebra, which has its origins in physics and topology, see [10, 16]. This algebra leads to another finite semigroup, the partition semigroup, usually denoted by Cn . Many ∗ †
E-mail: [email protected] E-mail: [email protected]
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classical semigroups, in particular, Sn , IS n , Bn and some others (again see Section 2) are subsemigroups in Cn . In the present paper we address the question of finding a presentation for some of the subsemigroups of Cn . As we have already mentioned, for Sn this is a famous and very important result, with the major role played by the so-called braid relations. Because of the “geometric” nature of the generators of the semigroups we consider, our initial motivation was that the additional relations for our semigroups would be some kind of “singular deformations” of the braid relations (analogous to the case of the singular braid monoid, see [1, 3], or to the known presentations of the Brauer algebra from [2, 4]). In particular, we wanted to get a complete list of “deformations” of the braid relations, which can appear in these cases. It turns out that all the semigroups we considered indeed have presentations, all ingredients of which are in some sense deformations or degenerations of the braid relations. As the main results of the paper we obtain a presentation for the semigroup Bn (see Section 3), its partial analogue PBn (which is also called the rook Brauer monoid, see Section 5), and a special inverse subsemigroup IT n of Cn , which is isomorphic to the greatest factorizable inverse submonoid of the dual symmetric inverse monoid, see Section 4 (another presentation for the latter monoid was obtained in [7]). The technical details in all cases are quite different, however, the general approach is the same. We first “guess” the relations and in the standard way obtain an epimorphism from the semigroup T , given by the corresponding presentation, onto the semigroup we are considering. It remains to show that this epimorphism is in fact a bijection. For this we compare the cardinalities of the semigroups. In all cases the symmetric group Sn is the group of units in T . The product Sn × Sn thus acts on T via multiplication from the left and from the right. The general idea is to then show that each orbit of this action contains a very special element, for which, using the relations, one can estimate the cardinality of the stabilizer. The necessary statement then follows by comparing the cardinalities. Note added in proofs: A presentation for Cn can be found in Theorem 1.11 of the paper [9].
2
Brauer type semigroups
For n ∈ N we denote by Sn the symmetric group of all permutations on the set {1, 2, . . . , n}. We will consider the natural right action of Sn on {1, 2, . . . , n} and the induced action on the Boolean of {1, 2, . . . , n}. For a semigroup, S, we denote by E(S) the set of all idempotents of S. Fix n ∈ N and let M = Mn = {1, 2, . . . , n}, M = {1 , 2 , . . . , n }. We will consider : M → M as a bijection, whose inverse we will also denote by . Consider the set Cn of all decompositions of M ∪ M into disjoint unions of subsets. Given α, β ∈ Cn , α = X1 ∪ · · · ∪ Xk and β = Y1 ∪ · · · ∪ Yl , we define their product γ = αβ as the unique element of Cn satisfying the following conditions: (P1) For i, j ∈ M the elements i and j belong to the same block of the decomposition γ
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415
if and only if they belong to the same block of the decomposition α or there exists a sequence, s1 , . . . , sm , where m is even, of elements from M such that i and s1 belong to the same block of α; s1 and s2 belong to the same block of β; s2 and s3 belong to the same block of α and so on; sm−1 and sm belong to the same block of β; sm and j belong to the same block of α. (P2) For i, j ∈ M the elements i and j belong to the same block of the decomposition γ if and only if they belong to the same block of the decomposition β or there exists a sequence, s1 , . . . , sm , where m is even, of elements from M such that i and s1 belong to the same block of β; s1 and s2 belong to the same block of α; s2 and s3 belong to the same block of β and so on; sm−1 and sm belong to the same block of α; sm and j belong to the same block of β. (P3) For i, j ∈ M the elements i and j belong to the same block of the decomposition γ if and only if there exists a sequence, s1 , . . . , sm , where m is odd, of elements from M such that i and s1 belong to the same block of α; s1 and s2 belong to the same block of β; s2 and s3 belong to the same block of α and so on; sm−1 and sm belong to the same block of α; sm and j belong to the same block of β. One can think about the elements of Cn as “microchips” or “generalized microchips” with n pins on the left hand side (corresponding to the elements of M ) and n pins on the right hand side (corresponding to the elements of M ). For α ∈ Cn we connect two pins of the corresponding chip if and only if they belong to the same set of the partition α. The operation described above can then be viewed as a “composition” of such chips: having α, β ∈ Cn we identify (connect) the right pins of α with the corresponding left pins of β, which uniquely defines a connection of the remaining pins - the left pins of α and the right pins of β. An example of multiplication of two chips from Cn is given on Figure 1. Note that, performing the operation we can obtain some “dead circles” formed by some identified pins from α and β. These circles should be disregarded (however they play an important role in representation theory as they allow one to deform the multiplication in the semigroup algebra). From this interpretation it is fairly obvious that the composition of elements from Cn defined above is associative. On the level of associative algebra, the partition algebra was defined in [16] and then studied by several authors especially in recent years, see for example [5, 17–19, 22, 24]. Purely as a semigroup it seems that Cn appeared in [21]. Let α ∈ Cn and X be a block of α. The block X will be called • a line provided that |X| = 2 and X intersects with both M and M ; • a generalized line provided that X intersects with both M and M ; • a bracket if |X| = 2 and either X ⊂ M or X ⊂ M ; • a generalized bracket if |X| ≥ 2 and either X ⊂ M or X ⊂ M ; • a point if |X| = 1. By a Brauer-type semigroup we will mean a “natural” subsemigroup of the semigroup Cn . Here are some examples: (E1) The subsemigroup, consisting of all elements α ∈ Cn such that each block of α is a line. This subsemigroup is canonically identified with Sn and is the group of units
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G. Kudryavtseva, V. Mazorchuk / Central European Journal of Mathematics 4(3) 2006 413–434
2→
• • •
3→
•
•
4→
•
•
1→
5→ 6→ 7→
•
@ B @B • • HH @ B@ B • • XXHHH B HH HH B • H•
•
#• # # • # • # #
•
•
•
•
•
•
• •
• bb
•
•
•
b b
•
•
•
•
•
% % • • % • %% •
•
•
•
=
Fig. 1 Multiplication of elements of Cn . of Cn . (E2) The subsemigroup, consisting of all elements α ∈ Cn such that each block of α is a either a line or a point. This subsemigroup is canonically identified with the symmetric inverse semigroup IS n . (E3) The subsemigroup Bn , consisting of all elements α ∈ Cn such that each block of α is a either a line or a bracket. This is the classical Brauer semigroup, see [11, 20]. (E4) The subsemigroup PBn , consisting of all elements α ∈ Cn such that each block of α is a either a line, a bracket or a point. This is the partial analogue of the Brauer semigroup, see [20]. (E5) The subsemigroup IP n , consisting of all α ∈ Cn such that each block of α is a generalized line. In this form the semigroup IP n appeared in [14, 15]. It is easy to see that the semigroup IP n is isomorphic to the dual symmetric inverse monoid ∗ from [8]. IM (E6) The subsemigroup IT n , consisting of all α ∈ Cn such that each block X of α is a generalized line and |X ∩ M | = |X ∩ M |. In this form the semigroup IT n appeared in [15]. The semigroup IT n is isomorphic to the greatest factorizable ∗ ∗ inverse submonoid FM of IM from [8]. All the semigroups described above are regular. Sn is a group. The semigroups ISn , IP n and IT n are inverse, while Cn , Bn and PBn are not. The partially ordered set consisting of these semigroups, with the partial order given by inclusions, is illustrated in Figure 2. In the following we shall need some easy combinatorial results for Brauer-type semigroups. For α ∈ Cn we define the rank rk(α) of α as the number of generalized lines in α, that is the number of blocks in α intersecting with both M and M . Note that for the semigroups Sn , IS n , Bn , PBn and Cn ranks of the elements classify the D-classes (this is obvious for Sn , for IS n this is an easy exercise, for Bn and PBn this can be found in [20], and for Cn it can be obtained by arguments similar to those from [20] for Bn ).
G. Kudryavtseva, V. Mazorchuk / Central European Journal of Mathematics 4(3) 2006 413–434
PB
C lll n OOOOO lll OOO l l l OOO lll l O l l
nF FF ww FF ww w FF w w F w w IS n WWWWW BnB WWWWW BB B WWWWW WWWWW BBB WWWWW B W
Sn
pp ppp p p pp ppp
417
IP n IT n
Fig. 2 Inclusions for classical Brauer-type semigroups For the semigroup IT n we need a different notion. Let X be a finite set and X = be a decomposition of X into a union of pairwise disjoint subsets. For each i, 1 ≤ i ≤ |X|, let mi denote the number of subsets of this decomposition, whose cardinality equals i. The tuple (m1 , . . . , m|X| ) will be called the type of the decomposition. Consider an element, α ∈ IT n . By definition α is a decomposition of M ∪ M into a disjoint union of subsets, whose intersections with M and M have the same cardinality. Let (m1 , . . . , m2n ) be the type of this decomposition (note that mi = 0 only if i is even). The element α induces a decomposition of M into disjoint subsets, whose blocks are intersections of the blocks of α with M . By the type of α we will mean the type of this decomposition of M , which is obviously equal to (m2 , m4 , . . . , m2n ). The types of elements from IT n correspond bijectively to partitions of n (a partition, λ n, of n is a tuple, λ = (λ1 , . . . , λk ), of positive integers such that λ1 ≥ λ2 ≥ · · · ≥ λk and λ1 + · · · + λk = n). The types of the elements classify the D-classes in IT n , see [8, Section 3]. For the semigroup PBn we need a more complicated technical tool. Although Dclasses are classified by ranks we will need to distinguish elements of a given rank, so we introduce the notion of a type. For α ∈ PBn let r denote the number of lines in α; b1 the number of brackets in α, contained in M ; b2 the number of brackets in α, contained in M ; p1 the number of points in α, contained in M ; p2 the number of points in α, contained in M . Obviously n = r + 2b1 + p1 = r + 2b2 + p2 . Define the type of α as follows: ∪ki=1 Xk
type(α) =
(b2 , b1 − b2 , 0, p1 ), b1 ≥ b2 ; (b1 , 0, b2 − b1 , p2 ), b2 > b1 .
We will need the following explicit combinatorial formulas for the number of elements of a given rank or type. Proposition 2.1. (a) For k ∈ {0, . . . , n} the number of elements of rank k in IS n equals n2 k!. k (b) For k ∈ {1, . . . , n} the number of elements of rank k in Bn equals 0 if n − k is odd 2 and 22l(n!) if n − k = 2l is even. (l!)2 k!
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(c) The number of elements of IT n of type (m1 , . . . , mn ) equals (n!)2 n (mi !(i!)2mi )
.
i=1
(d) For all non-negative integers k, m, t such that 2k +2m+t ≤ n the number of elements of the type (k, m, 0, t) in PBn is equal to the number of elements of the type (k, 0, m, t) in PBn and equals (n!)2 . k!2k (t + 2m)!(k + m)!2k+m t!(n − 2k − 2m − t)! Proof. This is a straightforward combinatorial calculation.
Remark 2.2. The semigroup Cn can be also connected to some other semigroups of binary relations. As we have already mentioned, the subsemigroup IP n of Cn is iso∗ morphic to the dual symmetric inverse monoid IM from [8], which is the semigroup of all difunctional binary relations under the operation of taking the smallest difunctional binary relations, containing the product of two given relations. The semigroup IT n is ∗ isomorphic to the greatest factorizable inverse submonoid of IM , that is to the semigroup ∗ E(IM )Sn . One can also deform the multiplication in Cn in the following way: given α, β ∈ Cn define γ = α β as follows: all blocks of γ are either points or generalized lines, and for i, j ∈ M the elements i and j belong to the same block of γ if and only if i belongs to some block X of α and j belongs to some block Y of β such that X ∩ M = (Y ∩ M ) . It is straightforward to show that this deformed multiplication is associative and hence we ˜ n . This semigroup is an inflation of Vernitski’s inverse semigroup get a new semigroup, C ˜ n in the natural way. An isomorphic object (DX , ), see [23], which is a subsemigroup of C can be obtained if instead of points one requires that γ contains at most one generalized bracket, which is a subset of M , and at most one generalized bracket, which is a subset of M .
3
Presentation for Bn
For i = 1, . . . , n − 1 we denote by si the elementary transposition (i, i + 1) ∈ Sn , and by πi the element {i, i + 1} ∪ {i , (i + 1) } ∪ j=i,i+1 {j, j } of Bn (the elementary atom from [20]). It is easy to see (and can be derived from the results of [20] and [13]) that Bn is generated by {si } ∪ {πi } as a monoid. Moreover, Bn is even generated by {si } and, for example, π1 . However, in the context being considered, the set {si } ∪ {πi } is more natural as a system of generators for Bn because, for example, the connection between Brauer and Temperley-Lieb algebras (and analogy with the singular braid monoid, see [1, 3]). In this section we obtain a presentation for Bn with respect to this system of generators (this resembles the presentation of the Brauer algebra in [4], see also [2]).
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Let T denote the monoid with the identity element e, generated by the elements σi , θi , i = 1, . . . , n − 1, subject to the following relations (where i, j ∈ {1, 2, . . . , n − 1}): σi2 = e; θi2
σi σj = σj σi , |i − j| > 1;
= θi ;
θi θj = θj θi , |i − j| > 1; θi σi = σi θi = θi , σi θj θi = σj θi ,
σi σj σi = σj σi σj , |i − j| = 1;
(1)
θi θj θi = θi , |i − j| = 1;
(2)
θi σj = σj θi , |i − j| > 1;
(3)
θi θj σi = θi σj , |i − j| = 1.
(4)
Theorem 3.1. The map σi → si and θi → πi , i = 1, . . . , n − 1, extends to an isomorphism, ϕ : T → Bn . The rest of the section will be devoted to the proof of Theorem 3.1. We start with the following easy observation, which will be used later in our computations: Lemma 3.2. Under the assumption that the relations (1)–(4) are satisfied, we have the following relations: σi θj σi = σj θi σj ,
θi σj θi = θi , |i − j| = 1;
σi σi+1 θi θi+2 = σi+2 σi+1 θi θi+2 .
(5) (6)
Proof. For i, j, |i − j| = 1, applying (4) twice we have σi θj σi = σj θi θj σi = σj θi σj . Applying (4), (3) and, finally, (2) we also have θi σj θi = θi θj σi θi = θi θj θi = θi . This gives (5). Analogously, applying (4), (1), (2) and (4) again gives σi+2 σi+1 θi θi+2 = σi+2 σi θi+1 θi θi+2 = σi σi+2 θi+1 θi+2 θi = σi σi+1 θi+2 θi , which implies (6).
It is a direct calculation to verify that the generators si and πi of Bn satisfy the relations, corresponding to (1)–(4). Thus the map σi → si and θi → πi , i = 1, . . . , n − 1, extends to an epimorphism, ϕ : T Bn . Hence, to prove Theorem 3.1 we have only to show that |T | = |Bn |. To do this we will have to study the structure of the semigroup T in detail. Let W denote the free monoid, generated by σi , θi , i = 1, . . . , n − 1, and ψ : W T denote the canonical projection. Let ∼ be the corresponding congruence on W , that is v ∼ w provided that ψ(v) = ψ(w). We start with the following description of the units in T : Lemma 3.3. The elements σi , i = 1, . . . , n−1, generate the group G of units in T , which is isomorphic to the symmetric group Sn .
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Proof. Let v, w ∈ W be such that v ∼ w. Assume further that v contains some θi . Since θ’s always occur on both sides in the relations (2)–(4) and do not occur in the relation (1), it follows that w must contain some θj . In particular, the submonoid, generated in W by σi , i = 1, . . . , n − 1, is a union of equivalence classes with respect to ∼. Using the wellknown Coxeter presentation of the symmetric group we obtain that σi , i = 1, . . . , n − 1, generate in T a copy of the symmetric group. All elements of this group are obviously units in T . On the other hand, if v, w ∈ W and v contains some θi , then vw contains θi as well. By the above arguments, vw cannot be equivalent to the empty word. Hence v is not invertible in T . The claim of the lemma follows. In the following we identify the group G of units in T with Sn via the isomorphism, which sends σi ∈ G to si . There is a natural action of Sn on T by inner automorphisms of T via conjugation: xg = g −1 xg for each x ∈ T , g ∈ Sn . Lemma 3.4. The Sn -stabilizer of θ1 is the subgroup H of Sn , consisting of all permutations, which preserve the set {1, 2}. This subgroup is isomorphic to S2 × Sn−2 . Proof. We have σj θ1 σj = θj , j = 2, by (3). Since σj , j = 2, generate H, we obtain that all elements of H stabilize θ1 . In particular, the Sn -orbit of θ1 consists of at most |Sn |/|H| = n2 elements. At the same time, it is easy to see that the Sn -orbit of ϕ(θ1 ) consists of exactly n2 different elements and hence H must coincide with the Sn -stabilizer of θ1 . Since Sn acts on T via automorphism and θ1 is an idempotent, all elements in the Sn -orbit of θ1 are idempotents. From Lemma 3.4 it follows that the elements of the Sn orbit of θ1 are in the natural bijection with the cosets H\Sn . By the definition of H, two elements, x, y ∈ Sn , are contained in the same coset if and only if x({1, 2}) = y({1, 2}). Lemma 3.5. The Sn -orbit of θ1 contains all θi , i = 1, . . . , n − 1. Moreover, for w ∈ Sn we have w−1 θ1 w = θi if and only if w({1, 2}) = {i, i + 1}. Proof. We use induction on i with the case i = 1 being trivial. Let i > 1 and assume that θi−1 is contained in our orbit. Then θi = σi−1 σi θi−1 σi σi−1 and hence θi is contained in our orbit as well. Hence all θi belong to the Sn -orbit of θ1 . The second claim follows from σi−1 σi σi−2 σi−1 · · · σ1 σ2 ({1, 2}) = {i, i + 1}, which is obtained through direct calculation. This completes the proof.
(7)
For w ∈ Sn such that w({1, 2}) = {i, j}, where i < j, we set i,j = w−1 θ1 w, which is well defined by Lemma 3.4. Lemma 3.6. Suppose {i, j} ∩ {p, q} = ∅. Then i,j p,q = p,q i,j .
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Proof. Since all elements i,j are obtained from θ1 via automorphisms, it is enough to show that θ1 commutes with all elements i,j such that {i, j} ∩ {1, 2} = ∅. Take any v ∈ Sn such that v({1, 2}) = {1, 2} and v({i, j}) = {3, 4} (such v obviously exists). Then θ1 commutes with i,j if and only if v −1 θ1 v = θ1 commutes with v −1 i,j v = θ3 . The statement now follows from (2). Lemma 3.7. Suppose {i, j} ∩ {p, q} = ∅. Then i,j p,q = uθ1 v for some u, v ∈ Sn . Proof. If {i, j} = {p, q} the statement is obvious as i,j is an idempotent. Assume |{i, j} ∩ {p, q}| = 1. Since all elements i,j are obtained from θ1 via automorphisms, it is enough to consider the case when {i, j} = {1, 2}, p = 2 and q > 2. Consider v ∈ Sn such that v(1) = 1, v(2) = 2 and v(q) = 3. Using (3), (1) and (5) we have v −1 θ1 p,q v = θ1 θ2 = θ1 σ1 θ2 σ1 σ1 = θ1 σ2 θ1 σ2 σ1 = θ1 σ2 σ1 .
The statement follows.
For each k, 1 ≤ k ≤ [ n2 ], set δk = θ1 θ3 . . . θ2k−1 . Set also δ0 = e. The elements δi , 0 ≤ i ≤ [ n2 ], will be called canonical. The group Sn × Sn acts naturally on T via (g, h)(x) = g −1 xh for x ∈ T and (g, h) ∈ Sn × Sn . Lemma 3.8. Every Sn × Sn -orbit contains a canonical element. Proof. Let x ∈ T . If x ∈ Sn the statement is obvious. Assume that x ∈ Sn . By Lemma 3.5 we can write x = wθ1 g1 θ1 g2 . . . θ1 gk for some k ≥ 1 and w, g1 , . . . , gk ∈ Sn . Moreover, we may assume that x cannot be written as a product of θ1 ’s and elements of Sn , which contains less than k occurrences of θ1 . We have x = w(g1 . . . gk )(g1 . . . gk )−1 θ1 (g1 . . . gk )· · (g2 . . . gk )−1 θ1 (g2 . . . gk ) . . . (gk−1 gk )−1 θ1 (gk−1 gk )gk−1 θ1 gk , (8) and hence we can write x = u i1 ,j1 . . . ik ,jk ,
(9)
where u = wg1 . . . gk and {it , jt }={(gt . . . gk )(1), (gt . . . gk )(2)}, 1 ≤ t ≤ k. Since x was chosen so that it cannot be reduced to an element of T which contains less that k entries of θ1 , from Lemmas 3.6 and 3.7 it follows that {it , jt } ∩ {is , js } = ∅ for any two factors it ,jt , is ,js in (9). This implies that the Sn × Sn -orbit of x contains i1 ,j1 . . . ik ,jk with {it , jt } ∩ {is , js } = ∅ for all s = t. Now consider some v ∈ Sn such that v(i1 ) = 1, v(j1 ) = 2, v(i2 ) = 3 and so on, v(jk ) = 2k. Then the element v −1 i1 ,j1 · · · ik ,jk v is canonical by definition. This completes the proof. Remark 3.9. From the proof of Lemma 3.8 it follows that each x ∈ T can be written in the form x = wθ1 g1 θ1 g2 . . . θ1 gk , where k ≤ n2 .
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Lemma 3.10. The Sn × Sn -orbit of the canonical element δk , 0 ≤ k ≤ [ n2 ], contains at most (n!)2 22k (k!)2 (n − 2k)! elements. Proof. It is enough to show that the stabilizer of δk under the Sn × Sn -action contains at least (k!)2 22k (n − 2k)! elements. Set Σ0i = σ2i σ2i−1 σ2i+1 σ2i , 1 ≤ i ≤ k − 1;
Σ1i = σ2i σ2i−1 σ2i+1 σ2i σ2i−1 , 1 ≤ i ≤ k − 1. Then both Σ0i and Σ1i swap the sets {2i−1, 2i} and {2i+1, 2i+2}. It follows that the group H, generated by all Σ0i , consists of all permutations of the set {1, 2}, {3, 4}, . . . , {2k−1, 2k} ˜ and is therefore isomorphic to the group Sk . Further, it is easy to see that the group H,
generated by all Σ0i and Σ1i , is isomorphic to the wreath product H S2 . From (6) and (3) it follows that the left multiplication with both Σ0i and Σ1i stabilizes δk . Therefore for ˜ the left multiplication with this element stabilizes δk as well. Similarly each element of H ˜ stabilizes δk . In one proves that the right multiplication with each element from H addition to this, from (3) we have that the conjugation by any element from the group H = σ2k+1 , . . . , σn−1 Sn−2k stabilizes δk . ˜ the right copy of H, ˜ and the Observe that the group, generated by the left copy of H, H is a direct product of these three components. Using the product rule we derive that the cardinality of the stabilizer of δk is at least (|H S2 |)2 |Sn−2k | = (k!)2 22k (n − 2k)!,
and the proof is complete. Corollary 3.11.
n
|T | ≤
2 k=0
(n!)2 . 22k (k!)2 (n − 2k)!
Proof. The proof follows from Lemma 3.10 and Remark 3.9 by a direct calculation. Proof (of Theorem 3.1). Comparing Corollary 3.11 and Proposition 2.1(b) we have |T | ≤ |Bn |. Since ϕ : T → Bn is surjective we have |T | ≥ |Bn |. Hence |T | = |Bn | and ϕ is an isomorphism.
4
Presentation for IT n
For i ∈ {1, 2, . . . , n−1} let i denote the element {i, i+1, i , (i+1) }∪ j=i,i+1 {j, j } ∈ IT n . By [15, Proposition 9], the elements {σi } and {i } generate IT n (and even {σi } and, say 1 , do).
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Let T denote the monoid with the identity element e, generated by the elements σi , τi , i = 1, . . . , n − 1, subject to the following relations (where i, j ∈ {1, 2, . . . , n − 1}): σi2 = e;
σi σj = σj σi , |i − j| > 1; τi2
= τi ;
σi σj σi = σj σi σj , |i − j| = 1;
τi τj = τj τi , i = j;
τi σi = σi τi = τi ;
τi σj = σj τi , |i − j| > 1;
σi τj σi = σj τi σj and τi σj τi = τi τj , |i − j| = 1.
(10) (11) (12) (13)
Theorem 4.1. The map σi → si and τi → i , i = 1, . . . , n−1, extends to an isomorphism, ϕ : T → IT n . The rest of the section will be devoted to the proof of Theorem 4.1. It is a direct calculation to verify that the generators si and i of IT n satisfy the relations, corresponding to (10)–(13). Thus the map σi → si and τi → i , i = 1, . . . , n−1, extends to an epimorphism, ϕ : T IT n . Hence, to prove Theorem 4.1 we have only to show that |T | = |IT n |. As in the previous section, to do this we will study the structure of T in detail. Let W denote the free monoid, generated by σi , τi , i = 1, . . . , n − 1, ψ : W T denote the canonical projection, and ∼ be the corresponding congruence on W . The first part of our argument is very similar to that from the previous Section. Lemma 4.2. The elements σi , i = 1, . . . , n−1, generate the group G of units in T , which is isomorphic to the symmetric group Sn (and will be identified with Sn in the following). Proof. Analogous to that of Lemma 3.3.
There are two natural actions on T : (I) The group Sn acts on T by inner automorphisms via conjugation. (II) The group Sn × Sn acts on T via (g, h)(x) = g −1 xh for x ∈ T and (g, h) ∈ Sn × Sn . Lemma 4.3. The Sn -stabilizer of τ1 is the subgroup H of Sn , consisting of all permutations, which preserve the set {1, 2}. This subgroup is isomorphic to S2 × Sn−2 . Proof. Analogous to that of Lemma 3.4.
Since Sn acts on T via automorphisms and τ1 is an idempotent, all elements in the Sn -orbit of τ1 are idempotents. From Lemma 4.3 it follows that the elements of the Sn orbit of τ1 are in the natural bijection with the cosets H\Sn . By the definition of H, two elements, x, y ∈ Sn , are contained in the same coset if and only if x({1, 2}) = y({1, 2}). Lemma 4.4. The Sn -orbit of τ1 contains all τi , i = 1, . . . , n − 1. Moreover, for w ∈ Sn we have w−1 τ1 w = τi if and only if w({1, 2}) = {i, i + 1}. Proof. Analogous to that of Lemma 3.5.
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Lemma 4.5. All elements in the Sn -orbit of τ1 commute. Proof. Since all elements in the Sn -orbit of τ1 are obtained from τ1 via automorphism, it is enough to show that τ1 commutes with all elements in this orbit. Let w ∈ Sn be such that w({1, 2}) = {i, j}. If {i, j} = {1, 2} then w−1 τ1 w = τ1 by Lemma 4.4 and hence we may assume {i, j} = {1, 2}. Take any v ∈ Sn such that • v({1, 2}) = {1, 2} and v({i, j}) = {3, 4} if {i, j} ∩ {1, 2} = ∅; • v({1, 2}) = {1, 2} and v({i, j}) = {2, 3} if {i, j} ∩ {1, 2} = ∅. (such v obviously exists). Then τ1 commutes with w−1 τ1 w if and only if v −1 τ1 v commutes with v −1 w−1 τ1 wv. Using our choice of v and Lemma 4.4 we have v −1 τ1 v = τ1 and v −1 w−1 τ1 wv = τj , where j = 3 if {i, j} ∩ {1, 2} = ∅, and j = 2 otherwise. The statement now follows from (11). For w ∈ Sn such that w({1, 2}) = {i, j}, where i < j, we set εi,j = w−1 τ1 w, which is well defined by Lemma 4.3. Lemma 4.6. Let {i, j, k} ⊂ {1, 2, . . . , n} and i < j < k. Then εi,j εj,k = εi,k εj,k = εi,j εi,k . Proof. We prove that εi,j εj,k = εi,k εj,k and the second equality is proved by analogous arguments. Let w ∈ Sn be such that w(i) = 1, w(j) = 2, w(k) = 3. Conjugating by w we reduce our equality to the equality τ1 τ2 = σ2 τ1 σ2 τ2 . Using (13) twice and (12) we have σ2 τ1 σ2 τ2 = σ1 τ2 σ1 τ2 = σ1 τ1 τ2 = τ1 τ2 .
The claim follows.
For i, j ∈ M set εi,i = e and εi,j = εj,i if j < i. For a non-empty binary relation, ρ, on M set ερ = εi,j . iρj
Corollary 4.7. Let ρ be non-empty binary relation on M and ρ∗ be the reflexive-symmetrictransitive closure of ρ. Then ερ = ερ∗ Proof. Follows easily from Lemma 4.5, Lemma 4.6 and the fact that all εi,j ’s are idempotents. Let λ : {1, . . . , n} = X1 ∪ · · · ∪ Xk be a decomposition of M into an unordered union of pairwise disjoint sets. With this decomposition we associate the equivalence relation ρλ on M , whose equivalence classes coincide with Xi ’s.
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Corollary 4.8. Let λ and μ be two decompositions of M as above. Assume that the types of λ and μ coincide. Then ερλ and ερμ are conjugate in T . Proof. Let v ∈ Sn be an element, which maps λ to μ (such element exists since the types of λ and μ are the same). One easily sees that v −1 ερλ v = ερμ . The statement follows. A decomposition, λ : {1, . . . , n} = X1 ∪ · · · ∪ Xk , is called canonical provided that (up to a permutation of the blocks) we have |X1 | ≥ |X2 | ≥ · · · ≥ |Xk |, X1 = {1, 2, . . . , l1 }, X2 = {l1 + 1, l1 + 2, . . . , l1 + l2 } and so on. Note that in this case λ can also be viewed as a partition of n. The element ερλ will be called canonical provided that λ is canonical. Lemma 4.9. Every Sn × Sn -orbit contains a canonical element. Proof. By Corollary 4.8 it is enough to show that every Sn × Sn -orbit contains ερλ for some decomposition λ. Let x ∈ T . If x ∈ Sn , then the statement is obvious. Let x ∈ T \ Sn . From Lemma 4.4 we have that the semigroup T is generated by Sn and τ1 . Hence we have x = wτ1 g1 τ1 g2 · · · τ1 gk for some w, g1 , . . . , gk ∈ Sn . Therefore x = w(g1 . . . gk )(g1 . . . gk )−1 τ1 (g1 . . . gk )· · (g2 . . . gk )−1 τ1 (g2 . . . gk ) . . . (gk−1 gk )−1 τ1 (gk−1 gk )gk−1 τ1 gk , and hence we can write x = uεi1 ,j1 . . . εik ,jk , where u = wg1 . . . gk and {it , jt } = {(gt . . . gk )(1), (gt . . . gk )(2)}, 1 ≤ t ≤ k. Define the equivalence relation ρ as the reflexive-symmetric-transitive closure of the relation {(i1 , j1 ), . . . , (ik , jk )} and let λ be the corresponding decomposition of {1, 2, . . . , n}. From Corollary 4.7 we get that the Sn × Sn -orbit of x contains ερ = ερλ . This completes the proof. Lemma 4.10. Let λ be a canonical decomposition of {1, 2, . . . , n}. For i = 1, . . . , n set λ(i) = |{j : |Xj | = i}|. Then the Sn × Sn -stabilizer of ερλ contains at least n (i) (λ(i) !(i!)2λ ) i=1
elements. Proof. Fix i ∈ {1, 2, . . . , n}. Let Xa , Xa+1 . . . , Xb be all blocks of λ of cardinality i. Then for any non-maximal element j of any of Xa , Xa+1 . . . , Xb , using Lemma 4.5, the definition of ερλ , and (12) we have σj ερλ = ερλ σj = ερλ . Moreover, for any w ∈ Sn , which stabilizes all elements outside Xa ∪ Xa+1 ∪ · · · ∪ Xb and maps each Xs to some Xt , we have (i) w(λ) = λ and hence w−1 ερλ w = ερλ . This gives us exactly λ(i) !(i!)2λ elements of the Sn × Sn -stabilizer. The statement of the lemma now follows by applying the product rule
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since for different i the nontrivial elements w above stabilize pairwise different subsets of {1, . . . , n}. Corollary 4.11. |T | ≤
(n!)2
λn
n (i) (λ(i) !(i!)2λ )
.
i=1
Proof. Canonical elements of T are in bijection with partitions λ n by construction. By Lemma 4.9, every Sn × Sn -orbit contains a canonical element. We have |Sn × Sn | = (n!)2 . By Lemma 4.10, the stabilizer of a canonical element, corresponding to λ, contains at (i) least ni=1 (λ(i) !(i!)2λ ) elements. The statement now follows by applying the sum rule. Proof (of Theorem 4.1.). Comparing Corollary 4.11 and Proposition 2.1(c) we have |T | ≤ |IT n |. Since ϕ : T → IT n is surjective we have |T | ≥ |IT n |. Hence |T | = |IT n | and ϕ is an isomorphism. Remark 4.12. From the above arguments it follows that the inequality obtained in Lemma 4.10 is in fact an equality. From the proof of Lemma 4.10 one easily derives that the Sn × Sn -stabilizer of ερλ is isomorphic to the direct product of wreath products Sλ(i) (Si × Si ). Remark 4.13. Following the arguments of the proof of Theorem 4.1 one easily proves the following presentation for the symmetric inverse semigroup IS n : IS n is generated, as a monoid, by σ1 , . . . , σn−1 , ϑ1 , . . . , ϑn subject to the following relations: σi2 = e;
σi σj = σj σi , |i − j| > 1; ϑ2i
= ϑi ;
σi σj σi = σj σi σj , |i − j| = 1;
ϑi ϑj = ϑj ϑi i = j;
σi ϑi = ϑi+1 σi ; σi ϑj = ϑj σi , j = i, i + 1;
ϑi σi ϑi = ϑi ϑi+1 .
(14) (15) (16)
The classical presentation for IS n usually involves only one additional generator (namely ϑ1 ) and can be found for example in [12, Chapter 9].
5
Presentation for PBn
For i ∈ {1, . . . , n} let ςi denote the element {i} ∪ {i } ∪ j=i {j, j }. Using [20], it is easy to see that PBn is generated by {σi } ∪ {πi } ∪ {ςi } (and even by {σi }, π1 and ς1 ). Let T denote the monoid with the identity element e, generated by the elements σi , θi , i = 1, . . . , n − 1, and ϑi , i = 1, . . . , n, subject to the relations (1)–(4), the relations
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from Remark 4.13, and the following relations (for all appropriate i and j): θi ϑj = ϑj θi , j = i, i + 1; θi ϑi = θi ϑi+1 = θi ϑi ϑi+1 , θi ϑ i θi = θi ,
ϑi θi = ϑi+1 θi = ϑi ϑi+1 θi ; ϑi θi ϑi = ϑi ϑi+1 .
(17) (18) (19)
Theorem 5.1. The map σi → si , θi → πi , i = 1, . . . , n − 1, and ϑi → ςi , i = 1, . . . , n, extends to an isomorphism, ϕ : T → PBn . We will again start with the following auxiliary technical statement, which we will need later: Lemma 5.2. Under the assumption that (1)–(4), (17)–(19) and the relations from Remark 4.13 are satisfied, one has the relation σi+2 σi+1 θi ϑi+2 ϑi+3 = σi σi+1 ϑi θi θi+2 ϑi+2 .
(20)
Proof. Using (4) twice and (1) we have σi+2 σi+1 θi ϑi+2 ϑi+3 = σi+2 σi θi+1 θi ϑi+2 ϑi+3 = = σi σi+2 θi+1 θi ϑi+2 ϑi+3 = σi σi+1 θi+2 θi+1 θi ϑi+2 ϑi+3 , and hence (20) reduces to θi+2 θi+1 θi ϑi+2 ϑi+3 = ϑi θi θi+2 ϑi+2 .
(21)
Using (17)-(19) and (2) we have θi+2 θi+1 θi ϑi+2 ϑi+3 = θi+2 ϑi+3 θi+1 ϑi+2 θi = θi+2 ϑi+2 θi+1 ϑi+1 θi = = θi+2 ϑi+1 θi+1 ϑi+1 θi = θi+2 ϑi+2 ϑi+1 θi = ϑi θi θi+2 ϑi+2 , which gives (21). The statement follows.
As in the previous section, one easily checks that this map extends to an epimorphism and hence to complete the proof one has to compare the cardinalities of T and PBn . Similar to the results of Section 4, using the presentation of IS n given in Remark 4.13, one proves that elements σi , i = 1, . . . , n − 1, generate the symmetric group Sn , and that the elements σi , i = 1, . . . , n − 1; ϑi , i = 1, . . . , n, generate the semigroup, which is isomorphic to IS n (and which will be identified with it). As in Section 4 we consider the natural action of Sn on T by inner automorphism of T via conjugation: xg = g −1 xg for each x ∈ T , g ∈ Sn . Set ξi = θi ϑi , ηi = ϑi θi , 1 ≤ i ≤ n − 1. Lemma 5.3. The Sn -stabilizer of each of θ1 , ξ1 , η1 is the subgroup H of Sn , consisting of all permutations, which preserve the set {1, 2}. This subgroup is isomorphic to S2 × Sn−2 .
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Proof. For θ1 this follows from Lemma 3.4. For each j ≥ 2 we have that σj commutes with both ξ1 and η1 by (3) and (16) respectively, and hence σj ξ1 σj = ξ1 and σj η1 σj = η1 . Let j = 1. Then σ1 ξ1 σ1 = σ1 θ1 ϑ1 σ1 = σ1 θ1 σ1 ϑ2 = θ1 ϑ2 = θ1 ϑ1 = ξ1 ; σ1 η1 σ1 = σ1 ϑ1 θ1 σ1 = ϑ2 σ1 θ1 σ1 = ϑ2 θ1 = ϑ1 θ1 = η1 by (16) and (3). Hence σ1 also stabilizes ξ1 and η1 . Since σj , j = 2, generate H, it follows that all elements of H stabilize ξ1 and η1 . In particular, the Sn -orbits of ξ1 and of η1 consist of at most |Sn |/|H| = n2 elements each. At the same time, the Sn -orbits of ϕ(ξ1 ) and ϕ(η1 ) consist of exactly n2 different elements and hence H must coincide with the Sn -stabilizer of both ξ1 and η1 . Since Sn acts on T via automorphisms and θ1 , ξ1 , η1 are idempotents, all elements in the Sn -orbits of θ1 , ξ1 , η1 are idempotents as well. From Lemma 5.3 it follows that the elements of the Sn -orbits of θ1 , ξ1 , η1 are in natural bijection with the cosets H\Sn . By the definition of H, two elements, x, y ∈ Sn , are contained in the same coset if and only if x({1, 2}) = y({1, 2}). Lemma 5.4. The Sn -orbits of θ1 , ξ1 , η1 contain all elements θi , ξi and ηi , i = 1, . . . , n−1, respectively. Moreover, for w ∈ Sn we have w−1 θ1 w = θi if and only if w({1, 2}) = {i, i + 1} and analogously for ξ1 and η1 . Proof. The proof for the Sn -orbit of θ1 is analogous to that of Lemma 3.5. We prove the statement for the Sn -orbit of ξ1 . For the Sn -orbit of η1 the arguments are analogous. We use induction on i with the case i = 1 being trivial. Let i > 1 and assume that ξi−1 is contained in our orbit. Then, using (16), (1) and (5), we compute ξi = θi ϑi = σi−1 σi θi−1 σi σi−1 ϑi = σi−1 σi θi−1 σi ϑi−1 σi−1 = σi−1 σi θi−1 ϑi−1 σi σi−1 = σi−1 σi ξi−1 σi σi−1 , and hence ξi is contained in our orbit as well. The second claim follows from (7). This completes the proof. For w ∈ Sn such that w({1, 2}) = {i, j}, where i < j, we set i,j = w−1 θ1 w, μi,j = w−1 ξ1 w, νi,j = w−1 η1 w. All these elements are well defined by Lemma 5.3. Lemma 5.5. (a) ϑi i,j = ϑj i,j = ϑi ϑj i,j = νi,j ; ϑk i,j = i,j ϑk , k ∈ {i, j}. (b) ϑi μi,j = ϑj μi,j = ϑi ϑj μi,j = ϑi ϑj ; ϑk μi,j = μi,j ϑk , k ∈ {i, j}. Proof. First we prove (a). By Lemma 5.4 it is enough to check that ϑ1 1,2 = ϑ2 1,2 = ϑ1 ϑ2 1,2 = ν1,2 and that ϑ3 1,2 = 1,2 ϑ3 . The latter equalities follow from (18) and (17). Now we prove (b). Again, because of Lemma 5.4 it is enough to check that ϑ1 μ1,2 = ϑ2 μ1,2 = ϑ1 ϑ2 μ1,2 = ϑ1 ϑ2 and that ϑ3 μ1,2 = μ1,2 ϑ3 . Using (19), (18) and (17) we have ϑ1 μ1,2 = ϑ1 θ1 ϑ1 = ϑ1 ϑ2 ;
ϑ1 μ2,3 = ϑ1 θ2 ϑ2 = θ2 ϑ1 ϑ2 = θ2 ϑ2 ϑ1 = μ2,3 ϑ1 ,
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as required.
Lemma 5.6. Suppose {i, j} ∩ {p, q} = ∅. Then i,j p,q = p,q i,j , μi,j μp,q = μp,q μi,j and i,j μp,q = μp,q i,j . Proof. Following the arguments from the proof of Lemma 3.6 it is enough to show that μ1,2 μ3,4 = μ3,4 μ1,2 and μ1,2 3,4 = 3,4 μ1,2 , that is that ξ1 ξ3 = ξ3 ξ1 and ξ1 θ3 = θ3 ξ1 . Using (17), (15) and (2) we have ξ1 ξ3 = θ1 ϑ1 θ3 ϑ3 = θ1 θ3 ϑ1 ϑ3 = θ3 θ1 ϑ3 ϑ1 = θ3 ϑ3 θ1 ϑ1 = ξ3 ξ1 , and using (17) and (2) we also obtain ξ1 θ3 = θ1 ϑ1 θ3 = θ1 θ3 ϑ1 = θ3 ξ1 , as required.
Lemma 5.7. Suppose {i, j} ∩ {p, q} = ∅. Then each of the elements i,j p,q , μi,j μp,q , i,j μp,q , μi,j p,q is equal to an element of the form uθ1 v for some u, v ∈ IS n . Proof. Using the argument from the proof of Lemma 3.7 it is enough to prove the statement only for the elements μ1,2 μ2,3 , μ1,2 2,3 , 1,2 μ2,3 . We have μ1,2 μ2,3 = ξ1 ξ2 = θ1 ϑ1 θ2 ϑ2 = θ1 ϑ2 θ2 ϑ2 = θ1 ϑ2 ϑ3 = ξ1 ϑ3 = θ1 ϑ1 ϑ3 by (18) and (19); and μ1,2 2,3 = θ1 ϑ1 θ2 = θ1 ϑ1 σ1 σ2 θ1 σ1 σ2 = θ1 σ1 ϑ2 σ2 θ1 σ1 σ2 = θ1 σ1 σ2 ϑ3 θ1 σ1 σ2 = θ1 σ1 σ2 θ1 ϑ3 σ1 σ2 = θ1 σ2 θ1 ϑ3 σ1 σ2 = θ1 ϑ3 σ1 σ2 by (1), (5), (3), (16). Finally, 1,2 μ2,3 = θ1 θ2 ϑ2 = θ1 σ1 σ2 θ1 σ2 σ1 ϑ2 = θ1 σ2 ϑ1 σ1 . using (1), (3) and (5). The statement follows.
For each subset {i1 , . . . , ik } of {1, 2, . . . , n} set ϑ({i1 , . . . , ik }) = ϑi1 . . . ϑik . Obviously, ϑ({i1 , . . . , ik }) is an idempotent and each idempotent of IS n has such a form. In the following we will use the obvious fact that each element of IS n can be written in the form uv, where u is an idempotent, and v ∈ Sn . As in the previous sections we consider the Sn × Sn -action on T given by (g, h)(x) = −1 g xh for x ∈ T and (g, h) ∈ Sn × Sn . Lemma 5.8. Every Sn × Sn -orbit contains either e or an element of the form ϑ(A)γi1 ,j1 . . . γis ,js , where A ⊂ {1, 2, . . . , n}, the sets {il , jl } are pairwise disjoint, and each γil ,jl equals either il ,jl or μil ,jl . Proof. The idea of the proof is analogous to that of Lemma 3.8. Let x ∈ T . If x ∈ Sn the statement is obvious so assume that x ∈ Sn . Since T is generated by IS n and θ1 we can write x = wuθ1 u1 g1 θ1 u2 g2 · · · θ1 uk gk (22)
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for some k ≥ 1, w, g1 , . . . , gk ∈ Sn and u, u1 , . . . , uk ∈ E(IS n ). Moreover, we may assume that x cannot be written as a product of θ1 ’s and elements of IS n , which contains less than k occurrences of θ1 . We claim that x can be written as x = wu γ11 g1 γ12 g2 · · · γ1k gk ,
(23)
where, w, g1 , . . . , gk ∈ Sn , u ∈ E(IS n ), and each γ1i is equal to either θ1 or ξ1 . Let us prove this by induction on k. Let k = 1 and x = wuθ1 u1 g1 . We know that u1 = ϑ(B) for some B ⊂ {1, . . . , n}. Let A = B \ {1, 2}. Using (17) and (18) we obtain that ⎧ ⎪ ⎨ wuu1 θ1 g1 , if B ∩ {1, 2} = ∅; x= ⎪ ⎩ wuϑ(A)ξ1 g1 , if B ∩ {1, 2} = ∅, as required. Now suppose k ≥ 2. Applying the basis of the induction to θ1 uk gk we obtain x = wuθ1 u1 g1 θ1 u2 g2 · · · θ1 uk−1 gk−1 θ1 uk gk = wuθ1 u1 g1 θ1 u2 g2 · · · θ1 uk−1 gk−1 uk γ1k gk , where uk is an idempotent of IS n and γ1k is either ξ1 or θ1 . Now, since uk−1 gk−1 uk ∈ IS n , for some gk−1 ∈ Sn and uk−1 ∈ E(IS n ). Now (23) we can write uk−1 gk−1 uk = uk−1 gk−1 follows by applying the inductive assumption to wuθ1 u1 g1 θ1 u2 g2 · · · uk−2 gk−2 θ1 uk−1 gk−1 . Similarly to (8) we can rewrite (23) as follows: x = wu (g1 · · · gk )(g1 · · · gk )−1 γ11 (g1 · · · gk )·
gk )−1 γ1k−1 (gk−1 gk )gk−1 γ1k gk , · (g2 · · · gk )−1 γ12 (g2 · · · gk ) · · · (gk−1
and therefore we can write x = vu γi1 ,j1 · · · γik ,jk ,
(24)
where v = wg1 · · · gk , {it , jt }={(gt · · · gk )(1), (gt · · · gk )(2)}, 1 ≤ t ≤ k, and each γil ,jl is equal to either il ,jl or μil ,jl . Since x was initially chosen so that it cannot be reduced to an element of T , which contains less that k entries of θ1 , from Lemma 5.7 it follows that {it , jt } ∩ {il , jl } = ∅ for any two factors γit ,jt , γil ,jl in (24). This implies that the Sn × Sn -orbit of x contains u γi1 ,j1 · · · γis ,js such that u ∈ E(IS n ), {it , jt } ∩ {il , jl } = ∅ for all l = t. The statement follows. Corollary 5.9. Any Sn × Sn - orbit contains either e or an element of the form ϑ(A)γi1 ,j1 · · · γis ,js , such that (i) the sets {il , jl } are pairwise disjoint; (ii) each γil ,jl equals to either il ,jl or μil ,jl or νil ,jl ; (iii) A ∩ {i1 , j1 , . . . is , js } = ∅. Proof. This follows from Lemmas 5.8 and 5.5.
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Now we introduce the notion of a canonical element. Let k, l, m, t be some nonnegative integers satisfying 2k + 2l + 2m + t ≤ n. Set δ(0, 0, 0, 0) = e and if at least one of k, l, m, t is not zero, set δ(k, l, m, t) = θ1 θ3 · · · θ2k−1 ξ2k+1 ξ2k+3 · · · ξ2k+2l−1 ν2k+2l+1 ν2k+2l+3 · · · · · ν2k+2l+2m−1 ϑ2k+2l+2m+1 ϑ2k+2l+2m+2 · · · ϑ2k+2l+2m+t . (25) The element δ(k, l, m, t) such that l = 0 or m = 0 will be called a canonical element of type (k, l, m, n). Corollary 5.10. Every Sn × Sn -orbit contains a canonical element. Proof. By Corollary 5.9 we have to prove that, the Sn ×Sn -orbit of the element ϑ(A)γi1 ,j1 · · · γis ,js , satisfying the conditions of Corollary 5.9, contains a canonical element. Using conjugation, we can always reduce ϑ(A)γi1 ,j1 · · · γis ,js to some δ(k, l, m, t). However, it might happen that both m and l are non-zero. Without loss of generality we may assume m ≥ l ≥ 1. Using (20) and conjugation we get that the Sn × Sn -orbit of the element μi,j νp,q contains i,j ϑp ϑq provided that {i, j} ∩ {p, q} = ∅. Hence the Sn × Sn -orbit of our δ(k, l, m, t) contains δ(k + 1, l − 1, m − 1, t + 2). Proceeding by induction we get that the Sn × Sn -orbit of our δ(k, l, m, t) contains δ(k + l, 0, m − l, t + 2l), which is canonical. This completes the proof. Lemma 5.11. The Sn × Sn -orbits of the canonical element δ(k, l, 0, t) and δ(k, 0, l, t) contain at most (n!)2 (k + l)!2k+l t!k!2k (2l + t)!(n − 2k − 2l − t)! elements. Proof. We will prove the statement for the element δ(k, l, 0, t). For δ(k, 0, l, t) the proof is analogous. We use the arguments similar to those from the proof of Lemma 3.10. It is enough to show that the stabilizer of δ(k, l, 0, t) under the Sn × Sn -action contains at least (k + l)!2k+l t!k!2k (2l + t)!(n − 2k − 2l − t)! elements. Set Σ0i = σ2i σ2i−1 σ2i+1 σ2i , 1 ≤ i ≤ k + l − 1;
Σ1i = σ2i σ2i−1 σ2i+1 σ2i σ2i−1 , 1 ≤ i ≤ k + l − 1. Then both Σ0i and Σ1i swap the sets {2i−1, 2i} and {2i+1, 2i+2}. It follows that the group H, generated by all Σ0i , consists of all permutations of the set {1, 2}, {3, 4}, . . . , {2k + 2l − 1, 2k + 2l} and is therefore isomorphic to the group Sk+l . Further, it is easy to ˜ generated by all Σ0i and Σ1i , is isomorphic to the wreath product see that the group H, H S2 . From (6) and (3) it follows that the left multiplications with Σ0i and Σ1i stabilizes ˜ stabilizes δ(k, l, 0, t) δ(k, l, 0, t). Therefore the left multiplication with each element of H as well. Now, from (16) and (18) it follows that σi ηi = σi ϑi ϑi+1 θi = ϑi+1 σi ϑi+1 θi = ϑi σi ϑi θi = ϑi ϑi+1 θi = ηi .
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for all i = 1, . . . , n − 1. Moreover, σi+1 ηi ηi+2 = σi+1 ϑi+1 θi ϑi+2 θi+2 = σi+1 ϑi+1 ϑi+2 θi θi+2 = ϑi+1 ϑi+2 θi θi+2 = ϑi+1 θi ϑi+2 θi+2 = ηi ηi+2 for all i = 1, . . . , n − 3 by (17) and (16) and σi+1 ηi ϑi+2 = σi+1 ϑi+1 θi ϑi+2 = σi+1 ϑi+1 ϑi+2 θi = ϑi+1 ϑi+2 θi = ηi ϑi+2 for all i = 1, . . . , n − 2 again by (17) and (16). Using this and the fact that ηi commutes with each of θj , ηj , ξj whenever |i−j| > 1 we see that each of the elements σi , 2k+2l−1 ≤ i ≤ 2k + 2l + t, stabilizes δ(k, l, 0, t) under the left multiplication. All these elements generate the group H0 St , which stabilizes δ(k, l, 0, t) and has trivial intersection with ˜ Let H1 = H0 × H. ˜ H. Analogously one shows that there is a group, H2 , isomorphic to the wreath product (Sk S2 ) × S2l+t , such that each element of this group stabilizes δ(k, l, 0, t) with respect to the right multiplication. In addition to this, from (3) we have that conjugation by any element from the group H3 = σ2k+2l+t+1 , . . . , σn−1 Sn−2k−2l−t stabilizes δ(k, l, 0, t). Observe that the group, generated by H1 , H2 and H3 , is a direct product of H1 , H2 and H3 . Hence, using the product rule we derive that the cardinality of the stabilizer of δ(k, l, 0, t) is at least (k + l)!2k+l t!k!2k (2l + t)!(n − 2k − 2l − t)!, and the proof is complete.
Proof (of Theorem 5.1). Comparing Lemma 5.11 and Proposition 2.1(d) we have |T | ≤ |Bn |. Since ϕ : T → Bn is surjective we have |T | ≥ |Bn |. Hence |T | = |Bn | and ϕ is an isomorphism.
Acknowledgment This paper was written during the visit of the first author to Uppsala University, which was supported by the Swedish Institute. The financial support of the Swedish Institute and the hospitality of Uppsala University are gratefully acknowledged. For the second author the research was partially supported by the Swedish Research Council. We thank Victor Maltcev for informing us about the reference [7]. We would also like to thank the referee for very helpful suggestions.
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DOI: 10.2478/s11533-006-0016-7 Research article CEJM 4(3) 2006 435–448
Accelerating the convergence of trigonometric series Anry Nersessian∗ , Arnak Poghosyan† Institute of Mathematics, National Academy of Sciences of Armenia, Yerevan 375019, Armenia
Received 1 July 2005; accepted 13 March 2006 Abstract: A nonlinear method of accelerating both the convergence of Fourier series and trigonometric interpolation is investigated. Asymptotic estimates of errors are derived for smooth functions. Numerical results are represented and discussed. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Convergence acceleration, Pade approximation, asymptotic error MSC (2000): 65B99, 42A10, 42A15, 41A21
1
Introduction
It is well known that, for f ∈ L2 (−1, 1), the series ∞
f (x) =
iπnx
fn e
n=−∞
is convergent by L2 -norm
||f || =
1 , fn = 2 1
−1
2
1
f (x)e−iπnx dx
(1)
−1
1/2
|f (x)| dx
.
For practical purposes, approximations are obtained by using only a finite number of Fourier coefficients {fn }, |n| ≤ N < ∞. As is also well known [32], when we approximate f by truncated Fourier series (partial sums) SN (f ) :=
N n=−N
∗ †
E-mail: [email protected] E-mail: [email protected]
fn eiπnx
(2)
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or by trigonometric interpolation IN (f ) :=
N n=−N
fn eiπnx ,
fn =
N 1 f (xk )e−iπnxk , 2N + 1 k=−N
xk =
2k , 2N + 1
(3)
the resulting error is strongly dependent on the smoothness of f . Approximating a 2periodic f ∈ C ∞ (R) function by SN or IN (N >> 1) is highly effective. When the approximated function has a point of discontinuity, the above mentioned approximations lead to the Gibbs phenomenon. The ”oscillations” caused by this phenomenon are typically propagated into regions away from the singularity and degrade the quality of the approximations. Different ways of treating this deficiency have been suggested in the literature (see, for example, [14–17]). The idea of increasing the convergence rate of Fourier series by subtracting a polynomial that represents the discontinuities in the function and some of its first derivatives was suggested by A.Krylov in 1906 [19] and later, in 1964, by Lanczos [20, 21] (see also [2, 23] and [18, with references]). The key problem lies in determining the singularity amplitudes. As formulated by Gottlieb [12], these amplitudes could be found from the first N Fourier coefficients. This idea has been realized by Eckhoff in a series of papers [5–8] where the values of the ”jumps” are solutions of the corresponding system of linear equations. Let us refer to this approach as the Krylov-Gottlieb-Eckhoff (KGE) method (see also [3, 11, 13, 22] and, for the multidimensional case, [25, 28]). Application of Pade approximants [1] to Fourier series has been studied by several investigators. The general form of Fourier-Pade representation has been suggested by Cheney [4], but he does not discuss algorithms for computing coefficients, rates of convergence, and so forth. Geer [10] introduced and studied a class of approximations to a periodic function f that uses the ideas of Pade (rational approximations). While these approximations do not ”eliminate” the Gibbs phenomenon, they do mitigate its effect. For eliminating the Gibbs phenomenon, algorithms based on Pade-type approximations were described and studied in [9, 26, 29, 30]. In [24], Pade approximants are applied to the asymptotic expansion of coefficients of Fourier series for piecewise smooth functions, leading to a new kind of approximation. In [27], the corresponding asymptotic estimates of errors of these approximations are investigated. Here, we extend the method to trigonometric interpolations. The proposed approximations are exact for a system of quasipolynomials while the KGE-method is exact for a subsystem of polynomials. Thus, we obtain a generalization of the latter. The quasipolynomial approach is nonlinear while the KGE-method is (given the exact jumps) linear. If the jumps and Fourier coefficients of the approximated function are known, then the KGE-method can be constructed without any additional calculations. However, for the quasipolynomial method, we also need the values of some parameters that can be determined from a nonlinear system of equations with jumps in the coefficients. This additional complexity in calculation yields round off errors of the approximations that are more precise and more stable. Theorems and numerical examples are presented. Moreover, comparisons between the quasipolynomial and the
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KGE-method are made. We expect that the proposed approximations, especially insofar as they are derived by a tool as flexible as the system of quasipolynomials, should result in new algorithms of increased precision and robustness.
2
KGE-method
We say that f ∈ C q [−1, 1], q ≥ 0 if f (q) is continuous in [−1, 1]. Denote Ak (f ) = f (k) (1) − f (k) (−1), k = 0, · · · , q. The idea of the KGE-method is to split the given function f ∈ C q [−1, 1] into two parts q−1 f (x) = F (x) + Ak (f )Bk (x), (4) k=0
where F is a relatively smooth function and Bk (x) are 2-periodic Bernoulli polynomials with Fourier coefficients ⎧ ⎪ ⎨ 0, n=0 (5) Bk,n = n+1 ⎪ ⎩ (−1) k+1 , n = ±1, ±2, ... 2(iπn)
Approximating the function F by truncated Fourier series leads to the KGE-approximation SN,q (f ) =
N
iπnx
Fn e
+
n=−N
q−1
Ak (f )Bk (x),
(6)
k=0
where the coefficients Fn can be expressed by fn and Bk,n from (1), (4), and (5). Similarly, approximating the function F by trigonometric interpolation leads to KGEinterpolation (see (3)) IN,q (f ) =
N
Fn e
iπnx
n=−N
+
q−1
Ak (f )Bk (x),
(7)
k=0
k )n from (4). where the discrete coefficients Fn can be expressed by fn and (B Theorem 2.1. Suppose f ∈ C q [−1, 1], q ≥ 1, f (q+1) ∈ L1 (−1, 1). Then RN,q (f ) := f (x) − SN,q (f ) = Aq (f )
(−1)n+1 eiπnx + o(N −q ), N → ∞. q+1 2(iπn)
(8)
|n|>N
Proof. By q-fold integration by parts in (1), we have that 1 q−1 (−1)n+1 Ak (f ) 1 fn = + f (q) (x)e−iπnx dx. k+1 q 2 (iπn) 2(iπn) −1 k=0
(9)
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Therefore,
RN,q (f ) =
Fn eiπnx ,
(10)
|n|>N
where (−1)n+1 Aq (f ) 1 + Fn = q+1 2 (iπn) 2(iπn)q+1
1
f (q+1) (x)e−iπnx dx.
(11)
−1
Note that, according to the well-known Riemann-Lebesgue theorem [32], the second term is o(n−q−1 ) as n → ∞. This concludes the proof. Similarly, we prove the following result: Theorem 2.2. Suppose f ∈ C q [−1, 1], q ≥ 1, f (q+1) ∈ L1 (−1, 1). Then rN,q (f ) := f (x) − IN,q (f ) = Aq (f )× |n|>N
N ∞ (−1)n+1 eiπnx (−1)n+s+1 eiπnx +o(N −q ), N → ∞. (12) + q+1 q+1 q+1 2(iπn) 2(iπ) (n + s(2N + 1)) n=−N s=−∞ s=0
Proof. Just note that, from (11), we have at least Fn = O(n−2 ), n → ∞, and so Fn =
∞
Fn+s(2N +1) .
s=−∞
3
Quasipolynomial (QP-) method
3.1 The essential features of quasipolynomial approximation [24] are both the application of Pade approximants to the asymptotic expansion of fn (see 9) and the solution of the corresponding system of nonlinear equations (a system that arises in the theory of Pade approximations). Consider a finite sequence of complex numbers θ := {θk }m k=1 , m ≥ 1, and denote k−1 Δ0n (θ) = An (f ), Δkn (θ) = Δk−1 n (θ) + θk Δn−1 (θ), k ≥ 1.
If n < 0, we set Δkn (θ) = 0. It is easy to verify that, for x = −1/θ1 , q−1
Aq−1 θ1 1 Ak x = x (Ak + θ1 Ak−1 )xk . + 1 + θ x 1 + θ x 1 1 k=0 k=0 q−1
k
q
(13)
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Note that, for θ1 = 0, the sum on the left side of (13) remains unchanged. Iterating this transformation up to m times yields the following formula (x = −1/θk ; k = 1, · · · , m; m ≤ q − 1): q−1
k
Ak x = x
k=0
q
m k=1
θk Δk−1 1 q−1 (θ) k + Δm k m k (θ)x . (1 + θ x) s s=1 s=1 (1 + θs x) k=0 q−1
(14)
Suppose f ∈ C q [−1, 1]. Applying transformation (14) to the first term of (9) with (iπn)−1 instead of x, we derive
where
and
fn = Qn + Pn , n = 0,
(15)
q−m−1 (−1)n+1 (iπn)m Δm k (θ) Qn = m 2 s=1 (iπn + θs ) k=0 (iπn)k+1
(16)
m k−1 (−1)n+1 θk Δq−1 (θ)(iπn)k Pn = + k 2(iπn)q+1 k=1 s=1 (iπn + θs )
1 q−1 (−1)n+1 (iπn)m Δm 1 k (θ) + + f (q) (t) e−iπnt dt. m k+1 q 2 k=1 (iπn + θk ) k=q−m (iπn) 2(iπn) −1
(17)
Inasmuch as (15) holds, we can split the function f into two parts f (x) = Q(x) + P (x), where Q(x) =
∞
∞
Qn eiπnx , P (x) =
n=−∞ n=0
Pn eiπnx , P0 = f0 .
(18)
(19)
n=−∞
Approximating P by the truncated Fourier series leads to SN,q,m (f ) = Q(x) +
N
(fn − Qn )eiπnx
(20)
n=−N
and approximating P by trigonometric interpolation leads to IN,q,m (f ) = Q(x) +
N
n )eiπnx . (fn − Q
(21)
n=−N
It is important to note that, for θ1 = θ2 = · · · = θm = 0, approximations SN,q,m (f ) and IN,q,m (f ) coincide with the KGE-approximation and the KGE- interpolation, respectively. Approximation properties of SN,q,m and IN,q,m are strongly connected with the smoothness of the function P or, put another way, with the rate of convergence of Pn to zero. This condition leads to the following system of nonlinear equations (see the second term in (17)) for the unknown vector θ: Δm k (θ) = 0, k = q − m, · · · , q − 1.
(22)
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Note that, if θ is a solution of (22) and f ∈ C q [−1, 1], f (q+1) ∈ L1 (−1, 1), then Pn = O(n−q−1 ), n → ∞. We call approximations (20) and (21), together with (22), QPapproximation and QP-interpolation, respectively. Actually, we apply the Pade approximation [q+m-1/m] to the sum on the left side of (14) [1].
3.2 We are interested in the asymptotic behavior of RN,q,m (f ) := f (x) − SN,q,m (f ) and rN,q,m (f ) := f (x) − IN,q,m (f ). By γk (m), k = 0, · · · , m we denote the coefficients of the polynomial m
(1 + θk x) ≡
k=1
γk (m)xk .
k=0
Note that Δm k (θ)
m
= Ak (f ) +
m
γs (m)Ak−s (f ).
s=1
Hence, the system (22) can be written in the form m
γs (m)Ak−s+q−m−1 (f ) = −Ak+q−m−1 (f ), k = 1, · · · , m.
(23)
s=1
Denote Urm = [Ak−s+r (f )], k, s = 1, · · · , m. Theorem 3.1. [27] Suppose f ∈ C q [−1, 1], q ≥ 1, f (q+1) ∈ L1 [−1, 1] and m detUq−m−1 = 0.
Then, with θ from (22), the following holds: RN, q,m (f ) = (−1)m
m+1 detUq−m (−1)n+1 eiπnx + o(N −q ), N → ∞. m detUq−m−1 2(iπn)q+1
(24)
|n|>N
Proof. From (18) and (20), we have RN, q,m (f ) =
Pn eiπnx ,
(25)
|n|>N
where (see (17)) (−1)n+1 Pn = 2(iπn)q+1
Aq (f ) +
m k=1
θk Δk−1 q−1 (θ) k θs 1 + s=1 iπn
+ o(n−q−1 ), n → ∞.
(26)
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It now follows that (−1)n+1 eiπnx + o(N −q ), N → ∞. 2(iπn)q+1
RN,q,m (f ) = Δm q (θ)
(27)
|n|>N
Here, we use the fact that m−1 m−2 Δm (θ) + θm Δm−1 (θ) + θm−1 Δm−2 q (θ) = Δq q−1 (θ) = Δq q−1 (θ)+
+θm Δm−1 q−1 (θ)
=
Δ0q (θ)
+
m
θk Δk−1 q−1 (θ)
= Aq (f ) +
k=1
m
θk Δk−1 q−1 (θ).
k=1
According to Cramer’s rule, γs (m) =
Ms , s = 1, · · · , m, m detUq−m−1
where {Ms } are the corresponding minors. Consequently, Δm q (θ) = Aq (f ) +
m
γs (m)Aq−s (f ) =
s=1
m m+1 1 m detUq−m Ms Aq−s (f ) = (−1) . = Aq (f ) + m m detUq−m−1 detUq−m−1 s=1
m Theorem 3.2. Suppose f ∈ C q [−1, 1], q ≥ 1, f (q+1) ∈ L1 [−1, 1], and detUq−m−1 = 0. Then, for θ from (22), the following holds: m+1 detUq−m × rN, q,m (f ) = (−1) m detUq−m−1 m
⎛
⎞ N
∞
(−1)n+s+1 eiπnx ⎜ (−1)n+1 eiπnx ⎟ −q + ⎝ ⎠ + o(N ), N → ∞. q+1 q+1 q+1 2(iπn) 2(iπ) (n + s(2N + 1)) n=−N s=−∞ |n|>N
s=0
(28) Proof. Using the relation (at least Pn = O(n−2 ), n → ∞) ∞
Pn =
Pn+s(2N +1) ,
s=−∞
we obtain rN,q,m (f ) =
|n|>N
iπnx
Pn e
−
N n=−N
iπnx
e
∞
Pn+s(2N +1) .
s=−∞ s=0
Proceeding as in the proof of theorem 3.1, this concludes the proof.
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3.3 For a practical realization of the QP-method, we need the explicit form of the function Q(x). Consider the case m + 1 ≤ q ≤ 2m and θs = 0, θs = θk , s, k = 1, · · · , m. Given the relation m (−1)m−k−1 θjm−k−1 (iπn)m−k−1 m m = , k = 0, · · · , q − m − 1, (iπn + θ ) (θ − θ ) j s j s=1 (iπn + θs ) s=1 j=1 s=j
we expand Qn into simple fractions q−m−1 m (−1)n 1 m−k m−k−1 m Δm θj . Qn = k (θ)(−1) 2 j=1 (iπn + θj ) s=1 (θs − θj ) k=0
(29)
s=j
According to the representation ∞ (−1)n eiπnx 1 −θj x = e iπn + θj shθj n=−∞
from (29), for the case m + 1 ≤ q < 2m, we derive Q(x) =
m j=1
2 sh(θj )
q−m−1
e−θj x m
s=1 (θs − θj ) s=j
m−k m−k−1 Δm θj . k (θ)(−1)
(30)
k=0
If q = 2m, then m m−1 Δm e−θj x m−1 (θ) m + Δm (θ)(−1)m−k θjm−k−1 . Q(x) = m 2 k=1 θk j=1 2sh θj s=1 (θs − θj ) k=0 k
(31)
s=j
The explicit form of Q(x) in other cases can be calculated similarly. In general, we have the following representation: Lemma 3.3. [24]. Let {αs } , s = 1, ..., , 1 ≤ < ∞ , be a finite set of complex numbers and Υ ⊆ {αs } a subset of integers. Then ‡ ∞ (−1)k+1 p(k)eiπkx p(z)eiπzx =π Res , βs βs z=αr sin(πz) (k − α (z − α ) s) s s=1 s=1 r=1 k=−∞ k∈Υ /
where {βs }, s = 1, ..., is a set of positive integers, p(z) is a polynomial of degree less than ‡ ‡ s=1 βs − 1, and x = (x + 1) (mod2) − 1, −1 < x < 1. Now it follows that, in general, Q(x) is a quasipolynomial of the form Q(x) = ak xpk eiωk x , k
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where ωk ∈ C and where {pk } is a set of nonnegative integers. We have calculated the explicit form of Q(x) for specific cases. Calculations are carried out using the MATHEMATICA software package [31]. For a given q and m, denote Qq,m (x) = Q(x). Some of them are A0 A0 −θ1 x − e , 2θ1 2 sh θ1
Q2,1 (x) = Q3,1 (x) = − Q3,2 (x) = −
A1 A1 + A0 θ1 A1 +x + e−θ1 x , 2 2θ1 2θ1 2 θ1 sh θ1
A0 θ 1 A0 θ 2 e−θ1 x + e−θ2 x , 2 (θ1 − θ2 ) sh θ1 2 (θ1 − θ2 ) sh θ2
A1 θ13 + A2 (θ12 − 6) x + Q4,1 (x) = − 12θ13 2
Q4,2 (x) =
A2 + A1 θ1 A2 A2 A0 − 2 + x2 − 2 e−θ1 x , θ1 4θ1 2 θ1 sh θ1
A 1 + A0 θ 2 A 1 + A0 θ 1 A1 + A0 (θ1 + θ2 ) e−θ1 x − e−θ2 x + , 2 (θ1 − θ2 ) sh θ1 2 (θ1 − θ2 ) sh θ2 2θ1 θ2
Q4,3 (x) = −
A0 θ12 A0 θ22 e−θ1 x − e−θ2 x − 2 (θ1 − θ2 )(θ1 − θ3 ) sh θ1 2 (θ2 − θ1 )(θ2 − θ3 ) sh θ2 −
A0 θ32 e−θ3 x . 2 (θ3 − θ1 )(θ3 − θ2 ) sh θ3
Similar calculations can be carried out for multiple θ. For example, when θ1 = θ2 = θ, q = 3, m = 2, we have Q3,2 (x) =
A0 (2θcth θ − 1 + 2xθ)e−θx . 2sh θ
If θ1 = θ2 = θ3 = θ, q = 4, m = 3, we derive Q4,3 (x) =
A0 e−θx (2xθ(5 − 3θcth θ) − 3x2 θ2 − 6θ2 cth2 θ + 3θ2 + 10θcth θ − 2). 4sh θ
For m = 1, system (22) can easily be solved symbolically. In particular, the explicit forms of some of the quasipolynomials are derived to be A0 AA1 x A20 0 e − , 1 2A1 2sh A A0 A2 A21 A21 A31 A0 x A1 Q3,1 (x) = − 2 + x − + e , A2 2A2 2 2A2 2A2 sh A1 Q2,1 (x) =
A4 A1 A22 Q4,1 (x) = − − 23 + +x 12 2A3 12A3
A3 A0 − 22 2 2A3
+x
2
A2 A1 − 2 4 4A3
A3
A3 e A 2 + 22 A3 . 2A3 sh A2 x
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Numerical results
For a given f , q, and m, we put m det(Uq−m−1 ) . aq,m (f ) = Aq det(U m+1 )
(32)
q−m
The constant aq,m (f ) describes the effectiveness of the QP-approximation (Theorem 3.1) compared to the KGE-approximation (Theorem 2.1) as well as the effectiveness of the QP-interpolation (Theorem 3.2) compared to the KGE-interpolation (Theorem 2.2) when N >> 1. Let us consider two typical examples. All calculations are carried out using MATHEMATICA software on a Pentium 4 computer. First, consider the Bessel function f (x) = J0 (14x − 1).
(33)
In Figure 1, the graphics of aq,m (f ) for (33) are represented when q = 8, 9, 10 and 1 ≤ m ≤ q −1. We observe that the QP-method is more precise than the KGE-method almost 250 times for q = 8; m = 4 and more than 300 times for q = 10; m = 4, 6. Figure 1 also shows the optimal values of m when parameter q is fixed.
a8,m
a9,m 250
250
a10,m 300
50
50
50 m
m 1
4
7
1
4 5
8
m 1
5
9
Fig. 1 Graphics of aq,m (f ) for (33) when q = 8, 9, 10 and 1 ≤ m ≤ q − 1. Results in Figure 1 are asymptotic (N 1). It is interesting to see the numerical behavior of the QP-method for both small and moderate values of N . We illustrate the results for just the QP-interpolation because our experiments show that, in general, the behavior of the QP-approximation (see [24] and [27] for details) is very similar to that of the QP-interpolation. The actual effectiveness (in a uniform metric) of the QPinterpolation compared to the KGE-interpolation can be represented by the ratio aN,q,m (f ) =
max|x|≤1 |rN,q (f )| . max|x|≤1 |rN,q,m (f )|
(34)
In Table 1, approximate values of aN,8,4 are shown for (33). Calculations are carried out with 64 digits of precision. Comparison with the theoretical value a8,4 = 271.1 shows that experimental and theoretical estimates are rather close for N ≥ 32. In Figure 2, the uniform errors are scaled logarithmically. Here, we compare the QP- and the KGE-interpolations for q = 8
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445
N
8
16
32
64
128
256
512
aN,8,4
97
206
250
264
269
270
271
14
20 17.5 15 12.5 10 7.5 5
lgerror
lgerror
Table 1 Approximate values of aN,8,4 for different N while interpolating the function (33).
KGE QP 0
100
200
300
400
12 10 8 KGE QP
6 4
500
0
100
200
N
300
400
500
N
Fig. 2 Uniform errors in log scale, f defined by (33), q = 8, m = 4, N ≤ 512. Left: with 64 digits of precision, Right: with standard precision. and m = 4. The left figure corresponds to calculations with 64 digits of precision; the right figure we obtain from standard MATHEMATICA precision calculations. We see that, even in standard machine precision, the QP-interpolation is much more precise than the KGE-interpolation. For N ≥ 100, the QP-method is nearly 103 (compare this with the theoretically-predicted value of 271) times more precise than the KGEmethod. Furthermore, the QP-method is less sensitive to round-off errors. Now consider the second example f (x) =
1 . 1.1 − x
(35)
This function has the greatest increase of Ak jumps within the class of analytic functions in the neighborhood of the interval [−1, 1]. In Figure 3, the graphics of aq,m are represented. Approximate values of aN,8,4 are displayed in Table 2. For this example, when N ≥ 256, experimental results aN,8,4 are close to the theoretical estimate a8,4 = 72.9. Table 2 Approximate values of aN,8,4 for different N while interpolating the function (35). N
8
16
32
64
128
256
512
1024
aN,8,4
3414
433
237
62
44
62
68
71
In Figure 4, the logarithms of the uniform errors are represented. The left figure corresponds to calculations with 64 digits of precision; the right figure is obtained from stan-
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a8,m 70
a10,m 250
a9,m 120 50
20
50 1
4
7
m
m
m 1
4 5
8
1
5
9
15 12.5 10 7.5 5 2.5 0
lgerror
lgerror
Fig. 3 The graphs of aq,m for fixed values of q (q = 8, 9, 10) and 1 ≤ m ≤ q when (35) is approximated.
KGE QP 0
100
200
300
400
500
12 10 8 6 4 2 0
KGE QP 0
100
N
200
300
400
500
N
Fig. 4 Uniform errors in log scale, f defined by (35), q = 8, m = 4, N ≤ 512. Left: with 64 digits of precision, Right: with standard precision. dard precision calculations. With standard precision and N ≥ 200 the QP-interpolation is 105 times more precise than the KGE-interpolation. For practical application of the QP-method, the numerical values of jumps Ak (f ) are also needed. These values can be recovered from Fourier coefficients or from discrete Fourier coefficients as shown in [5–8]. Numerical experiments show [24] that the application of this procedure to the QP-method is acceptable and, in general, maintains all characteristic features.
Acknowledgment The authors were supported in part by ISTC Grant A-823
References [1] G.A. Baker and P. Graves-Morris: Pade Approximants. Encyclopedia of mathematics and its applications, 2nd ed., Cambridge Univ. Press, Cambridge, 1996.
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DOI: 10.2478/s11533-006-0019-4 Research article CEJM 4(3) 2006 449–506
Local geometry of orbits for an ordinary classical Lie supergroup Tomasz Przebinda∗ Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA
Received 22 February 2006; accepted 10 May 2006 Abstract: In this paper we identify a real reductive dual pair of Roger Howe with an Ordinary Classical Lie supergroup. In these terms we describe the semisimple orbits of the dual pair in the symplectic space, a slice through a semisimple element of the symplectic space, an analog of a Cartan subalgebra, the corresponding Weyl group and the corresponding Weyl integration formula. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Dual pairs, Lie supergroups, orbits, integration formulas MSC (2000): 17B05, 17B75, 22E15
Introduction The purpose of this article is to present a few elementary facts about the local structure of orbits in the symplectic space under the action of a real reductive dual pair, see [6] and [7]. We shall use this material later to study the characters of the representations which occur in Howe’s correspondence. The corresponding facts for the adjoint action of a real reductive group on its Lie algebra is essentaily contained in section one (eleven pages) of part one of [11]. The main results are presented as quickly as possible, with the proofs deffered to further sections. These proofs, based on elementary linear algebra, are rather noninteresting, but had to be included. Some of the material included here is contained in an unpublished work of Howe, [5]. However our approach through the Lie superalgebras seems more akin to the standard theory, [11]. ∗
E-mail: [email protected]
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1
A slice through a point
Let M be a manifold and let G be a Lie group acting on M . Let x ∈ M and let Gx be the stabilizer of x in G. Assume that the orbit Gx ⊆ M is a regularly embedded submanifold. A connected submanifold U ⊆ M is called an admissible slice through x if and only if
x ∈ U,
(1.1)
U is G − stable, x
(1.2)
the tangent space Tx (M ) = Tx (U ) ⊕ Tx (Gx),
(1.3)
if g ∈ G and u, u ∈ U are such that gu = u then g ∈ G ,
(1.4)
the map G × U (g, u) → gu ∈ M is a submersion.
(1.5)
x
The condition (1.3) implies that the map μ : GU gu → gx ∈ Gx
(1.6)
is well defined. As shown in [11, part I, pages 15, 16], μ is a locally trivial fibration with the fiber U . In other words, for every point gx ∈ Gx there is an open neighborhood W ⊆ Gx, and a diffeomorphism φ such that the following diagram commutes: φ
W × U −−−→ μ−1 (W ) ⏐ ⏐ ⏐ ⏐ μ W
=
−−−→
(1.7)
W,
where the left vertical arrow is the projection on the first component. Let N ⊆ M be a complete metric subspace. Suppose N is the union of a finite set of G-orbits. Then, as shown in [12, 8.A.4.5], we can label the orbits O1 , O2 , ...Ok so that for 1 ≤ j ≤ k the set k Ol (1.8) Nj = l=j
is closed in N . Suppose x ∈ Oj for some 1 ≤ j ≤ k. A connected manifold U ⊆ M is called a weakly admissible slice through x if and only if the conditions (1.0), (1.2), (1.5) hold and the intersection of the image of the map (1.5) with Nj (1.9) is equal to Oj , and U ∩ Oj = {x}.
(1.10)
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Ordinary classical Lie supergroups and dual pairs
Let D = R, C or H, and let V0 , V1 be two finite dimensional left vector spaces over D. Set V = V0 ⊕ V1
(2.1)
and define an element S ∈ End(V ) by S(v0 + v1 ) = v0 − v1 Set
(vo ∈ V0 , v1 ∈ V1 ).
(2.2)
End(V )0 = {x ∈ End(V ); Sx = xS}, End(V )1 = {x ∈ End(V ); Sx = −xS},
(2.3)
GL(V )0 = GL(V ) ∩ End(V )0 . The real vector space End(V )0 is a Lie algebra, with the usual commutator [x, y] = xy − yx. The adjoint action of GL(V )0 on End(V ) Ad(g)x = gxg −1
(g ∈ GL(V )0 , x ∈ End(V ))
preserves both End(V )0 and End(V )1 . Furthermore the anticommutator End(V )1 × End(V )1 (x, y) → {x, y} = xy + yx ∈ End(V )0
(2.4)
is R-bi-linear and GL(V )0 -equivariant. Set x, y = trD/R {Sx, y}
(x, y ∈ End(V )).
(2.4’)
(Here trD/R (y) is the trace of y ∈ End(V ) viewed as an endomorphism of V over R.) It is easy to see that the form , is preserved under the action of GL(V )0 . Lemma 2.1. The restriction of the bilinear form , to End(V )1 is symplectic and non-degenerate. Moreover, the group homomorphism Ad : G → Sp(End(V )1 , , ) maps the groups G0 = {g ∈ GL(V )0 ; g|V1 = 1}, and G1 = {g ∈ GL(V )0 ; g|V0 = 1} injectively onto an irreducible dual pair of type II in the symplectic group Sp(End(V )1 , , ). Proof. The following map Hom(V0 , V1 ) ⊕ Hom(V1 , V0 ) (A, B) → xA,B ∈ End(V )1 xA,B (v0 + v1 ) = Bv1 + Av0
(v0 ∈ V0 , v1 ∈ V1 ),
(2.5)
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is an R-linear bijection. Furthermore, for v0 ∈ V0 and v1 ∈ V1 , and for any A, A ∈ Hom(V0 , V1 ) and B, B ∈ Hom(V1 , V0 ) we have xA,B xA ,B (v0 + v1 ) = BA v0 + AB v1 , and therefore S(xA,B xA ,B − xA ,B xA,B )(v0 + v1 ) = (BA − B A)v0 + (A B − AB )v1 . Hence, xA,B , xA ,B = trD/R (SxA,B xA ,B + xA ,B SxA,B ) = trD/R (S(xA,B xA ,B − xA ,B xA,B )) = trD/R (BA − B A) + trD/R (A B − AB )
(2.6)
= 2trD/R (BA − B A). Thus the form , is symplectic. It is easy to check that if trD/R (BA − B A) = 0 for all A and B then A = 0 and B = 0. Thus the form , is non-degenerate. The groups G0 and G1 are isomorphic to GL(V0 ) and GL(V1 ) by restriction. Further, the action of the groups G0 and G1 on Hom(V0 , V1 ) induced by the isomorphism (2.5), embeds these groups into GL(Hom(V0 , V1 )). It is not hard to check that G0 and G1 are mutual centralizers in GL(Hom(V0 , V1 )), and hence form a dual pair of type II in the symplectic group. Let ι be a possibly trivial involution on D. Let τ0 be a non-degenerate ι-hermitian form on V0 , and let τ1 be a non-degenerate ι-skew-hermitian form on V1 . Set τ = τ0 ⊕ τ1 . Then τ (u, v) = ι(τ (v, Su)) (u, v ∈ V ). (2.7) Define
g(V, τ )0 = {x ∈ End(V )0 ; τ (xu, v) = τ (u, −xv), u, v ∈ V }, g(V, τ )1 = {x ∈ End(V )1 ; τ (xu, v) = τ (u, Sxv), u, v ∈ V },
(2.8)
G(V, τ )0 = {g ∈ GL(V )0 ; τ (gu, gv) = τ (u, v), u, v ∈ V }. Clearly, G(V, τ )0 is a Lie subgroup of GL(V )0 , with the Lie algebra g(V, τ )0 . Moreover, it is easy to check that the anticommutator (2.4) maps g(V, τ )1 × g(V, τ )1 into g(V, τ )0 . Furthermore, the adjoint action of G(V, τ )0 preserves g(V, τ )0 , g(V, τ )1 , and the form , . Lemma 2.2. The restriction of the bilinear form , to g(V, τ )1 is symplectic and non-degenerate. Moreover, Ad : G(V, τ )0 → Sp(g(V, τ )1 , , ) maps the groups G0 = {g ∈ G(V, τ )0 ; g|V1 = 1}, and G1 = {g ∈ G(V, τ )0 ; g|V0 = 1} injectively onto an irreducible dual pair of type I in the symplectic group Sp(g(V, τ )1 , , ).
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Proof. Recall the map End(V0 ) A → A ∈ End(V0 ), τ0 (Au0 , v0 ) = τ0 (u0 , A v0 )
(A ∈ End(V0 )).
Let v1 , v2 , v3 , ... be a basis of V0 , and let v1 , v2 , v3 , ... be the dual basis, in the sense that τ0 (vi , vj ) = δij . (Here δij is the Kronecker delta: δij = 1 if i = j, and δij = 0 if i = j.) Then for A ∈ End(V )0 τ0 (Avi , vi ) = τ0 (vi , A vi ) = ι(τ0 (A vi , vi )). i
i
i
Thus trD/R (A) = trD/R (A )
(A ∈ End(V0 )).
(2.9)
Define the following maps Hom(V0 , V1 ) w → w∗ ∈ Hom(V1 , V0 ), τ1 (wv0 , v1 ) = τ0 (v0 , w∗ v1 )
(v0 ∈ V0 , v1 ∈ V1 ),
∗
Hom(V1 , V0 ) w → w ∈ Hom(V0 , V1 ), τ0 (wv1 , v0 ) = τ1 (v1 , w∗ v0 ) Then
(2.10)
(v0 ∈ V0 , v1 ∈ V1 ).
τ1 (wv0 , v1 ) = τ0 (v0 , w∗ v1 ) = ι(τ0 (w∗ v1 , v0 )) = ι(τ1 (v1 , w∗∗ v0 )) = τ1 (−w∗∗ v0 , v1 ).
Thus (as is well known, [5]) w∗∗ = −w
(w ∈ Hom(V0 , V1 )).
(2.11)
For x ∈ g(V, τ )1 let wx ∈ Hom(V0 , V1 ) be the restriction of x to V0 . Then x(v0 + v1 ) = wx∗ v1 + wx v0
(v0 ∈ V0 , v1 ∈ V1 ).
Since for x, y ∈ g(V, τ )1 x, y = trD/R (S(xy − yx)), the form , is symplectic. Furthermore, by (2.12), Sxy(v0 + v1 ) = wx∗ wy v0 − wx wy∗ v1
(v0 ∈ V0 , v1 ∈ V1 ).
Thus S(xy − yx)(v0 + v1 ) = (wx∗ wy − wy∗ wx )v0 + (wy wx∗ − wx wy∗ )v1 . Hence, x, y = trD/R (wx∗ wy − wy∗ wx ) + trD/R (wy wx∗ − wx wy∗ ) = 2trD/R (wx∗ wy ) − 2trD/R (wx wy∗ ).
(2.12)
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But, by (2.9), (2.10) and (2.11), trD/R (wx wy∗ ) = trD/R (wy∗ wx ) = trD/R ((wy∗ wx ) )
= trD/R (wx∗ wy∗ ∗ ) = −trD/R (wx∗ wy ).
Thus x, y = 4trD/R (wx∗ wy )
(x, y ∈ g(V, τ )1 ).
(2.13)
As shown in [5], the right hand side of (2.13) defines a non- degenerate symplectic form on Hom(V0 , V1 ). Since (2.12) defines an R-linear bijection g(V, τ )1 x → wx ∈ Hom(V0 , V1 ),
(2.13)
the first part of the Lemma follows. The groups G0 , G1 defined in (b), are isomorphic to the isometry groups G(V0 , τ0 ), G(V1 , τ1 ), by restriction. As is well known, [5], these isometry groups form an irreducible dual pair of type I in the symplectic group on Hom(V0 , V1 ), equipped with the symplectic form defined by the right hand side of (2.13). Definition 2.3. An irreducible ordinary classical Lie supergroup is a pair (G, g) with g = g0 ⊕ g1 , where either G = GL(V )0 , g0 = End(V )0 , g1 = End(V )1 , as in (2.3),
(II)
G = G(V, τ )0 , g0 = g(V, τ )0 , g1 = g(V, τ )1 , as in (2.9).
(I)
or The pair (G, g) is a supergroup of type II in the case (II) and of type I in the case (I). The space V shall be called the defining module or the defining space for (G, g). If needed, we shall indicate this by writing G = G(V ) and g = g(V ). For the general theory of Lie superalgebras and Lie supergroup see, [8] and [9]. The following Proposition is easy to verify. Proposition 2.4. The restriction of the form , , (see (2.4’)), to g0 is symmetric, non-degenerate and G-invariant. Furthermore, the spaces g0 and g1 are orthogonal. If we identify g with the dual g∗ by y(x) = y, x
(x, y ∈ g)
(a)
then, for x ∈ g1 , the map g1 z → {x, z} ∈ g0
(b)
g0 y → [x, y] ∈ g1 .
(c)
is adjoint to the map In other words, {x, z}, y = z, [x, y]
(y ∈ g0 , x, z ∈ g1 ).
(d)
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Theorem 2.5. Let (G, g) be an irreducible ordinary classical Lie supergroup. Up to conjugation by G there is exactly one automorphism θ of g such that θ|g0 is a Cartan involution on g0 and θ|g1 is a positive compatible complex structure on g1 . The automorphism θ may be realized as follows. Let V = V0 ⊕ V1 be the defining module for (G, g). Then there is a positive definite hermitian form η on V such that V0 is orthogonal to V1 with respect to η, and if End(V ) x → x† ∈ End(V ) is the adjoint with respect to η, (η(xu, v) = η(u, x† v)), then −x† if x ∈ g0 , θ(x) = Sx† if x ∈ g1 . Moreover, if the Lie supergroup (G, g) is of type I and (D, ι) = (C, 1), then there is an element T ∈ G, unique up to conjugation, such that η( , ) = τ (T , ). Proof. The existence of θ is known (see, for example [3, 8.1, 10.2]). We shall verify the uniqueness. Since θ is an automorphism of g, the restriction θ|g1 is invertible in End(g1 ) and [θy, x] = θ[y, θ−1 x]
(y ∈ g0 , x ∈ g1 ).
In other words, ad(θy)|g1 = (θ|g1 )(ad(y)|g1 )(θ|g1 )−1
(y ∈ g0 ).
Suppose θ1 is another automorphism of g such that θ1 |g0 is a Cartan involution on g0 and θ1 |g1 is a positive compatible complex structure on g1 . Since the Cartan involution on g0 is unique up to conjugation by a element of G, [12, 2.3.2], we may assume that θ1 |g0 = θ|g0 . Then for y ∈ g0 , (θ|g1 )(ad(y)|g1 )(θ|g1 )−1 = ad(θy)|g1 = ad(θ1 y)|g1 = (θ1 |g1 )(ad(y)|g1 )(θ1 |g1 )−1 . Hence, (θ|g1 )−1 (θ1 |g1 ) ∈ Sp(g1 , , )ad(g0 ) . (Here, X Y is the centralizer of Y in X.) But we know from the structure of dual pairs, [7], that this last set is contained in (the centralizer of the identity component of Ad(G) in) Ad(G). For i = 1, 2, 3, ..., n, let (G(i) , g(i) ) be irreducible ordinary classical Lie supergroups, not necessarily of the same type, with the defining modules V (i) . The group G = G(1) × G(2) × G(3) × ... × G(n) and the Lie superalgebra g = g(1) ⊕ g(2) ⊕ g(3) ⊕ ... ⊕ g(n)
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act on the vector space V = V (1) ⊕ V (2) ⊕ V (3) ⊕ ... ⊕ V (n) componentwise. The resulting pair (G, g) shall be called the direct product of the ordinary classical Lie supergroups (G(i) , g(i) ), with the defining module V . An ordinary classical Lie supergroup (G, g) is a finite direct product of irreducible ordinary classical Lie supergroups, as defined above. Notice that the group G corresponds to the unique reductive dual pair obtained via the action of G on the symplectic space g1 . This correspondence is bijective.
3
The tangent space to the G-orbit through a point x ∈ g1
Let (G, g) be an irreducible ordinary classical Lie supergroup and let x ∈ g. Since the derivative of the adjoint action of the group G on g coincides with the adjoint action of the Lie algebra g0 on g, we may identify the tangent space to Gx at x with [g0 , x] = {[y, x]; y ∈ g0 }.
(3.1)
g1 = {z ∈ g1 ; {x, z} = 0}.
(3.2)
For x ∈ g1 let x
This is the anticommutant of x in g1 . For any subset W ⊆ g1 let W ⊥ = {y ∈ g1 ; y, z = 0 for all z ∈ W }.
(3.3)
Since our symplectic form , is non-degenerate, we have W ⊥⊥ = W,
(3.4)
if W is a vector subspace of g1 . Lemma 3.1. For any x ∈ g1 we have [g0 , x] = (x g1 )⊥ . Proof. Let z ∈ g1 . Then by (2.4.d), z ∈ [g0 , x]⊥ if and only if {x, z} = 0. Thus [g0 , x]⊥ = x g1 , and our claim follows from (3.4). Let θ be as in the Theorem 2.5. Then the formula (x, y) = − x, θy
(x, y ∈ g)
(3.5)
defines a symmetric positive definite form on g. In particular, for any x ∈ g1 , the orthogonal complement to [g0 , x] in g1 , with respect to the form (3.5), is equal to θ(x g1 ). The form (3.5) restricts to any subspace of g, and induces a positive definite form on the quotient of g by any subspace. Let U, V be two vector spaces, over the reals, of the same dimension. Suppose U, V are subspaces or quotients of g (one could be a subspace and the other one the quotient).
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Let L : U → V be a linear map. We define the absolute value of the determinant of L, |det(L)|, to be the absolute value of the determinant of the matrix of L with respect to any orthonormal basis of U, V, with respect to the form induced by (3.5). Fix x ∈ g1 . The derivative of the map G/Gx gGx → gx ∈ g1
(3.6)
at x may be identified with the following linear map g0 /gx0 y + gx0 → [y, x] ∈ [g0 , x].
(3.7)
Denote by J(x)
(x ∈ g1 )
the absolute value of the determinant of the map (3.7). We shall give a formula for the function J(x), for x ∈ g1 semisimple, in Corollary 6.9.
4
A G-equivariant localization in g1
In this section we state several theorems which shall be verified later. As in the previous section, let (G, g) be an irreducible ordinary classical Lie supergroup and let x ∈ g1 . The element x ∈ g1 is called semi-simple (or nilpotent) if and only if x is semi-simple (or nilpotent) as an endomorphism of V . Theorem 4.1. Let x ∈ g1 and let x = xs + xn be the Jordan decomposition of x, as an element of End(V ). (Here xs stands for the semisimple part of x, and xn for the nilpotent part of x.) Then xs and xn belong to g1 . Furthermore, an element y ∈ g1 anti-commutes with x if and only if y anti-commutes with xs and xn . Theorem 4.2. For any x ∈ g1 the semisimple part xs belongs to Cl(Gx), the closure of the orbit Gx. Theorem 4.3. For any x ∈ g1 , x is semisimple if and only if the orbit Gx is closed. 2
Notice that if x ∈ g1 , then x2 = 12 {x, x} ∈ g0 . Let Gx ⊆ G denote the centralizer of 2 2 2 x2 , and let gx , gx0 , gx1 denote the centralizer of x2 in g, g0 , g1 respectively. Theorem 4.4. Let x ∈ g1 be semisimple. 2 2 (a) Suppose ker(x) = 0. Then (Gx , gx ) is the direct product of the irreducible ordinary classical Lie supergroups, with the corresponding dual pairs isomorphic either to (Un , Un ) or to (GLn (D), GLn (D)), where the division algebra D may be different than the division algebra over which the defining module V for the supergroup (G, g) was defined. 2 The restriction of the symplectic form , to gx1 is non-degenerate and 2
gx1 = x g1 ⊕ gx1
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is a complete polarization. (Here gx1 = {y ∈ g1 ; xy = yx}.) (b) Let V 0 = ker(x) and let V + = xV . Then V =V0⊕V+ is a direct sum (orthogonal in the type I case) decomposition into graded non-zero subspaces preserved by x. Moreover, 2
Gx = G(V 0 ) × G(V + )x
2
and 2
2
g1 x = g1 (V 0 ) ⊕ g1 (V + )x , where the sum is orthogonal, with respect to the symplectic form , , and g1 (V 0 ) = 0 unless V 0 ∩ V0 = 0 and V 0 ∩ V1 = 0. Moreover, the double anticommutant of x in g1 coincides with the double commutant of x in g1 (V + ): (x g1 )
g1 = g1 (V + )(g1 (V
+ )x )
. x
(c) The maximal possible dimension of the real vector space ( g1 ) g1 is equal to the minimum of the rank of G(V0 ) and the rank of G(V1 ), viewed as real reductive Lie groups. For the x x such that the dimension of ( g1 ) g1 is maximal, we have V 0 ⊆ V0 or V 0 ⊆ V1 , so that g1 (V 0 ) = 0, and (gx ) (x g1 ) g1 = g1 1 . x
An explicit description of the double anticommutant ( g1 ) g1 will be given in the proof of the theorem (see (13.13.1), (13.22.1), (13.31.1), (13.42.1), (13.47.1), (13.47.2), and (13.53.1)). Theorem 4.5. Let x ∈ g1 be semisimple. Then gx1 has a basis for the Gx -invariant neighborhoods of x consisting of admissible slices Ux through x, such that for i = 0, 1, the map 2 Ux y → y 2 |Vi ∈ g0 (Vi )x is an (injective) immersion, (see [10] Vol 1, for the definition of an immersion.) Theorem 4.6. [3] The set of nilpotent G-orbits in g1 is finite. Theorem 4.7. Let x ∈ g1 be nilpotent and let W ⊆ g1 be a subspace such that g1 = [g0 , x] ⊕ W. Then the affine space x + W ⊆ g1 has a basis for the neighborhoods of x consisting of weakly admissible slices through x. The following theorem has a substantial overlap with the Proposition 8.2 in [5]
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Theorem 4.8. The map g1 ⊇ Gx → Gx2 ⊆ g0 is injective on the set of semisimple orbits. (Here, in order to simplify the notation we write Gx rather than Ad(G)x, and similarly for Gx2 .)
5
The G-orbits in g1
We retain the notation of the previous section. An element x ∈ g1 , or the pair (x, V ), is called decomposable if and only if there are two non-zero Z/2Z-graded subspaces V , V ⊆ V , (which are orthogonal if (G, g) is of type I), preserved by x and such that V = V ⊕ V . In this case we say that (x, V ) is the direct sum of the elements (x|V , V ) and (x|V , V ). The element (x, V ) is called indecomposable if and only if (x, V ) is not decomposable. Theorem 5.1. For any x ∈ g1 , (x, V ) is the direct sum of indecomposable elements. Proof (a reduction to the case when x is semisimple). Let x = xs + xn be the Jordan decomposition of x. Then, as we know from Theorem 4.1, xn ∈ g1 . Suppose xn = 0. There is a decomposition V = V 1 ⊕ V 2 ⊕ V 3 ⊕ ... into indecomposables with respect to xn , [3, sections 5 and 6]. Since x commutes with xn , each V j is x-invariant, and indecomposable with respect to x (because xn is a polynomial of x). Thus we may assume that x is semisimple. We shall consider this case and complete the argument in sections 8 and 9. Let (G, g), (G , g ) be two irreducible ordinary classical Lie supergroups with the defining spaces V , V respectively. We’ll say that two elements x ∈ g1 , x ∈ g1 are similar if and only if the supergroups (G, g), (G , g ) are of the same type and there is a Z/2Z-graded linear bijection φ : V → V (an isometry in the type I case) such that x = φ−1 x φ. In particular if V = V and (G, g) = (G , g ) then x is similar to x if and only if x and x are in the same G-orbit. In that case we shall write x ≈ x . Theorem 5.2. Let (G, g) be of type I. The following is a complete list of all non-zero semisimple indecomposable elements (x, V ), x ∈ g1 , up to similarity. In each case we indicate which elements of the list are similar, describe an element g ∈ G which provides the similarity, and list the eigenvalues of x.
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V0 = Dv0 ⊕ Dv0 , V1 = Dv1 ⊕ Dv1 ;
τ (v0 , v0 ) = τ (v0 , v0 ) = τ (v1 , v1 ) = τ (v1 , v1 ) = 0, τ (v0 , v0 ) = τ (v1 , v1 ) = 1;
if ι = 1 then let ξ ∈ D \ 0, if ι = 1 then let ξ ∈ C ⊆ D and ξ 2 ∈ / iR; x = x(ξ) : v0 → ξv1 , v1 → ξv0 , v0 → −ι(ξ)v1 , v1 → ι(ξ)v0 ; if D = R then x(ξ) ≈ x(−ξ) has eigenvalues ξ, iξ, −ξ, −iξ; if D = C and ι = 1 then x(ξ) ≈ x(iξ) ≈ x(−ξ) ≈ x(−iξ) has eigenvalues ξ, iξ, −ξ, −iξ; if D = C and ι = 1 then x(ξ) ≈ x(−ξ) has eigenvalues ξ, iι(ξ), −ξ, −iι(ξ);
(a)
if D = H then x(ξ) ≈ x(−ξ) ≈ x(ι(ξ)) ≈ x(−ι(ξ)) has eigenvalues ξ, ι(ξ), −ξ, −ι(ξ), iξ, iι(ξ), −iξ, −iι(ξ); for all D, gx(ξ)g −1 = x(−ξ) if g : v0 → −v0 , v1 → v1 , v0 → −v0 , v1 → v1 ; for D = C and ι = 1, gx(ξ)g −1 = x(iξ) if
g : v0 → −iv0 , v1 → −v1 , v0 → iv0 , v1 → v1 ;
for D = H, gx(ξ)g −1 = x(ι(ξ)) if
g : v0 → jv0 , v1 → jv1 , v0 → jv0 , v1 → jv1 ;
V0 = Dv0 , V1 = Dv1 , C ⊆ D, ι = 1; τ (v0 , v0 ) = = ±1, τ (v1 , v1 ) = δi = ±i; ξ 2 ∈ iR \ 0, sgn(im(ξ 2 )) = − δ; x = x(ξ) : v0 → ξv1 , v1 → ξv0 ;
(b)
x(ξ) ≈ x(−ξ) has eigenvalues ξ, −ξ; gx(ξ)g −1 = x(−ξ) if g : v0 → −v0 , v1 → v1 ;
V0 = Rv0 ⊕ Rv0 , V1 = Rv1 ⊕ Rv1 , D = R;
τ (v0 , v0 ) = τ (v0 , v0 ) = = ±1, τ (v1 , v1 ) = τ (v1 , v1 ) = 0, τ (v0 , v0 ) = 0, τ (v1 , v1 ) = 1; ξ ∈ R \ 0; x = x(ξ) : v0 → ξ(v1 − v1 ), v1 → ξ(v0 − v0 ),
v0 → ξ(v1 + v1 ), v1 → ξ(v0 + v0 );
x(ξ) ≈ x(−ξ) has eigenvalues ξ(1 − i), ξ(1 + i), −ξ(1 − i), −ξ(1 + i); gx(ξ)g −1 = x(−ξ) if g : v0 → −v0 , v1 → v1 , v0 → −v0 , v1 → v1 ;
(c)
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V0 = (Ru0 ⊕ Rv0 ) ⊕ (Ru0 ⊕ Rv0 ),
V1 = (Ru1 ⊕ Rv1 ) ⊕ (Ru1 ⊕ Rv1 ), D = R,
the spaces in parenthesis are isotropic, and τ (u0 , u0 ) = τ (v0 , v0 ) = τ (u1 , u1 ) = τ (v1 , v1 ) = 1;
ξ, η ∈ R, ξ 2 = η 2 , ξη = 0; x = x(ξ, η) :
u0 → ξu1 + ηv1 , u1 → ξu0 + ηv0 , v0 → −ηu1 + ξv1 , v1 → −ηu0 + ξv0 , u0 → −ξu1 + ηv1 , u1 → ξu0 − ηv0 , v0 → −ηu1 − ξv1 , v1 → ηu0 + ξv0 ;
(d)
x(ξ, η) ≈ x(−ξ, η) ≈ x(ξ, −η) ≈ x(−ξ, −η) ≈ x(η, ξ) ≈ x(−η, ξ) ≈ x(η, −ξ) ≈ x(−η, −ξ) has eigenvalues ξ + iη, ξ − iη, −ξ + iη, −ξ − iη, η + iξ, −η + iξ, η − iξ, −η − iξ; gx(ξ, η)g −1 = x(−ξ, η) if g : u0 → −u0 , v0 → v0 , u0 → −u0 , v0 → v0 , u1 → u1 , v1 → −v1 , u1 → u1 , v1 → −v1 ;
gx(ξ, η)g −1 = x(η, ξ) if g : u0 → u0 , v0 → v0 , u0 → u0 , v0 → v0 , u1 → v1 , v1 → −u1 , u1 → −v1 , v1 → u1 ;
Theorem 5.3. Let (G, g) be of type II. The following is a complete list of all non-zero semisimple indecomposable elements (x, V ), x ∈ g1 , up to similarity. In each case we indicate which elements of the list are similar, describe an element g ∈ G which provides the similarity, and list the eigenvalues of x. V0 = Dv0 , V1 = Dv1 ; ξ ∈ D \ 0; x = x(ξ) : v0 → ξv1 , v1 → ξv0 ; if D = H then x(ξ) ≈ x(−ξ) has eigenvalues ξ, −ξ; if D = H then x(ξ) ≈ x(−ξ) ≈ x(ι(ξ)) ≈ x(−ι(ξ))
(a)
has eigenvalues ξ, −ξ, ι(ξ), −ι(ξ); gx(ξ)g −1 = x(−ξ) if g : v0 → −v0 , v1 → v1 ; gx(ξ)g −1 = x(ι(ξ)) if D = H and g : v0 → jv0 , v1 → jv1 ; V0 = Rv0 , V1 = Rv1 ; ξ ∈ R \ 0; x = x(ξ) : v0 → ξv1 , v1 → −ξv0 ; x(ξ) ≈ x(−ξ) has eigenvalues ± iξ; gx(ξ)g −1 = x(−ξ) if g : v0 → −v0 , v1 → v1 ;
(a’)
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V0 = Ru0 ⊕ Rv0 , V1 = Ru1 ⊕ Rv1 , D = R; ξ, η ∈ R \ 0; x = x(ξ, η) : u0 → ξu1 + ηv1 , u1 → ξu0 + ηv0 , v0 → −ηu1 + ξv1 , v1 → −ηu0 + ξv0 ; x(ξ, η) ≈ x(−ξ, η) ≈ x(ξ, −η) ≈ x(−ξ, −η)
(b)
has eigenvalues ξ + iη, ξ − iη, −ξ + iη, −ξ − iη; gx(ξ, η)g −1 = x(−ξ, η) if g : u0 → −u0 , v0 → v0 , u1 → u1 , v1 → −v1 ; gx(ξ, η)g −1 = x(ξ, −η) if g : u0 → u0 , v0 → −v0 , u1 → u1 , v1 → −v1 . Proof (of Theorem 4.1). Let x ∈ g1 and let x = xs +xn be the Jordan decomposition of x, as an element End(V ). Suppose (G, g) is of type II. Notice that Sxs S −1 is semisimple and Sxn S −1 is nilpotent, and that these elements commute. Moreover, Sxs S −1 + Sxn S −1 = SxS −1 = −x = −xs − xn . Thus the uniqueness of the Jordan decomposition in End(V ) implies that Sxs S −1 = −xs and Sxn S −1 = −xn . In other words, xs , xn ∈ g1 . Suppose (G, g) is of type I. Then, as shown above, xs , xn ∈ End(V )1 . Consider the map End(V )1 y → y ∈ End(V )1 , τ (yu, v) = τ (u, y v)
(u, v ∈ V ).
Then y is semisimple if and only if y is semisimple, and y is nilpotent if and only if y is nilpotent. In particular, xs is semisimple and xn is nilpotent. The Theorem 5.1 (for semisimple elements) and Theorem 5.2 imply that Sxs is semisimple. It is easy to check that Sxn is nilpotent and that Sxs Sxn = Sxn Sxs . Since x ∈ g1 , we have, by the definition (2.8), x = Sx = Sxs + Sxn . Thus the uniqueness of the Jordan decomposition in End(V ) implies xs = Sxs and xn = Sxn . Hence, xs , xn ∈ g1 . Let y ∈ g1 . If y anticommutes with xs and xn then clearly y anticommutes with x = xs + xn . Conversely, suppose yx = −xy. Then, with S as in (2.2), the following computation holds in End(V ): x(Sy) = −Sxy = (Sy)x. Thus x commutes with Sy. Therefore xs and xn commute with Sy. Hence, for z = xs or xn , zy = zS 2 y = −Sz(Sy) = −S(Sy)z = −yz. In other words, xs and xn anti-commute with y.
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Let
k(k−1)/2
δ(k) = (−1)
=
1 if k ∈ 4Z or 4Z + 1, −1 if k ∈ 4Z + 2 or 4Z + 3.
463
(5.1’)
Theorem 5.4. [3] Let (G, g) be of type I. The following is a complete list of all non-zero nilpotent indecomposable elements (x, V ), x ∈ g1 , up to similarity. m ∈ 4Z; m Dvk , veven ∈ V0 , vodd ∈ V1 ; V = k=0 k
vk = x v0 = 0, 0 ≤ k ≤ m, xvm = 0; τ (vk , vl ) = 0 if l = m − k, τ (vk , vm−k ) = δ(k)δ(
(a) m )sgn(τ0 ), 2
where sgn(τ0 ) = 1 if D = C and ι = 1; m ∈ 4Z, D = R, ι = 1; m+1 Dvk , veven ∈ V0 , vodd ∈ V1 ; V = k=1
vk+1 = xk v1 = 0, 0 ≤ k ≤ m, xvm+1 = 0;
(b)
τ (vk , vl ) = 0 if l = m + 2 − k, τ (vk , vm+2−k ) = δ(k)τ (v1 , vm+1 ), m τ (v1 , vm+1 ) = i sgn(−iτ1 )δ(1 + ) if D = C; 2 τ (v1 , vm+1 ) = j if D = H; m ∈ 4Z, D = H, ι = 1; m+1 V = (Dvk ⊕ Dvk ), veven , veven ∈ V0 , vodd , vodd ∈ V1 ; k=1 = xk v1 = 0, 0 ≤ k ≤ m, xvm+1 = 0, xvm+1 = 0; vk+1 = xk v1 = 0, vk+1
(c)
τ (vk , vl ) = τ (vk , vl ) = 0, 1 ≤ k, l ≤ m + 1,
τ (vk , vl ) = τ (vk , vl ) = 0, l = m + 2 − k,
τ (vk , vm+2−k ) = −τ (vk , vm+2−k ) = δ(k), 1 ≤ k ≤ m + 1;
m ∈ 2Z \ 4Z, D = R, ι = 1; m Dvk , veven ∈ V0 , vodd ∈ V1 ; V = k=0 k
vk = x v0 = 0, 0 ≤ k ≤ m, xvm = 0; τ (vk , vl ) = 0 if l = m − k, τ (vk , vm−k ) = δ(k)isgn(−iτ1 ), (here − iτ1 is hermitian);
(d)
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m ∈ 2Z \ 4Z; m+1 V = Dvk , veven ∈ V0 , vodd ∈ V1 ; k=1
vk+1 = xk v1 = 0, 0 ≤ k ≤ m, xvm+1 = 0;
(e)
τ (vk , vl ) = 0 if l = m + 2 − k, τ (vk , vm+2−k ) = δ(k)τ (v1 , vm+1 ), m τ (v1 , vm+1 ) = δ(1 + )sgn(τ0 ); 2 m ∈ 2Z \ 4Z, D = H, ι = 1; m V = (Dvk ⊕ Dvk ), veven , veven ∈ V0 , vodd , vodd ∈ V1 ; k=0 k
= 0; vk = x v0 = 0, vk = xk v0 = 0, 0 ≤ k ≤ m, xvm = 0, xvm
τ (vk , vl ) = τ (vk , vl ) =
τ (vk , vl ) τ (vk , vl )
τ (vk , vm−k )
=
(f)
= 0, 0 ≤ k, l ≤ m, = 0, l = m − k,
−τ (vk , vm−k )
= δ(k), 0 ≤ k ≤ m;
m ∈ 2Z + 1; m V = (Dvk ⊕ Dvk+1 ), veven , veven ∈ V0 , vodd , vodd ∈ V1 ; k=0 k
vk = x v0 = 0, vk+1 = xk v1 = 0, 0 ≤ k ≤ m, xvm = 0, xvm+1 = 0;
(g)
, vl+1 ) = 0, 0 ≤ k, l ≤ m, τ (vk , vl ) = τ (vk+1
τ (vk , vl+1 ) = τ (vk+1 , vl ) = 0, l = m − k,
) = δ(k)(−1)k , τ (vk+1 , vm−k ) = δ(k)δ(m), 0 ≤ k ≤ m; τ (vk , vm+1−k
The following theorem is well known, and goes back to Jordan. See [3] for details. Theorem 5.5. Let (G, g) be of type II. The following is a complete list of all non-zero nilpotent indecomposable elements (x, V ), x ∈ g1 , up to similarity. V =
m
Dvk , veven ∈ V0 , vodd ∈ V1 ;
k=0 k
(a)
vk = x v0 = 0, 0 ≤ k ≤ m, xvm = 0; V =
m+1
Dvk , veven ∈ V0 , vodd ∈ V1 ;
k=1
(b)
vk+1 = x v1 = 0, 0 ≤ k ≤ m, xvm+1 = 0; k
For a nilpotent element x ∈ g1 the height of x, or the height of (x, V ), is the integer m ≥ 0 such that xm = 0 and xm+1 = 0. In particular x = 0 if and only if the height of x is 0. The pair (x, V ) is called uniform if ker(xm ) = im(x) (= x(V )). These notions are adopted from [1], and have been used in [3].
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Let x ∈ g1 and let x = xs + xn be the Jordan decomposition of x. Let V = V (1) ⊕ V (2) ⊕ ...
(5.2)
be the decomposition of V into a direct (and orthogonal in the type I case) sum, such that each (xn , V (j) ) is uniform (see [3]). Then each V (j) is preserved by xs . As one can see from the proof of Theorems 5.12 and 6.1 in [3], there is a graded xs - invariant subspace F (j) such that V (j) = F (j) ⊕ xn F (j) ⊕ x2n F (j) ⊕ ..., where, in the type I case, xkn F (j) ⊥ xln F (j) , for k + l ≤ mj − 1, and mj is the height of (xn , V (j) ). Since the xn and xs commute, the action of xs on V (j) is determined by the action on the F (j) . This space is equipped with the form j τmj ,j (u, v) = τ (u, xm n v)
(u, v ∈ F (j) ),
(5.3)
in the type I case. As an endomorphism of F (j) , xs is of degree one (i.e. the restriction of xs to F (j) is in End(F (j) )1 ) and (u, v ∈ F (j) ),
τmj ,j (xs u, v) = τmj ,j (u, Sxs v)(−1)mj
(5.4)
If mj ∈ 4Z, then τmj ,j = τmj ,j |F (j) ⊕ τmj ,j |F (j) 0
(5.5)
1
with τmj ,j |F (j) hermitian and τmj ,j |F (j) skew-hermitian. 0 1 If mj ∈ 2Z\4Z, then (5.5) holds with τmj ,j |F (j) skew-hermitian and τmj ,j |F (j) hermitian. 0
1
Thus for mj even we know how to decompose (xs , F (j) ) into indecomposables. Recall the function δ, (5.1’). For mj ∈ 2Z + 1 τmj ,j (u, v) = −δ(mj )ι(τmj ,j (v, u))
(u, v ∈ F (j) ),
τmj ,j |F (j) = 0, and τmj ,j |F (j) = 0. 0
(5.6)
1
Let us write m = mj , τm = τmj ,j and F = F (j) in the last case. We may rewrite (5.6) and (5.4) as τm (u, v) = −δ(m)ι(τm (v, u)), (5.7) F0 , F1 are isotropic subspaces of F, τm (xs u, v) = −τm (u, Sxs v)
(u, v ∈ F ).
As an easy consequence of Theorem 5.5 we deduce the following fact. Theorem 5.6. Suppose (xs , F ), described in (5.7), is indecomposable and non-zero. Then ι = 1 and, up to similarity, F = Dv0 ⊕ Dv1
(v0 ∈ F0 , v1 ∈ F1 ),
x s : v 0 → a 1 v 1 , v 1 → a0 v 0 , where a0 = δ(m)ι(a0 ) and a1 = −δ(m)ι(a1 ) if D = R.
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A Cartan subspace, the Weyl group and an integration formula
Definition 6.1. An element x ∈ g1 is regular if and only if the G-orbit through x is of maximal possible dimension. A Cartan subspace h1 ⊆ g1 is the double anticommutant x h1 = ( g1 ) g1 of a regular semisimple element of x ∈ g1 . The Weyl group W (G, h1 ) is the quotient of the stabilizer of h1 in G by the subgroup which acts trivially on h1 . (We shall identify the Weyl group W (G, h1 ) with its image in GL(h1 ).) Proposition 6.2. The following is a complete list of the Cartan subspaces h1 ⊆ g1 and the Weyl groups W (G, h1 ), up to conjugation by an element of G, such that h1 contains a non-zero regular semisimple indecomposable element: Type I V0 = Dv0 ⊕ Dv0 , V1 = Dv1 ⊕ Dv1 ;
τ (v0 , v0 ) = τ (v0 , v0 ) = τ (v1 , v1 ) = τ (v1 , v1 ) = 0, τ (v0 , v0 ) = τ (v1 , v1 ) = 1; x(a) : v0 → av1 , v1 → av0 , v0 → −ι(a)v1 , v1 → ι(a)v0 , a ∈ D; if D = R then h1 = {x(a); a ∈ R}, |W (G, h1 )| = 2, the non-trivial element of W (G, h1 ) maps x(a) to x(−a); if D = C and ι = 1 then h1 = {x(a); a ∈ C}, |W (G, h1 )| = 4,
(a)
the non-trivial elements of W (G, h1 ) map x(a) to x(−a), x(ia), x(−ia); if D = C and ι = 1 then h1 = {x(a); a ∈ C}, |W (G, h1 )| = 2, the non-trivial element of W (G, h1 ) maps x(a) to x(−a); if D = H then h1 = {x(a); a ∈ C}, |W (G, h1 )| = 4, the non-trivial elements of W (G, h1 ) map x(a) to x(−a), x(ι(a)), x(−ι(a)); V0 = Dv0 , V1 = Dv1 , C ⊆ D, ι = 1; τ (v0 , v0 ) = = ±1, τ (v1 , v1 ) = δi = ±i; x(a) : v0 → av1 , v1 → av0 , a ∈ C;
(b)
h1 = {x(a); a = − δiι(a) ∈ C}, |W (G, h1 )| = 2; the non-trivial element of W (G, h1 ) maps x(a) to x(−a); V0 = Rv0 ⊕ Rv0 , V1 = Rv1 ⊕ Rv1 , D = R;
τ (v0 , v0 ) = τ (v0 , v0 ) = = ±1, τ (v1 , v1 ) = τ (v1 , v1 ) = 0, τ (v0 , v0 ) = 0, τ (v1 , v1 ) = 1;
x = x(a) : v0 → a(v1 − v1 ), v1 → a(v0 − v0 ),
v0 → a(v1 + v1 ), v1 → a(v0 + v0 ), a ∈ R; h1 = {x(a); a ∈ R}, |W (G, h1 )| = 2;
the non-trivial element of W (G, h1 ) maps x(a) to x(−a);
(c)
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V0 = (Ru0 ⊕ Rv0 ) ⊕ (Ru0 ⊕ Rv0 ),
V1 = (Ru1 ⊕ Rv1 ) ⊕ (Ru1 ⊕ Rv1 ), D = R,
the spaces in parenthesis are isotropic, and τ (u0 , u0 ) = τ (v0 , v0 ) = τ (u1 , u1 ) = τ (v1 , v1 ) = 1; x = x(a, b) : u0 → au1 + bv1 , u1 → au0 + bv0 , v0 → −bu1 + av1 , v1 → −bu0 + av0 ,
(d)
u0 → −au1 + bv1 , u1 → au0 − bv0 ,
v0 → −bu1 − av1 , v1 → bu0 + av0 , a, b ∈ R; h1 = {x(a, b); a, b ∈ R}, |W (G, h1 )| = 8; the non-trivial elements of W (G, h1 ) map x(a, b) to
x(−a, b), x(a, −b), x(−a, −b), x(b, a), x(−b, a), x(b, −a), x(−b, −a); Type II V0 = Dv0 , V1 = Dv1 ; x(a) : v0 → av1 , v1 → av0 , a ∈ D; if D = R or C, then h1 = {x(a); a ∈ D}, |W (G, h1 )| = 2; the non-trivial element of W (G, h1 ) maps x(a) to x(−a);
(e)
if D = H, then h1 = {x(a); a ∈ C}, |W (G, h1 )| = 4; the non-trivial elements of W (G, h1 ) map x(a) to x(−a), x(ι(a)), x(−ι(a));
V0 = Rv0 , V1 = Rv1 ; x(a) : v0 → av1 , v1 → −av0 ; h1 = {x(a); a ∈ R}, |W (G, h1 )| = 2;
(f)
the non-trivial element of W (G, h1 ) maps x(a) to x(−a);
V0 = Ru0 ⊕ Rv0 , V1 = Ru1 ⊕ Rv1 , D = R; x = x(a, b) : u0 → au1 + bv1 , u1 → au0 + bv0 , v0 → −bu1 + av1 , v1 → −bu0 + av0 ; h1 = {x(a, b); a, b ∈ R}, |W (G, h1 )| = 4;
(g)
the non-trivial elements of W (G, h1 ) map x(a, b) to x(−a, b), x(a, −b), x(−a, −b); Proof. This Proposition is a straightforward consequence of Theorems 5.2, 5.3 and the proof of Theorem 4.4, (see the section 13).
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In general, a Cartan subspace h1 ⊆ g1 induces a direct sum decomposition V = V 0 ⊕V 1 ⊕ V 2 ⊕ ... ⊕ V i1 ⊕V i1 +1 ⊕ V i1 +2 ⊕ ... ⊕ V i2
(6.1)
... ⊕V ik−1 +1 ⊕ V ik−1 +2 ⊕ ... ⊕ V ik , orthogonal in the type I case, into graded subspaces preserved by h1 , such that (a) h1 (V 0 ) = 0, (b) for each 0 ≤ i ≤ ik there is x ∈ h1 such that (x, V i ) is indecomposable, (c) there is x ∈ h1 such that the elements (x, V j ), (x, V k )
(6.2)
are indecomposable and similar if and only if there is 1 ≤ l ≤ k − 1, with il < j ≤ il+1 and il < k ≤ il+1 . Then the Weyl group W (G, h1 ) = (Si1 (W (G(V 1 ), h1 (V 1 )) × W (G(V 2 ), h1 (V 2 )) × ... × W (G(V i1 ), h1 (V i1 )))) ×(Si2 −i1 (W (G(V i1 +1 ), h1 (V i1 +1 )) × W (G(V i1 +2 ), h1 (V i1 +2 )) × ... × W (G(V i2 ), h1 (V i2 ))))
(6.3)
... ×(Sik −ik−1 (W (G(V ik−1 +1 ), h1 (V ik−1 +1 )) × W (G(V ik−1 +2 ), h1 (V ik−1 +2 )) × ... × W (G(V ik ), h1 (V ik )))), with the action on h1 compatible with the decomposition (6.3). (Here Sm stands for the group of all permutations of m objects.) Proposition 6.2, together with (6.1), imply that there are finitely many conjugacy classes of Cartan subspaces in g1 and that any Cartan subspace consists of elements which commute in End(V ). Also, it is easy to see from Definition 6.1 that each semisimple element of g1 belongs to a Cartan subspace of g1 . Also, the set or regular elements coincides with the set where certain determinants don’t vanish, hence it is open and dense. We record these facts in the following proposition. Proposition 6.3. There are finitely many G-conjugacy classes of Cartan subspaces in g1 . Every semisimple element of g1 belongs to the G-orbit through an element of a Cartan subspace. The set of regular semisimple elements is dense in g1 . Any two elements of a Cartan subspace h1 ⊆ g1 commute as endomorphisms of V . The following lemma shall be verified at the end of section 13. Lemma 6.4. For any two commuting regular semisimple elements x, y ∈ g1 we have 2
2
2
2
(a) gx1 = gy1 ; (b) gx1 = gy1 ; (c) x g1 = y g1 ; (d) gx0 = gy0 ; (e) gx0 = gy0 .
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Let h1 ⊆ g1 be a Cartan subspace and let hreg 1 ⊆ h1 be the subset of regular elements. 2 Denote by h1 = {h1 , h1 } the linear span of the elements {x, y}, where x, y ∈ h1 . Define y y h2 h1 y g1 = g1 , gh11 = g1 , gi 1 = gi , (i = 0, 1). (6.4) y∈h1
y∈h1
y∈h21
Lemma 6.5. For any x ∈ hreg 1 , (a)
h1
h2
g1 = x g1 , (b) gh11 = gx1 = h1 , (c) gi 1 = gxi
2
(i = 0, 1).
Proof. In the definition (6.4), it suffices to take the finite intersection over the y’s which form a basis of the corresponding linear space. For the equations (a), (b) we may choose a basis of h1 consisting of regular elements. Then the equalities follow from Lemma 6.4. Since the elements of h1 commute, the space h21 is spanned by the squares y 2 , y ∈ h1 . Thus we may choose a basis y12 , y22 ,..., of h21 such that each yi ∈ hreg 1 . Then the equality (c) also follows from Lemma 6.4. 0 + Proposition 6.6. Let x ∈ hreg = xV , as in Theorem 4.4(b). 1 . Set V = ker(x) and V Then
h1 = g1 (V + )h1 = g1 (V + )x ; h1
g1 = h21
h1
g1 (V ); h21
g1 = g1 (V + ) h2 g01
(a)
+
(b) x2
= g1 (V + ) ;
(c)
= g0 (V 0 ) ⊕ h0 ,
(d) + h21
+
where h0 = g0 (V )
is a Cartan subalgebra of g0 (V ),
and the sum is orthogonal; gh01
= g0 (V 0 ) ⊕ h21 ;
(e)
the restriction of the form , to h0 is non-degenerate and h0 =
Sh21
⊕
h21
(f)
is a complete polarization.
Proof. Parts (a), (b), (c) are immediate from Theorem 4.4 and Lemma 6.5. Similarly we have the orthogonal decomposition in (d). A straightforward computation based on Theorem 4.4(a) shows that 2
dim g0 (V + )x = 2 dim g1 (V + )x . By Theorem 4.4(a) and 4.4(c), dim g1 (V + )x = min{rank g0 (V0+ ), rank g0 (V1+ )}. Since g0 (V + ) = g0 (V0+ ) ⊕ g0 (V1+ ), we see that the restriction of x2 to V + is a regular element of g0 (V + ). Hence, h0 is a Cartan subalgebra of g0 (V + ) and (d) follows.
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For (e) we may assume that V 0 = 0 and that (x, V ) is indecomposable. Then the equality follows from Proposition 6.2 via a case by case analysis. Similarly we check (f). Lemma 6.7. For any x ∈ hreg the following map is a linear bijection 1 (Sgh01 )⊥ y → [y, x] ∈ (h1 g1 )⊥ .
(a)
The map (a) intertwines the adjoint action of h21 on both spaces. The following are direct sum decompositions into the trivial and the non-trivial h21 -components: (Sgh01 )⊥ = Sh21 ⊕ g0 (V 0 )⊥ ∩ h⊥ 0, h2
(h1 g1 )⊥ = h1 g1 ⊕ (g11 )⊥ .
(b)
(c)
The map (a) restricts to bijections Sh21 y → [y, x] ∈ h1 g1 ,
(d) h2
1 ⊥ g0 (V 0 )⊥ ∩ h⊥ 0 y → [y, x] ∈ (g1 ) .
(e)
Proof. We see from Proposition 6.6(e) that (Sgh01 )⊥ = g0 (V 0 )⊥ ∩ (Sh21 )⊥ . The h21 -trivial component of this space is g0 (V 0 )⊥ ∩ (Sh21 )⊥ ∩ h0 = Sh21 . This verifies (b). We see from (b), and from Proposition 6.6(e), that g0 = (Sgh01 )⊥ ⊕ gh01 . Since h1 g1 = x g1 , Lemma 3.1 implies that the map (a) is well defined and surjective. Let z ∈ h1 and let x, y be as in (a). Then [z 2 , [y, x]] = [[z 2 , y], x] + [y, [z 2 , x]] = [[z 2 , y], x], because [z 2 , x] = 0. Hence, the map (a) is h21 -intertwining, and the proof of (a) is complete. Part (c) is clear from Theorem 4.4. Parts (d) and (e) follow from the intertwining property of the map (a). The derivative of the map h1 x → x2 ∈ h21
(6.5)
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at x ∈ h1 , coincides with the following linear map h1 y → {y, x} ∈ h21 ,
(6.6)
which, by Proposition 2.4, is adjoint to the map Sh21 y → [x, y] ∈ Sh1 .
(6.7)
(Notice that the range of the map (6.7) is contained in Sh1 . Indeed, let x, z ∈ hreg 1 . Then 2 2 2 2 2 2 [Sz , x] ∈ [g0 , g1 ] ⊆ g1 = Sg1 and [Sz , x] = S(z x+xz ) = S2z x. Clearly z x commutes with x. Thus [Sz 2 , x] ∈ Sgx1 . Furthermore z 2 x|V 0 = 0. Therefore [Sz 2 , x] ∈ Sh1 .) Hence, |det(h1 y → {y, x} ∈ h21 )| = |det(Sh21 y → [y, x] ∈ Sh1 )|.
(6.8)
Define polynomials Dj (x), x ∈ g1 , by 2
det(tI − ad(x )|g1 ) =
R
(x ∈ g1 ),
tj Dj (x)
(6.9)
j=r
where R = dim(g1 ), DR = 1, and r ≥ 0 is the smallest integer such that Dr is not identically equal zero. Lemma 6.8. For x ∈ h1 we have |Dr (x)| = |det(ad(x2 )|
h2
(g1 1 )⊥
)| h2
1 ⊥ 2 = |det(g0 (V 0 )⊥ ∩ h⊥ 0 y → [y, x] ∈ (g1 ) )|
= |det(ad(x2 )|g0 (V 0 )⊥ ∩h⊥0 )|.
Proof. The first equality is clear from (6.9). The map ad(x2 )|
h2 (g1 1 )⊥
h2
h2
: (g1 1 )⊥ → (g1 1 )⊥
coincides with (−1 times) the composition of the following two maps: h2
(g1 1 )⊥ y → {y, x} ∈ g0 (V 0 )⊥ ∩ h⊥ 0, and
h2
1 ⊥ g0 (V 0 )⊥ ∩ h⊥ 0 y → [y, x] ∈ (g1 ) ,
(6.10)
(6.11)
which, by Proposition 2.4, are adjoint to each other. Hence the second equality follows. Similarly the map 0 ⊥ ⊥ ad(x2 ) : g0 (V 0 )⊥ ∩ h⊥ 0 → g0 (V ) ∩ h0 is (−1 times) the composition of the maps (6.10) and (6.11), and the third equality follows.
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Recall the function J(x), equal to the absolute value of the determinant of the map (3.7): J(x) = |det(g0 /gx0 y + gx0 → [y, x] ∈ [g0 , x])|, (x ∈ g1 ). (6.12) Corollary 6.9. For x ∈ h1 we have J(x) = |det(h1 y → {y, x} ∈ h21 )| · |Dr (x)|1/2 . Proof. Let x ∈ hreg 1 . By Proposition 6.6(e), gx0 = g0 (V 0 ) ⊕ h21 . Therefore Proposition 6.6(e) and Proposition 6.6(f)) imply (Sgx0 )⊥ = (g0 (V 0 ) ⊕ Sh21 )⊥ = g0 (V 0 )⊥ ∩ (Sh21 )⊥ 2 = (g0 (V 0 )⊥ ∩ h⊥ 0 ) ⊕ Sh1 .
But we see from Proposition 6.6(d) and Proposition 6.6(f) that 2 2 g0 = g0 (V 0 ) ⊕ (g0 (V 0 )⊥ ∩ h⊥ 0 ) ⊕ Sh1 ⊕ h1
2 = (g0 (V 0 ) ⊕ h21 ) ⊕ (g0 (V 0 )⊥ ∩ h⊥ 0 ⊕ Sh1 ),
where the middle direct sum is orthogonal. Thus, x g0 = gx0 ⊕ (gx0 )⊥ = (Sh21 ⊕ g0 (V 0 )⊥ ∩ h⊥ 0 ) ⊕ g0 .
(6.13)
Furthermore, by (3.1), (4.4) and Proposition 6.6(b), h2
[g0 , x] = (x g1 )⊥ = x g1 ⊕ g1 (V 0 ) = x g1 (V + ) ⊕ g1 (V 0 ) = h1 g1 ⊕ (g11 )⊥ . Hence, by Lemma 6.7, J(x) is the absolute value of the determinant of the map Lemma 6.7(d) times the absolute value of the determinant of the map Lemma 6.7(e): h2
1 ⊥ J(x) = |det(Sh21 y → {y, x} ∈ Sh1 )| · |det(g0 (V 0 )⊥ ∩ h⊥ 0 y → [y, x] ∈ (g1 ) )|.
Hence our formula for J(x) follows from (6.8) and Lemma 6.8.
Example 6.10. The dual pair O2n (C), Sp2n (C) For i = 0, 1 choose a basis vi1 , vi2 , ..., vin , vi1 , vi2 , ..., vin
of the vector space Vi such that τ (vik , vik )=1
(k = 1, 2, 3, ..., n)
and all the other pairings are zero. Let h1 be the Cartan subspace consisting of elements x(a), a ∈ Cn , such that x(a) :v0k → ak v1k , v1k → ak v0k , → −ak v1k , v1k → ak v0k , v0k
(k = 1, 2, 3, ..., n).
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Then det(h1 y → {y, x} ∈
h21 )
=
(2ak ),
k=1
Dr (x(a)) =
n
2 n √ (a2j − a2k )(a2j + a2k ) · ( 2ak ) , and r = 4n. j=k
k=1
Example 6.11. The dual pair GLn (C), GLn (C). For i = 0, 1 choose a basis vi1 , vi2 , ..., vin of the vector space Vi . Let h1 be the Cartan subspace consisting of elements x(a), a ∈ Cn , such that x(a) : v0k → ak v1k , v1k → ak v0k ,
(k = 1, 2, 3, ..., n).
Then det(h1 y → {y, x} ∈ Dr (x(a)) =
h21 )
=
n
(2ak ),
k=1
2 (a2j − a2k ) , and r = 2n. j=k
reg Corollary 6.12. For a Cartan subspace h1 ⊆ g1 let h+ be a (measurable) funda1 ⊆ h1 mental domain for the action of the Weyl group W (G, h1 ). Let Q : g1 x → x2 ∈ g0 . Let greg,ss ⊂ g1 denote the subset of regular semisimple elements. Then for f ∈ Cc (greg,ss ) 1 1
1 f (x) dx = |W (G, h1 )| g1 h1 = J(x) h1
=
h+ 1
Qh+ 1
Qh+ 1
hreg 1
J(x)
.
G/Gh 1
f (gx) dg dx
.
G/Gh 1 −1
f (gx) dg dx 1/2
|Dr (Q (x))|
G/Gh 1
.
f (gQ−1 (x)) dg dx,
where the summation is over a maximal family of mutually non-conjugate Cartan subspaces h1 ⊆ g1 . Proof. The first equality follows from the fact that the absolute value of Jacobian of the map reg,ss h1 G/Gh1 × hreg 1 (gG , x) → gx ∈ g1
at (gGh1 , x) is equal to J(x). The second equality is obvious and the third one follows from Corollary 6.21.
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A canonical complementary subspace to the tangent space of an orbit
Here we adopt the view point of Harish-Chandra, [4, section 14], that the complementary subspace in Theorem 4.7 should be the orthogonal with respect to a natural positive definite form, (see (7.13) below). Recall the symmetric positive definite form ( , ) = − , θ on g defined in (3.5), and the adjoint map: End(g) A → A† ∈ End(g), (A(x), y) = (x, A† (y)), x, y ∈ g.
(7.1)
Lemma 7.1. For the adjoint representation ad : g → End(g), we have −ad(θ(x)) if x ∈ g0 , ad(x)† = ad(θ(x)) if x ∈ g1 . Proof. Let x, y, z ∈ g0 . Then (ad(x)y, z) = − [x, y], θz = − y, −[x, θz] = − y, −θ[θx, z] = (y, −ad(θx)z). Let x ∈ g0 , and let y, z ∈ g1 . Then the above computation applies without any change. Hence the formula follows for x ∈ g0 (as is well known [4, Lemma 27]). If x ∈ g1 , and either y, z ∈ g0 or y, z ∈ g1 , then all the pairings in question are zero. Let x ∈ g1 , y ∈ g1 and z ∈ g0 . Then, by Proposition 2.4, (ad(x)y, z) = − {x, y}, θz = − y, [x, θz] = − y, θ[θx, z] = (y, ad(θx)z). Let x ∈ g1 , y ∈ g0 and z ∈ g1 . Then, by Proposition 2.4, (ad(x)y, z) = − [x, y], θz = θz, [x, y] = {x, θz}, y = − y, {x, θz} = − y, θ{θx, z} = (y, ad(θx)z).
This verifies the second formula.
Let x ∈ g0 . Then the ( , )-orthogonal complement to [g0 , x] in g0 is equal to θ(gx0 ) = Thus (7.2) g0 = [g0 , x] ⊕ θ(gx0 ).
θ(x) g0 .
In particular, if h0 ⊆ g0 is a Cartan subalgebra, then g0 = [g0 , x] ⊕ θ(h0 )
(x ∈ hreg 0 ),
(7.3)
which provides a geometric interpretation for the notion of a θ-stable Cartan subalgebra. If x ∈ g0 is nilpotent and such that {x, y = −θ(x), h = [x, y]} is a Cayley triple, [2], then (7.2) may be rewritten as g0 = [g0 , x] ⊕ gy0 . (7.4)
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Moreover, if l = Rx ⊕ Ry ⊕ Rh, then by Lemma 7.1, ad(l) ⊆ End(g0 ) is a self adjoint family of operators. Hence g0 decomposes into a direct, ( , )- orthogonal sum of irreducible components, which have managable structure because the Lie algebra l is isomorphic to sl(2, R). Furthermore, g0 = [g0 , x] ⊕ gx0 (7.5) is another ( , )-orthogonal decomposition, and the map [g0 , x] × gy0 (v, w) → [v, x] + w ∈ g0
(7.6)
is a linear bijection. Consider an element x ∈ g1 . Then θ(x g1 ) = θ(x) g1 is the ( , )-orthogonal complement of [g0 , x] in g1 . Thus g1 = [g0 , x] ⊕ θ(x g1 ). (7.7) In particular, if h1 ⊆ g1 is a Cartan subspace, then g1 = [g0 , x] ⊕ θ(h1 g1 )
(x ∈ hreg 1 ).
(7.8)
Moreover, if ( )⊥ denotes the orthogonal complement with respect to the form , , then θ(x) θ((gx0 )⊥ ) = (g0 )⊥ is the ( , )-orthogonal complement of gx0 , so that g0 = θ((gx0 )⊥ ) ⊕ gx0 ,
(7.9)
and the following map is a linear bijection θ((gx0 )⊥ ) × θ(x g1 ) (v, w) → [v, x] + w ∈ g1 .
(7.10)
As shown by Harish-Chandra, [4, sections 13 and 14], the Jacobson-Morozov theorem and a theorem of Mostow imply that for a nilpotent orbit O ⊆ g there is an element x ∈ O such that the Lie algebra generated by x and θ(x) is isomorphic to sl(2, R). This may be deduced directly from the classification Theorems 5.4 and 5.5, and motivates the following problem. Problem 7.2. Let O ⊆ g∞ be a non-zero nilpotent orbit. For an element x ∈ O let s(x) ⊆ g be the Lie sub- superalgebra generated by x and θ(x). Let nO = min{dim s(x); x ∈ O}. Describe all the s(x) with dim s(x) = nO . Remark 7.3. With the notation of Problem 7.2, suppose (x, V ) is indecomposable. Then one can show, using the classification Theorems 4.4 and 4.5, that for x = 0, the height of x is even if and only if there is (a possibly different) x ∈ O such that [{x, θ(x)}, x] = x. Consequently s(x) is isomorphic to (o1 , sp2 (R)) as a dual pair, and (x2 , −θ(x)2 , [x2 , −θ(x)2 ]) is a Cayley triple.
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Notice that the adjoint representation maps g into the ortho- symplectic Lie subsuperalgebra osp(g) ⊆ End(g)0 ⊕ End(g)1 , defined as in (2.8) with the τ replaced by , . Let x ∈ g1 be nilpotent. Then ad(x) ∈ osp(g)1 is nilpotent. Hence the classification Theorems 4.4 and 4.5, applied to osp(g) provide a decomposition of g into a direct orthogonal sum of ad(x)-indecomposables. Moreover θ((gx0 )⊥ ) = θ((g0 ∩ ker(ad(x))⊥ ) = θ((g0 ∩ ad(x)(g)) = ad(θ(x))(g1 ) = {θ(x), g1 }.
(7.11)
Therefore (7.11) may be rewritten as {θ(x), g1 } × θ(x g1 ) (v, w) → [v, x] + w ∈ g1 .
(7.12)
The space W of Theorem 4.7 may be taken to be θ(x g1 ) and the local coordinates around x are provided by the map {θ(x), g1 } × θ(x g1 ) (v, w) → [v, x + u] + w ∈ g1
8
Let
(u ∈ θ(x g1 )).
(7.13)
A proof of Theorem 5.1 for x semisimple and (G, g) of type II, and a proof of Theorem 5.3 ⎧ C ⎪ ⎨ V , the complexification of V, if D = R, U = V, if D = C, ⎪ ⎩ V, viewed as a vector space over C, if D = H.
Since x is semisimple we have a direct sum decomposition into eigenspaces: U= U λ , xu = λu, u ∈ U λ .
(8.1)
λ
Let L be a set of eigenvalues of x such that L ∩ (−L) = ∅ and L ∪ (−L) is the set of all non-zero eigenvalues of x. Since SU λ = U −λ , (see (2.2) for S), U = U0 ⊕ (U λ ⊕ U −λ ) (8.2) λ∈L
is a direct sum decomposition into Z/2Z-graded subspaces preserved by x. The space U 0 is either zero or decomposes into a direct sum of one dimensional graded subspaces U= U 0,k . (8.3) k
For each λ ∈ L let Uλ =
l
U λ,l
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be a direct sum decomposition into one-dimensional subspaces. Then U λ ⊕ SU λ = (U λ,l ⊕ SU λ,l )
477
(8.4)
l
is a direct sum decomposition into graded x-invariant subspaces, which does not admit any finer decomposition of this type. Thus each term U λ,l ⊕ SU λ,l is indecomposable under the action of x and S. This verifies Theorem 5.1 for (G, g) of type II and D = C. Fix λ and l as in (8.4), and let uλ ∈ U λ,l be a non-zero vector. Set v0 = uλ + Suλ and v1 = uλ − Suλ . Then Sv0 = v0 , Sv1 = −v1 , xv0 = λv1 , xv1 = λv0 .
(8.5)
Thus Theorem 5.3, for D = C, follows. Let D = R. Let U u → u ∈ U be the complex conjugation with respect to the real form V ⊆ U . Then for each eigenvalue λ of x we have U λ = U λ.
(8.6)
We may split the set of eigenvalues of x into a disjoint union {0} ∪ LR ∪ (−LR ) ∪ LC ∪ (−LC ), where the elements λ ∈ LR are such that λ2 ∈ R, and the elements λ ∈ LC are such that λ2 ∈ C \ R, so that the four complex numbers λ, −λ, λ, −λ are distinct. Thus U = U0 ⊕ (U λ ⊕ U −λ ) ⊕ (U λ ⊕ U −λ ⊕ U λ ⊕ U −λ ). (8.7) λ∈LR
λ∈LC
Each summand in (8.7) is invariant under x, S, and the complex conjugation. The terms U 0 and U λ ⊕ U −λ , with λ ∈ R \ 0, may be treated as in the case D = C. The indecomposable summands of U λ ⊕ U −λ are described in part (a) of Theorem 5.3. Suppose λ = iξ ∈ iR \ 0. The, with the notation (8.5), x : v0 + v 0 → ξi(v1 − v 1 ) → −ξ(v0 + v 0 ). Hence, in this case, the indecomposable summands of U λ ⊕ U −λ are described in part (a’) of Theorem 5.3. Consider λ ∈ LC . Let Uλ = U λ,l l
be a direct sum decomposition into one dimensional subspaces. Then U λ ⊕ U −λ ⊕ U λ ⊕ U −λ = (U λ,l ⊕ U −λ,l ⊕ U λ,l ⊕ U −λ,l )
(8.8)
l
is a direct sum decomposition into (x, S, u → u invariant) subspaces of minimal possible dimension (equal 4). This verifies Theorem 5.1 for for (G, g) of type II and D = R.
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Fix λ ∈ LC and l as in (8.8). Let uλ ∈ U λ,l be a non-zero vector. Set uλ = uλ , u−λ = Suλ , and u−λ = u−λ . Then xuλ = λuλ , xu−λ = −λu−λ , xuλ = λuλ , xu−λ = −λu−λ .
(8.9)
Let λ = ξ + iη, with ξ, η ∈ R. Then ξη = 0. We see from (8.9) that x :uλ + u−λ + uλ + u−λ → λ(uλ − u−λ ) + λ(uλ − u−λ ), uλ − u−λ + uλ − u−λ → λ(uλ + u−λ ) + λ(uλ + u−λ ), uλ + u−λ − uλ − u−λ → λ(uλ − u−λ ) − λ(uλ − u−λ ),
(8.10)
uλ − u−λ − uλ + u−λ → λ(uλ + u−λ ) − λ(uλ + u−λ ). Set
u0 = uλ + u−λ + uλ + u−λ , v0 = i(uλ + u−λ − uλ − u−λ ), u1 = uλ − u−λ + uλ − u−λ , v1 = i(uλ − u−λ − uλ + u−λ ).
We see from (8.10) that x :u0 → ξu1 + ηv1 , u1 → ξu0 + ηv0 , v0 → −ηu1 + ξv1 , v1 → −ηu0 + ξv0 .
(8.11)
The formulas (8.11) are consistent with the formulas of part (b) of Theorem 5.3. This verifies Theorem 5.3 for D = R. Let D = H. Since jU λ = U λ , each summand in the decomposition (8.7) is a vector space over H. Let λ ∈ LR ∪ LC . Pick a non-zero vector uλ ∈ U λ . Let v0 = uλ + Suλ , v1 = uλ − Suλ . Then xv0 = λv1 , xv1 = λv0 ,
(8.12)
Hv0 + Hv1 ⊆ U
(8.13)
and is a graded x-invariant subspace over H. The non-zero part of the right hand side of (8.7) may be grouped into a direct sum of spaces of the form (8.13). This verifies Theorems 5.1 and 5.3 for D = H.
9
A proof of Theorem 5.1 for x semisimple and (G, g) of type I, and a proof of Theorem 5.2
We consider four cases: (D = C, ι = 1), (D = C, ι = 1), (D = H, ι = 1) and (D = R, ι = 1). Case (D = C, ι = 1). Let V = Vλ (9.1) λ
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be the decomposition of V into the eigenspaces for x (xv = λv for v ∈ V λ ). For two eigenvalues λ, μ and the corresponding eigenvectors v λ , v μ , we have μ2 τ (v μ , v λ ) = τ (x2 v μ , v λ ) = τ (v μ , −x2 v λ ) = τ (v μ , v λ )(−λ2 ). Thus V λ ⊥ V μ if μ2 + λ2 = 0.
(9.2)
Let us decompose the set of non-zero eigenvalues of x into a disjoint union L ∪ (−L) ∪ iL ∪ (−iL). Then, by (9.2), V =V0⊕
(V λ ⊕ V −λ ⊕ V iλ ⊕ V −iλ )
(9.3)
λ∈L
is a direct sum orthogonal decomposition into graded subspaces preserved by x. For λ ∈ L, pick a non-zero vector v λ ∈ V λ , and a non-zero vector v iλ ∈ V iλ . Let v0 = v λ + Sv λ , v1 = v λ − Sv λ , v0 = i(v iλ + Sv iλ ), v1 = v iλ − Sv iλ . Then x : v0 → λv1 , v1 → λv0 , v0 → −λv1 , v1 → λv0 .
(9.4)
Hence, λτ (v1 , v1 ) = τ (xv0 , v1 ) = τ (v0 , Sxv1 ) = τ (v0 , Sλv0 ) = τ (v0 , v0 )λ. Thus τ (v1 , v1 ) = τ (v0 , v0 ).
(9.5)
We may multiply v0 and v1 by the same complex number to ensure τ (v1 , v1 ) = τ (v0 , v0 ) = 1.
(9.6)
Notice that τ (v0 , v0 )λ = τ (v0 , xv1 ) = τ (−Sxv0 , v1 ) = τ (−Sλv1 , v1 ) = λτ (v1 , v1 ) and λτ (v0 , v0 ) = τ (xv1 , v0 ) = τ (v1 , Sxv0 ) = τ (v1 , Sλv1 ) = τ (v1 , v1 )(−λ). Hence τ (v0 , v0 ) = τ (v1 , v1 ) = 0. Similarly τ (v0 , v0 )λ = τ (v0 , xv1 ) = τ (−Sxv0 , v1 ) = τ (Sλv1 , v1 ) = −λτ (v1 , v1 ) and λτ (v0 , v0 ) = τ (xv1 , v0 ) = τ (v1 , Sxv0 ) = τ (v1 , −Sλv1 ) = τ (v1 , v1 )λ.
(9.7)
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Hence τ (v0 , v0 ) = τ (v1 , v1 ) = 0.
(9.8)
Cv0 ⊕ Cv0 ⊕ Cv1 ⊕ Cv1 ⊆ V λ ⊕ V −λ ⊕ V iλ ⊕ V −iλ
(9.9)
The subspace is graded and x-invariant and the action of x on this subspace is consistent with the formulas of part (a) of Theorem 5.2. Moreover it is clear that we may decompose the right hand side of (9.9) into direct sum of such subspaces. This verifies Theorems 5.1 and 5.2. Case (C, ι = 1). Here, instead of (9.2) we have 2
V λ ⊥ V μ if μ2 + λ = 0,
(9.10)
where λ = ι(λ). We decompose the set of non-zero eigenvalues of x into a disjoint union LiR ∪ (−LiR ) ∪ LC ∪ (−LC ) ∪ iLC ∪ (−iLC ), where λ2 ∈ iR \ 0 for λ ∈ LiR , and λ2 ∈ C \ iR for λ ∈ LC . Then by (9.10), V =V0⊕ (V λ ⊕ V −λ ) ⊕ (V λ ⊕ V −λ ⊕ V iλ ⊕ V −iλ ) λ∈LiR
(9.11)
λ∈LC
is a direct sum orthogonal decomposition into graded x-invariant subspaces. For λ ∈ LiR , pick a non-zero vector v λ ∈ V λ . Let v0 = v λ + Sv λ , v1 = v λ − Sv λ . Then x : v0 → λv1 , v1 → λv0 .
(9.12)
Moreover, λτ (v0 , v0 ) = τ (xv1 , v0 ) = τ (v1 , Sxv0 ) = τ (v1 , Sλv1 ) = τ (v1 , v1 )(−λ). Thus
λ τ (v1 , v1 ) = − τ (v0 , v0 ), λ
where −
λ λ2 = − 2 = −i sgn(im(λ2 )). |λ| λ
(9.13)
(9.14)
Since τ0 is hermitian, τ (v0 , v0 ) ∈ R \ 0. Thus we may multiply v0 and v1 by the same positive real number so that (by (9.13) and (9.14)) τ (v0 , v0 ) = = ±1, τ (v1 , v1 ) = δi = ±i, with − δ = sgn(im(λ2 )).
(9.15)
Clearly Cv0 ⊕ Cv1 ⊆ V λ ⊕ V −λ
(9.16)
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is a graded, x-invariant subspace described in part (b) of Theorem 5.2. This subspace is indecomposable and the right hand side of (9.16) decomposes into an orthogonal direct sum of such subspaces. Let λ ∈ LC and let v λ ∈ V λ be a non-zero vector. Set v0 = v λ + Sv λ , v1 = v λ − Sv λ , v0 = i(v iλ + Sv iλ ), v1 = v iλ − Sv iλ . Then x : v0 → λv1 , v1 → λv0 , v0 → −λv1 , v1 → λv0 .
(9.17)
Furthermore λτ (v1 , v1 ) = τ (xv0 , v1 ) = τ (v0 , Sxv1 ) = τ (v0 , v0 )λ so that τ (v1 , v1 ) = τ (v0 , v0 ). As before we may scale the vectors v0 and v1 by the same number so that τ (v1 , v1 ) = τ (v0 , v0 ) = 1.
(9.18)
Moreover,
2
λ2 τ (v0 , v0 ) = τ (x2 v0 , v0 ) = τ (v0 , −x2 v0 ) = τ (v0 , v0 )(−λ ) and
2
λ2 τ (v1 , v1 ) = τ (x2 v1 , v1 ) = τ (v1 , −x2 v1 ) = τ (v1 , v1 )(−λ ), so that
2
2
(λ2 + λ )τ (v0 , v0 ) = (λ2 + λ )τ (v1 , v1 ) = 0, which implies τ (v0 , v0 ) = τ (v1 , v1 ) = 0.
(9.19)
τ (v0 , v0 ) = τ (v1 , v1 ) = 0.
(9.20)
Cv0 ⊕ Cv0 ⊕ Cv1 ⊕ Cv1 ⊆ V λ ⊕ V −λ ⊕ V iλ ⊕ V −iλ
(9.21)
Similarly, Clearly, is a graded, x-invariant subspace, as in part (a) of Theorem 5.2. This subspace is indecomposable and the right hand side of (9.21) decomposes into an orthogonal direct sum of such subspaces. Case (D = H, ι = 1). Here we view V as a vector space over C ⊆ H. Then the decomposition (9.1) holds and jV λ = V λ , (9.22) where λ = ι(λ). Furthermore, 2
V λ ⊥ V μ if μ2 + λ2 = 0 and μ2 + λ = 0.
(9.23)
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Indeed, let v μ ∈ V μ , v λ ∈ V λ and let τ (v μ , v λ ) = α + jβ, with α, β ∈ C. Then μ2 (α + jβ) = μ2 τ (v μ , v λ ) = τ (x2 v μ , v λ ) = τ (v μ , −x2 v λ ) = τ (v μ , −λ2 v λ ) 2
2
= τ (v μ , v λ )(−λ ) = (α + jβ)(−λ ). Hence,
2
(μ2 + λ )α = 0 and (μ2 + λ2 )jβ = 0, and (9.23) follows. We decompose the set of non-zero eigenvalues of x into a disjoint union LiR ∪ (−LiR ) ∪ LR ∪ (−LR ) ∪ LC ∪ (−LC ),
(9.24)
where λ2 ∈ iR \ 0 for λ ∈ LiR , λ2 ∈ R \ 0 for λ ∈ LR , and λ2 ∈ C \ (iR ∪ R) for λ ∈ LC . Then by (9.22) and (9.23), (V λ ⊕ V −λ ⊕ V λ ⊕ V −λ ) V =V0⊕ λ∈LiR
⊕
(V λ ⊕ V −λ ⊕ V iλ ⊕ V −iλ )
(9.25)
λ∈LR
⊕
(V λ ⊕ V −λ ⊕ V iλ ⊕ V −iλ ⊕ V λ ⊕ V −λ ⊕ V −iλ ⊕ V iλ )
λ∈LC
is a direct sum orthogonal decomposition into graded x-invariant subspaces over H. Let λ ∈ LiR . Then (9.12) holds, and since τ (v0 , v0 ) ∈ R, (9.13) holds too. Therefore (9.15) holds and, instead of (9.16), we see that Hv0 ⊕ Hv1 ⊆ V λ ⊕ V −λ ⊕ V λ ⊕ V −λ
(9.26)
is a graded x-invariant subspace, as in part (c) of Theorem 5.2. This subspace is indecomposable and the right hand side of (9.26) decomposes into an orthogonal direct sum of such subspaces. For λ ∈ LR ∪ LC the argument (9.17)-(9.21) carries over. This completes the proof of both Theorems 5.1 and 5.2 in the case D = H. Case (D = R, ι = 1). Let VC = V C,λ (9.27) λ C
be the decomposition of V , the complexification of V , into the eigenspaces for x. Let V C v → v ∈ V C be the complex conjugation with respect to the real form V ⊆ V C . The form τ extends uniquely to a complex linear form on V C . As before we check that V C,μ ⊥ V C,λ if μ2 + λ2 = 0.
(9.28)
Let LiR , LR , LC be as in (9.24). Then V C = V C,0 ⊕ (V C,λ ⊕ V C,−λ ⊕ V C,λ ⊕ V C,−λ ) ⊕
λ∈LiR
(V C,λ ⊕ V C,−λ ⊕ V C,iλ ⊕ V C,−iλ )
λ∈LR
⊕
λ∈LC
(V C,λ ⊕ V C,−λ ⊕ V C,iλ ⊕ V C,−iλ ⊕ V C,λ ⊕ V C,−λ ⊕ V C,−iλ ⊕ V C,iλ )
(9.29)
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483
is an orthogonal direct sum decomposition into graded x-invariant subspaces invariant under the complex conjugation, V C u → u ∈ V C with respect to the real form V . Let λ ∈ LiR and let v λ ∈ V C,λ be a non-zero vector. Let u = v λ + Sv λ and v = v λ − Sv λ . Then x : u → λv, v → λu, u → λv, v → λu. Hence
(9.30)
λτ (u, u) = τ (xv, u) = τ (v, Sxu) = τ (v, −λv) = τ (v, v)(−λ), λτ (v, v) = τ (xu, v) = τ (u, Sxv) = τ (u, λu) = τ (u, u)λ.
Therefore τ (u, u) = τ (v, v) = τ (u, u) = τ (v, v) = 0.
(9.31)
Furthermore λτ (v, v) = τ (xu, v) = τ (u, Sxv) = τ (u, u)λ, so that τ (v, v) = τ (u, u)(−i)sgn(im(λ2 )).
(9.32)
Multiplying u and v by the same non-zero real number does not change (9.30), (9.31), (9.32). Thus we may assume τ (u, u) = /2, (9.33) where = ±1. Set v0 = u + u, v0 = i(u − u), v1 = v + v, v1 = i(v − v).
(9.34)
Then, by (9.31) and (9.33), τ (v0 , v0 ) = τ (u, u) + τ (u, u) = , τ (v0 , v0 ) = −τ (u, −u) − τ (−u, u) = ,
τ (v0 , v0 ) = τ (u, −iu) + τ (u, iu) = 0, τ (v1 , v1 ) = τ (v, v) + τ (v, v) = 0,
(9.35)
τ (v1 , v1 ) = τ (iv, −v) + τ (−v, iv) = 0,
τ (v1 , v1 ) = τ (v, −iv) + τ (v, iv) = −2iτ (v, v)
= −2τ (u, u)sgn(im(λ2 )) = −τ (v0 , v 0 )sgn(im(λ2 )).
We see from (9.30) and (9.34) that, with λ = ξ + iη, ξ, η ∈ R, x :v0 → ξv1 + ηv1 , v1 → ξv0 + ηv0 ,
v0 → −ηv1 + ξv1 , v1 → −ηv0 + ξv0 .
(9.36)
Notice that im(λ2 ) = 2ξη.
(9.37)
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Thus the last formula in (9.35) may be rewritten as τ (v1 , v1 ) = δ, −δ = sgn(ξη).
(9.38)
Thus λ = ξ(1 − iδ) and if we replace v1 by δv1 then (9.36) coincides with x :v0 → ξ(v1 − v1 ), v1 → ξ(v0 − δv0 ),
v0 → δξ(v1 + v1 ), v1 → ξ(v0 + δv0 ).
(9.39)
Let g : v0 → v0 , v0 → δv0 , v1 → v1 , v1 → v1 . Then g ∈ G and gxg −1 acts according to the formula (c) of Theorem 5.2. The subspace Rv0 ⊕ Rv0 ⊕ Rv1 ⊕ Rv1 ⊆ V
(9.40)
is graded, x-invariant and indecomposable. Let λ ∈ LR . Then either λ ∈ R \ 0 or λ ∈ iR \ 0. Suppose λ ∈ R \ 0. Then the eigenspace V C,λ is closed under the complex conjugation. Hence we may chose a non-zero vector v λ ∈ V ∩ V C,λ . Let v iλ ∈ V ∩ V C,iλ be a non-zero vector such that v iλ = Sv iλ . Set v0 = v λ + Sv λ , v1 = v λ − Sv λ , v0 = v iλ + Sv iλ , v1 = −i(v iλ − Sv iλ ).
(9.41)
Then v0 = v0 , v1 = v1 , v0 = v0 , v1 = v1 ,
(9.42)
x : v0 → λv1 , v1 → λv0 , v0 → −λv1 , v1 → λv0 .
(9.43)
and Furthermore, λτ (v1 , v1 ) = τ (xv0 , v1 ) = τ (v0 , Sxv1 ) = τ (v0 , v0 )λ, so that τ (v1 , v1 ) = τ (v0 , v0 ).
(9.44)
Similarly we check that the vectors v0 , v0 , v1 , v1 are isotropic. Multiplying v0 and v1 by the same number we may assume that τ (v1 , v1 ) = τ (v0 , v0 ) = 1.
(9.45)
Rv0 ⊕ Rv0 ⊕ Rv1 ⊕ Rv1 ⊆ V
(9.46)
The subspace is graded, x-invariant and indecomposable. The formulas (9.43) are compatible with the formulas of part (a) of Theorem 5.2. Finally, let λ ∈ LC . Choose non-zero vectors v λ ∈ V C,λ , v iλ ∈ V C,iλ and let u = v λ + Sv λ , v = v λ − Sv λ , u = v iλ + Sv iλ , v = v iλ − Sv iλ .
(9.47)
T. Przebinda / Central European Journal of Mathematics 4(3) 2006 449–506
Then
x :u → λv, v → λu, u → λv, v → λu, u → iλv , v → iλu , u → iλ v , v → iλ u .
We see from (9.28) and (9.24) that C,λ V + V C,−λ ⊥ V C,−iλ + V C,iλ , and C,λ C,λ C,−λ C,−λ V ⊥ V . +V +V
485
(9.48)
(9.49)
Thus by (9.29), the restriction of the form τ to V C,λ + V C,−λ + V C,iλ + V C,−iλ is non-degenerate. Hence we may choose the vectors v λ , v iλ so that τ (u, u ) = 0.
(9.50)
The following calculation λτ (v, v ) = τ (xu, v ) = τ (u, xv ) = τ (u, iλu ) = τ (u, u )iλ shows that τ (v, v ) = τ (u, u ). As before we may assume that 1 τ (v, v ) = τ (u, u ) = . (9.51) 2 Since τ is the complexification of a real form the usual calculation using (9.24), (9.28) and (9.51) implies 1 τ (u, u ) = τ (u, u ) = , 2 τ (u, u) = τ (u, u) = τ (u, u ) = τ (u , u ) = 0, τ (u, u) = τ (u , u ) = 0, 1 τ (v, −iv ) = = , 2 τ (v, v) = τ (v, v) = τ (v, v ) = τ (v, v )
(9.52)
τ (v, iv )
= τ (v , v ) = τ (v , v ) = 0. Set
u0 = u + u, v0 = i(u − u), u0 = u + u , v0 = −i(u − u ),
u1 = v + v, v1 = i(v − v), u1 = −i(v − v ), v1 = −v − v .
(9.53)
Let λ = ξ + iη, ξ, η ∈ R, Then the formulas of part (d) of Theorem 5.2 hold. Furthermore the subspace Ru0 ⊕ Ru0 ⊕ Rv0 ⊕ Rv0 ⊕ Ru1 ⊕ Ru1 ⊕ Rv1 ⊕ Rv1 ⊆ V C,λ ⊕ V C,−λ ⊕ V C,iλ ⊕ V C,−iλ ⊕ V C,λ ⊕ V C,−λ ⊕ V C,−iλ ⊕ V C,iλ is graded, x-invariant and indecomposable, and the space on the right hand side of the inclusion decomposes into a direct sum of such subspaces. This completes our proof.
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A proof of Theorem 4.2
We may assume that (xn , V ) is uniform and that V = F ⊕ xn F ⊕ x2n F ⊕ ...xm n F, where the subspace F is graded and xs -invariant. For each set of positive numbers a0 , a1 , ..., am (such that ak am−k = 1, 0 ≤ k ≤ m, in the type I case) the formula g(u0 + xn u1 + ... + xm n um ) (u0 , u1 , ..., um ∈ F )
= a0 u0 + a1 xn u1 + ... + am xm n um defines an element g ∈ Gxs . Furthermore, ak+1 k+1 x ui gxn g −1 : xkn ui → ak k
(ui ∈ F, 0 ≤ i, k ≤ m)
(10.1)
(10.2)
(l)
An elementary argument shows that there is a sequence ak , l = 1, 2, 3, ... of positive numbers such that for all 0 ≤ k ≤ m, (l)
→ lim
l→∞
ak+1 (l)
ak
= 0.
(10.3)
(l)
Let g (l) ∈ Gxs be as in (10.1), for the sequence ak . Then, by (10.2) and (10.3), g (l) xg (l)−1 = xs + g (l) xn (g (l) )−1 → xs as l → ∞.
11
A proof of Theorem 4.8
Let x1 ∈ g1 be semisimple and let V = V (0) ⊕ V (1) ⊕ V (2) ⊕ ... be such that each x1 |V (0) = 0 and for j ≥ 1, (x1 , V (j) ) is indecomposable, with x1 |V (j) = 0. We assume that the sum is orthogonal in the type I case. Then, by Theorems 5.2 and (j) (j) 5.3, (x21 , V0 ) and (x21 , V1 ), j ≥ 1, are indecomposable. Let x2 ∈ g1 be another semisimple elements such that Gx21 = Gx22 . Then, by the above argument, we may assume that x2 |V (0) = 0 and that for j ≥ 1 (x2 , V (j) ) is indecomposable with x2 |V (j) = 0. Hence we may assume that (x1 , V (j) ) and (x2 , V (j) ), j ≥ 1, are indecomposable. But then the Theorem 4.8 follows from Theorems 5.2 and 5.3 by inspection.
12
A proof of Theorem 4.3
Suppose X ∈ G1 is not semisimple. Then in the Jordan decomposition X = XS + XN , XN = 0. By Theorem 4.2, XS ∈ Cl(Gx). But XS ∈ / Gx. Thus the orbit Gx is not closed. Suppose X ∈ G1 is semisimple. Since the Gl(V )-orbit through X in End(V ) is closed, we see that Cl(Gx) is a union of semisimple orbits. Since Gx2 ⊆ G0 is closed, Theorem 4.8 implies that Cl(Gx) is a single orbit, hence is equal to Gx.
T. Przebinda / Central European Journal of Mathematics 4(3) 2006 449–506
13
487
A proof of Theorem 4.4
Let x ∈ g1 be semisimple. We’ll say that V is isotypic for x, or that (x, V ) is isotypic, if (x, V ) decomposes into mutually similar indecomposable pieces. Two isotypic elements (x, V ) and (x , V ) are of different types if the indecomposable pieces of (x, V ) are not similar to the indecomposable pieces of (x , V ). Lemma 13.1. Let (x, V ) and (x , V ) be two semisimple isotypic elements of different types. Then the only y ∈ Hom(V , V ) such that xy + yx = 0 is y = 0. Proof. We may assume that the elements (x, V ), (x , V ) are indecomposable. We need to check that the map Hom(V , V ) y → xy + yx ∈ Hom(V , V ) is injective. The eigenvalues of this map are sums of the eigenvalues of x and x . By Theorems 5.2 and 5.3 these sums are not zero. Lemma 13.2. The anticommutant of g1 in g1 is zero:
g1
g1 = 0.
Proof. Suppose the Lie superalgebra g is of type II. Then g1 = Sg1 . Let x ∈ g1 g1 and let y ∈ g1 . Then Sy ∈ g1 and therefore {Sy, x} = 0. In particular, 0 = tr{Sy, x} = y, x . Thus x is orthogonal to g1 , and since the form , is non-degenerate, we see that x = 0. Suppose the Lie superalgebra g is of type I. In this case g1 ∩ Sg1 = 0, so we are forced to use a different argument. We may assume that g is complex (and of type I). Then dim V0 ≥ 2 or dim V1 ≥ 2. Consider the case dim V0 ≥ 2. The second one is analogous. Let x ∈ g1 g1 , and let y ∈ g1 . Then, in terms of (2.12), we have (xy + yx)(v0 + v1 ) = (wx∗ wy + wy∗ wx )v0 + (wx wy∗ + wy wx∗ )v1 . Thus the condition xy +yx = 0 translates to wx∗ wy +wy∗ wx = 0. Hence, for any v0 , v0 ∈ V0 , τ0 (v0 , wx∗ wy v0 ) + τ0 (v0 , wy∗ wx v0 ) = 0, or equivalently, τ1 (wx v0 , wy v0 ) + τ1 (wy v0 , wx v0 ) = 0. Now fix v0 ∈ V0 \ 0 and let v0 ∈ V0 \ 0 be such that the vectors v0 , v0 are linearly independent. Then {w(v0 ); w(v0 ) = 0, w ∈ Hom(V0 , V1 )} = V1 . Hence, wx (v0 ) is orthogonal to V1 with respect to the form τ1 . Since this form is nondegenerate, wx = 0, and therefore x = 0. Let V = V 0 ⊕ V 1 ⊕ V 2 ⊕ ...
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be the isotypic decomposition of V , with respect to x, with V 0 = ker(x). Then xV = V 1 ⊕ V 2 ⊕ ... and the first part of (b) follows. Let (G(V i ), g(V i )) be the restriction of (G, g) to V i . Then 2 2 2 2 Gx = G(V 0 )x × G(V 1 )x × G(V 2 )x × ..., 2
2
2
2
gx = g(V 0 )x ⊕ g(V 1 )x ⊕ g(V 2 )x ⊕ ..., where the summands are orthogonal with respect to the symplectic form , on g1 . 2 2 Moreover, G(V 0 )x = G(V 0 ) and g(V 0 )x = g(V 0 ). Hence, 2
2
2
Gx = G(V 0 ) × G(V 1 )x × G(V 2 )x × ..., 2
2
2
gx = g(V 0 ) ⊕ g(V 1 )x ⊕ g(V 2 )x ⊕ ... . This verifies the second part of (b). We shall see in section 13 () that there is y ∈ x g1 such that for all i = j greater than or equal to 1, the restrictions (y, V i ), (y, V j ) are isotropic of different types. Hence the lemmas 13.1 and 13.2 imply (x g1 )
g1 = (
x g (V 1 )) 1
g1 (V 1 ) ⊕ (
x g (V 2 )) 1
g1 (V 2 ) ⊕ ... .
Hence the proof of the last formula in (b) is reduced to the case when (x, V ) is isotypic and non-zero. Also, the above formula reduces the proof of (c), which we leave to the reader, to the case when (x, V ) is isotypic and non-zero. From now on, we assume that (x, V ) is isotypic and x = 0. We proceed via a case by case analysis according to Theorems 5.2 and 5.3. Case 5.2.a. Here D is arbitrary and the vector spaces Vi (i = 0, 1) have basis vi,1 , vi,2 , vi,3 , ..., vi,n , such that τ (v0,k , v0,k ) = τ (v1,k , v1,k )=1
(k = 1, 2, 3, ..., n)
(13.1)
and all the other pairings are zero. Further, → −ι(ξ)v1,k , v1,k → ι(ξ)v0,k . x = x(ξ) : v0,k → ξv1,k , v1,k → ξv0,k , v0,k
(13.2)
x2 : v0,k → ξ 2 v0,k , v1,k → ξ 2 v1,k , v0,k → −ι(ξ)2 v0,k , v1,k → −ι(ξ)2 v1,k .
(13.3)
Thus
2
Since ξ 2 = −ι(ξ)2 , the group Gx preserves each of the isotropic subspaces n
Dvi,k ⊆ Vi ,
k=1
n
Dvi,k ⊆ Vi
(i = 0, 1).
(13.4)
k=1
By Witt’s Theorem the restriction x2 n
G |
k=1
Dvi,k
n 2 = GL( Dvi,k )x k=1
(i = 0, 1).
(13.5)
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Hence, G
x2
x2
x2
= G |V0 × G |V1 =
GLn (D) × GLn (D) if D = H or D = H and ξ 2 ∈ R, GLn (C) × GLn (C) if D = H and ξ 2 ∈ / R.
(13.6) Suppose w ∈ Hom(V0 , V1 ) commutes with x . Then by (13.3), there are elements ∗ ∗ , wkl , wkl ∈ D commuting with ξ 2 and such that wkl , wkl 2
n
w(v0,k ) = ∗
w (v1,k ) =
wkl v1,l ,
l=1 n
w(v0,k )
=
n
wkl v1,l ,
l=1 ∗ wkl v0,l ,
w
∗
(v1,k )
=
l=1
n
(13.7) ∗ wkl v0,l .
l=1
By (2.10) and (13.1) we have ∗ ) ι(wpk
= τ0 (v0,k ,
n
∗ wpl v0,l ) = τ0 (v0,k , w∗ (v1,p ))
l=1
=
τ1 (w(v0,k ), v1,p )
n = τ1 ( wkl v1,l , v1,p ) = wkp , l=1
and ∗ ) ι(wpk
=
∗ τ0 (v0,k , v0,k )ι(wpk )
=
τ0 (v0,k ,
n
∗ wpl v0,l ) = τ0 (v0,k , w∗ (v1,p ))
l=1
=
τ1 (w(v0,k ), v1,p )
=
wkp τ1 (v1,p , v1,p )
= −wkp .
Hence, ∗ ∗ wpk = −ι(wkp ), wpk = ι(wkp ).
(13.8)
2
For y, z ∈ gx1 let w = y|V0 and let u = z|V0 . Then w, u ∈ Hom(V0 , V1 ) commute with x2 and, by (2.13), (13.7) and (13.8), 1 y, z = trD/R (w∗ u) 4 n n ∗ ∗ = trD/R (ukl wlk + ukl wlk ) = trD/R (ukl ι(wkl ) − ukl ι(wkl )). k,l=1
(13.9)
k,l=1
The formula (2.12), (13.2), (13.7) and (13.8) imply yx : v0,k → −
n
ξι(wlk )v0,l ,
v0,k
l=1
xy : v0,k →
n l=1
Hence,
wkl ξv0,l , v0,k →
→−
n
n
ι(ξ)ι(wlk )v0,l ,
l=1
(13.10)
wkl ι(ξ)v0,l .
l=1
y ∈ gx1 if and only if wkl = −ι(ξ)ι(wlk )ι(ξ)−1 ,
y ∈ x g1 if and only if wkl = ι(ξ)ι(wlk )ι(ξ)−1 .
(13.11)
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Suppose y, z ∈ gx1 . Then by (13.9) and (13.11), n 1 trD/R (−ι(ξ)ι(ulk )ι(ξ)−1 ι(wkl ) + ukl ξ −1 wlk ξ) y, z = 4 k,l=1
=− =−
n
k,l=1 n
−1
trD/R (wkl ξ ulk ξ) + trD/R (wkl ξulk ξ −1 ) +
k,l=1
n
k,l=1 n
trD/R (ulk ξ −1 wkl ξ)
(13.12)
trD/R (ulk ξ −1 wkl ξ) = 0,
k,l=1
where the equation ξ −1 ulk ξ = ξulk ξ −1 follows from the fact that ulk commutes with ξ 2 . The computation (13.12) shows that gx1 is an isotropic subspace of g1 . Suppose y ∈ gx1 and z ∈ x g1 . Then wkl = ι(ξ −1 wlk ξ) and, as in (13.12), we show that n 1 trD/R (ulk ξwkl ξ −1 ), y, z = −2 4 k,l=1
(13.13)
which implies that the symplectic form , provides a non-degenerate pairing between 2 2 2 gx1 and x g1 . The supergroup (Gx , gx ) is irreducible, of type II, and the ranks of Gx |V0 2 and Gx |V1 are equal. Furthermore a straightforward computation shows that (x(ξ) g1 )
x(ξ)
(g
g1 = g1 1
)
= {x(ζ); ζ 2 ∈ D(D
ξ2 )
}.
(13.13.1)
Case 5.2.b Here D ⊇ C and the vector spaces Vi (i = 0, 1) have basis vi,1 , vi,2 , vi,3 , ..., vi,n , such that τ (v0,k , v0,k ) = = ±1, τ (v1,k , v1,k ) = δi = ±i
(k = 1, 2, 3, ..., n)
(13.14)
and all the other pairings are zero. Furthermore, x = x(ξ) : v0,k → ξv1,k , v1,k → ξv0,k .
(13.15)
x2 : v0,k → ξ 2 v0,k , v1,k → ξ 2 v1,k .
(13.16)
Thus Since ξ 2 ∈ iR \ 0, 2
2
2
Gx = Gx |V0 × Gx |V1 = Un (C) × Un (C).
(13.17)
∗ Suppose w ∈ Hom(V0 , V1 ) commutes with x2 . Then, by (13.16), there are wkl , wkl ∈D 2 commuting with ξ and such that
w(v0,k ) =
n l=1
∗
wkl v1,l , w (v1,k ) =
n l=1
∗ wkl v0,l .
(13.18)
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By (2.10) and (13.14) we have ∗ ) ι(wpk
= τ0 (v0,k ,
n
∗ wpl v0,l ) = τ0 (v0,k , w∗ (v1,p ))
l=1
= τ1 (w(v0,k ), v1,p ) = wkp τ1 (v1,p , v1,p ) = wkp δi. Thus, ∗ wpk = − δiι(wkp ).
(13.19)
2
For y, z ∈ gx1 let w = y|V0 and let u = z|V0 . Then w, u ∈ Hom(V0 , V1 ) commute with x2 and, by (2.13), 1 y, z = trD/R (w∗ u) 4 n n ∗ = trD/R (ukl wlk ) = trD/R (− δiι(wkl )ukl ). k,l=1
(13.20)
k,l=1
The formulas (2.12) and (13.15) imply yx : v0,k →
n
∗ ξwlk v0,l ,
l=1
xy : v0,k →
n
(13.20’) wkl ξv0,l .
l=1
Furthermore, since ξ 2 ∈ R \ 0, the centralizer of ξ 2 in D coincides with the the centralizer of ξ in D. In particular, wkl ξ = ξwkl . By combining this with (13.19) and (13.20’) we see that y ∈ gx1 if and only if wkl = − δiι(wlk ), (13.21) y ∈ x g1 if and only if wkl = δiι(wlk ). Suppose y, z ∈ x g1 . Then (13.20) and (13.21) imply n 1 trD/R (−ξ −1 wlk ξukl ) y, z = 4 k,l=1 n
1 = trD/R (− δiι(ulk )wkl ) = z, y . 4 k,l=1
(13.22)
Thus y, z = 0. Hence x g1 is an isotropic subspace of g1 . Similarly we check that gx1 is an isotropic subspace of g1 , and that the symplectic form provides a non-degenerate 2 2 pairing between x g1 and gx1 . The dual pair corresponding to the supergroup (Gx , gx ) is isomorphic to (Un , Un ). Furthermore a straightforward computation shows that (x(ξ) g1 )
x(ξ)
(g
g1 = g1 1
)
= {x(ζ); ζ 2 ∈ iR, sgn(im(ζ 2 )) = sgn(im(ξ 2 ))}.
(13.22.1)
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Case 5.2.c. Here D = R and the vector spaces Vi (i = 0, 1) have basis vi,1 , vi,2 , vi,3 , ..., vi,n , vi,1 , vi,2 , vi,3 , ..., vi,n ,
such that with = ±1 τ (v0,k , v0,k ) =
(13.23)
τ (v0,k , v0,k )
τ (v1,k , v1,k )
= ,
=1
(k = 1, 2, 3, ..., n)
and all the other pairings are zero. Further, with ξ ∈ R \ 0, x = x(ξ, ) :v0,k → ξ(v1,k − v1,k ), v1,k → ξ(v0,k − v0,k ),
→ ξ(v1,k + v1,k ), v1,k → ξ(v0,k + v0,k ). v0,k
Therefore,
, v1,k → −2 ξ 2 v1,k , x2 :v0,k → −2ξ 2 v0,k → 2ξ 2 v0,k , v1,k → 2 ξ 2 v1,k . v0,k
(13.24)
(13.25)
Since ξ = 0, (13.25) implies 2
2
2
Gx = Gx |V0 × Gx |V1 = Un × Un .
(13.26)
Suppose w ∈ Hom(V0 , V1 ) commutes with x2 . Then, by (13.25), there are elements ∗ ∗ , wkl , wkl ∈ R such that wkl , wkl w(v0,k ) =
n
(wkl v1,l − wkl v1,l ),
l=1
w(v0,k )=
n
∗
w (v1,k ) =
(wkl v1,l + wkl v1,l ),
l=1 n
(13.27) ∗ (wkl v0,l
−
∗ wkl v0,l ),
l=1
w∗ (v1,k )=
n
∗ ∗ (wkl v0,l + wkl v0,l ).
l=1
From (13.23) and (13.27) we see that ∗ ∗ wpk = τ0 (v0,k , wpk v0,k ) = τ0 (v0,k , w∗ (v1,p )) = τ1 (w(v0,k ), v1,p ) = τ1 (wk,p v1,p , v1,p ) = wkp ,
and
∗ ∗ wpk = τ0 (v0,k , wpk v0,k ) = τ0 (v0,k , w∗ (v1,p ))
= τ1 (w(v0,k ), v1,p ) = τ1 (−wkp v1,p , v1,p ) = wkp .
Thus, ∗ ∗ wpk = wkp , wpk = wkp .
(13.28)
2
For y, z ∈ gx1 let w = y|V0 and let u = z|V0 . Then w, u ∈ Hom(V0 , V1 ) commute with x2 and, by (2.13), (13.27) and (13.28), 1 y, z = tr(w∗ u) 4 n n ∗ ∗ =2 (ukl wlk − ukl wlk ) = 2 (ukl wkl − ukl wkl ). k,l=1
k,l=1
(13.29)
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We calculate using (13.24), (13.27) and (13.28): 1 ( wlk − wlk )v0,l − (wlk + wlk )v0,l , yx :v0,k → ξ l=1 l=1 n
v0,k
1 xy :v0,k ξ v0,k
Hence,
n
n n → ( wlk + wlk )v0,l − (wlk − wlk )v0,l , l=1
l=1
l=1
l=1
l=1
l=1
n n → (wkl − wkl )v0,l − (wkl + wkl )v0,l ,
(13.30)
n n → ( wkl + wkl )v0,l − ( wkl − wkl )v0,l
y ∈ gx1 if and only if wkl = wlk ,
y ∈ x g1 if and only if wkl = − wlk .
(13.31)
It is clear from (13.29) and (13.31) that the spaces x g1 , gx1 are isotropic, and that the sym2 2 plectic form provides a non-degenerate pairing between them. The supergroup (Gx , gx ) is irreducible, of type I, and the corresponding dual pair is isomorphic to (Un , Un ). Furthermore a straightforward computation shows that (x(ξ,) g1 )
x(ξ,)
(g
g1 = g1 1
)
= {x(ζ, ); ζ ∈ R}.
(13.31.1)
Case 5.2.d. Here D = R and the vector spaces Vi (i = 0, 1) have basis ui,1 , ui,2 , ui,3 , ..., ui,n , ui,1 , ui,2 , ui,3 , ..., ui,n , vi,1 , vi,2 , vi,3 , ..., vi,n , vi,1 , vi,2 , vi,3 , ..., vi,n ,
such that τ (ui,k , ui,k ) = τ (vi,k , vi,k ) = 1,
(13.32)
(i = 0, 1; k = 1, 2, 3, ..., n)
and all the other pairings are zero. Moreover, x = x(ξ, η) :u0,k → ξu1,k + ηv1,k , u1,k → ξu0,k + ηv0,k , v0,k → −ηu1,k + ξv1,k , v1,k → −ηu0,k + ξv0,k , u0,k → −ξu1,k + ηv1,k , u1,k → ξu0,k − ηv0,k ,
(13.33)
→ −ηu1,k − ξv1,k , v1,k → ηu0,k + ξv0,k , v0,k
Therefore, with α = ξ 2 − η 2 and β = 2ξη, x2 :u0,k → αu0,k + βv0,k , u1,k → αu1,k + βv1,k , v0,k → −βu0,k + αv0,k , v1,k → −βu1,k + αv1,k , u0,k → −αu0,k + βv0,k , u1,k → −αu1,k + βv1,k ,
(13.34)
→ −βu0,k − αv0,k , v1,k → −βu1,k − αv1,k . v0,k
Since α, β = 0, (13.34) and (13.32) imply 2
2
2
Gx = Gx |V0 × Gx |V1 = GLn (C) × GLn (C).
(13.35)
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Suppose w ∈ Hom(V0 , V1 ) commutes with x2 . Then, by (13.34), w maps the span of the u0,k , v0,k to the span of the u1,k , v1,k and the span of the u0,k , v0,k to the span of the ∗ ∗ ∗ ∗ u1,k , v1,k . More precisely, there are numbers wkl , w˜kl , wkl , w˜kl , wkl , w˜kl , wkl , w˜kl ∈ R such that n w(u0,k ) = (wkl u1,l + w˜kl v1,l ), l=1
w(v0,k ) =
n
(−w˜kl u1,l + wkl v1,l ),
l=1
n w(u0,k ) = (wkl u1,l + w˜kl v1,l ),
(13.36)
l=1
)= w(v0,k
n
(−w˜kl u1,l + wkl v1,l ).
l=1
and ∗
w (u1,k ) =
n
∗ ∗ (wkl u0,l + w˜kl v0,l ),
l=1
n ∗ ∗ ∗ w (v1,k ) = (−w˜kl u0,l + wkl v0,l ), l=1
w
∗
(u1,k )
=
n
(13.37) ∗ (wkl u0,l
+
∗ w˜kl v0,l ),
l=1
n ∗ ∗ ∗ w (v1,k ) = (−w˜kl u0,l + wkl v0,l ). l=1
Using (13.32), (13.36) and (13.37) we show that ∗ ∗ ∗ ∗ = −wlk , wkl = wlk , w˜kl = w˜lk , w˜kl = −w˜lk . wkl
(13.38)
2
For y, z ∈ gx1 let w = y|V0 and let u = z|V0 . Then 1 y, z = tr(w∗ u) 4 n ∗ ∗ ∗ ∗ (ukl wlk − u˜kl w˜lk + ukl wlk − u˜kl w˜lk ) =2 =2
k,l=1 n k,l=1
(wkl ukl + w˜kl u˜kl − wkl ukl − w˜kl u˜kl )
(13.39)
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Furthermore, yx :u0,k
n → ((−ξwlk − ηw ˜lk )u0,l + (ξ w˜lk − ηwlk )v0,l ),
v0,k →
l=1 n
((ηwlk − ξ w˜lk )u0,l + (−η w˜lk − ξwlk )v0,l ),
l=1
u0,k
n → ((−ξwlk + η w˜lk )u0,l + (ξ w˜lk + ηwlk )v0,l ),
(13.40)
l=1
v0,k →
n
((−ηwlk − ξ w˜lk )u0,l + (η w˜lk − ξwlk )v0,l ),
l=1
and xy :u0,k →
n ((ξwkl − η w ˜kl )u0,l + (ηwkl + ξ w˜kl )v0,l ), l=1
v0,k → u0,k
→
n
((−ξ w˜kl − ηwkl )u0,l + (−η w˜kl + ξwkl )v0,l ),
l=1 n
(13.41)
((ξwkl
+
ηw ˜kl )u0,l
+
(−ηwkl
+
ξ w˜kl )v0,l ),
l=1
v0,k →
n
((−ξ w˜kl + ηwkl )u0,l + (η w ˜kl + ξwkl )v0,l ).
l=1
Hence,
y ∈ gx1 if and only if ξwkl − η w˜kl + ξwlk + ηw ˜lk = 0,
ηwkl + ξ w˜kl + ηwlk − ξ w˜lk = 0;
− ηw ˜lk = 0, y ∈ x g1 if and only if ξwkl − η w˜kl − ξwlk ηwkl + ξ w˜kl − ηwlk + ξ w˜lk = 0.
Thus
y ∈ gx1 if and only if wkl = −wlk , w˜kl = w˜lk ;
y ∈ x g1 if and only if wkl = wlk , w˜kl = −w˜lk .
(13.42)
It is easy to see from (13.39) and (13.42) that the spaces x g1 , gx1 are isotropic, and that the symplectic form provides a non-degenerate pairing between them. The supergroup 2 2 (Gx , gx ) is irreducible, of type II, and as a dual pair is isomorphic to (GLn (C), GLn (C)). Furthermore a straightforward computation shows that (x(ξ,η) g1 )
x(ξ,η)
(g
g1 = g1 1
)
= {x(ζ, γ); ζ, γ ∈ R}.
(13.42.1)
Case 5.3.a The spaces V0 , V1 have basis vi,1 , vi,2 , vi,3 , ..., vi,n
(i = 0, 1)
such that x(ξ) : v0,k → ξv1,k , v1,k → ξv0,k .
(13.43)
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Hence, x2 : v0,k → ξ 2 v0,k , v1,k → ξ 2 v1,k .
Therefore G
x2
x2
x2
= G |V0 × G |V1 =
(13.44)
GLn (D) × GLn (D) if ξ 2 ∈ R, GLn (C) × GLn (C) if ξ 2 ∈ / R.
(13.45)
Suppose A ∈ Hom(V0 , V1 ) and B ∈ Hom(V1 , V0 ) commute with x2 . Then there are elements ak,l , bk,l ∈ D commuting with ξ 2 such that A : v0,k →
n
ak,l v1,l , B : v1,k →
l=1
n
bk,l v0,l .
l=1
Furthermore the formula (v0 ∈ V0 , v1 ∈ V1 )
y(v0 + v1 ) = Bv1 + Av0 2
2
defines an element y ∈ gx1 and all elements of gx1 may be described as above. Suppose y ∈ 2 gx1 . Let A , B be the corresponding elements of Hom(V0 , V1 ), Hom(V1 , V0 ) respectively. Then, by (2.6), n 1 trD/R (bk,l al,k − bk,l al,k ). y, y = trD/R (BA − B A) = 2 k,l=1
(13.46)
We see from (13.43) that y ∈ gx1 if and only if bk,l = ξak,l ξ −1 ,
y ∈ x g1 if and only if bk,l = −ξak,l ξ −1 .
(13.47)
It is clear from (13.46) and (13.47) that x g1 and gx1 are isotropic subspaces of g1 and that the symplectic form , provides a non-degenerate pairing between them. The 2 2 Lie supergroup (Gx , gx ) is of type II, is irreducible and the corresponding dual pair is isomorphic to (GLn (C), GLn (C)) or (GLn (D), GLn (D)), as indicated in (13.45). Furthermore a straightforward computation shows that (x(ξ) g1 )
x(ξ)
(g
g1 = g1 1
)
= {x(ζ); ζ 2 ∈ D(D
ξ2 )
}.
(13.47.1)
Case 5.3.a’ Here the spaces V0 , V1 have basis vi,1 , vi,2 , vi,3 , ..., vi,n
(i = 0, 1),
x = x(ξ) : v0,k → ξv1,k , v1,k → −ξv0,k , (ξ ∈ R \ 0), (x(ξ) g1 )
x(ξ)
(g
g1 = g1 1
and the proof is as in the previous case.
)
= {x(ζ); ζ ∈ R}.
(13.47.2)
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Case 5.3.b Here the spaces V0 , V1 have basis ui,1 , ui,2 , ui,3 , ..., ui,n ; vi,1 , vi,2 , vi,3 , ..., vi,n such that
(i = 0, 1)
x = x(ξ, η) :u0,k → ξu1,k + ηv1,k , u1,k → ξu0,k + ηv0,k , v0,k → −ηu1,k + ξv1,k , v1,k → −ηu0,k + ξv0,k
(13.48)
Therefore, with α = ξ 2 − η 2 and β = 2ξη, x2 :u0,k → αu0,k + βv0,k , u1,k → αu1,k + βv1,k , v0,k → −βu0,k + αv0,k , v1,k → −βu1,k + αv1,k
(13.49)
Hence, 2
2
2
Gx = Gx |V0 × Gx |V1 = GLn (C) × GLn (C).
(13.50)
2
Suppose y ∈ gx1 . Then there are numbers ak,l , a ˜k,l , bk,l , ˜bk,l in R such that y :u0,k →
n n (ak,l u1,l + a ˜k,l v1,l ), u1,k → (bk,l u0,l + ˜bk,l v0,l ), l=1
v0,k →
n
l=1 n
(−˜ ak,l u1,l + ak,l v1,l ), v1,k →
l=1
(13.51) (−˜bk,l u0,l + bk,l v0,l ).
l=1
2
If y ∈ gx1 , then (with the notation (13.51)), n 1 (ak,l bl,k − a ˜k,l˜bl,k − ak,l bl,k + a ˜k,l˜bl,k ). y, y = 2 k,l=1
(13.52)
Furthermore yx :u0,k →
n
((ξbk,l − η˜bk,l )u0,l + (ξ˜bk,l + ηbk,l )v0,l ),
l=1
v0,k →
n
((−ηbk,l − ξ˜bk,l )u0,l + (−η˜bk,l + ξbk,l )v0,l )
l=1
and xy :u0,k →
n
((ξak,l − η˜ ak,l )u0,l + (ξ˜ ak,l + ηak,l )v0,l ),
l=1
v0,k
n → ((−ξ˜ ak,l − ηak,l )u0,l + (ξak,l − η˜ ak,l )v0,l ) l=1
Therefore, y ∈ gx1 if ak,l = bk,l , a ˜k,l = ˜bk,l ; y ∈ x g1 if ak,l = −bk,l , a ˜k,l = −˜bk,l .
(13.53)
It is clear from (13.52) and (13.53) that x g1 and gx1 are isotropic subspaces of g1 and that the form , provides a non-degenerate pairing between them. The Lie supergroup
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2
(Gx , gx ) is irreducible, of type II and the corresponding dual pair is isomorphic to (GLn (C), GLn (C)). Furthermore a straightforward computation shows that (x(ξ,η) g1 )
x(ξ,η)
(g
g1 = g1 1
)
= {x(ζ, γ); ζ, γ ∈ R}.
(13.53.1)
This completes the proof of Theorem 4.4. Proof (of Lemma 6.4). Since the elements x and y commute, they preserve the same isotypic decomposition of V . For a fixed isotypic component, all the sets which occur in Lemma 6.4 (a), (b) and (c) are described in (13.11), (13.21), (13.31), (13.42), (13.47) and (13.53). One checks the equalities (a), (b), (c) via a case by case analysis. Similarly one verifies (d) and (e).
14
A proof of Theorem 4.5
Consider the map G × gx1 (g, y) → gy ∈ g1 .
(14.1)
The derivative of (14.1) at (g, y) coincides with the following linear map g0 × gx1 (A, B) → [A, gy] + gB ∈ g1 .
(14.2)
The range of the map (14.2) is equal to [g0 , gy] + g(gx1 ) = g([g0 , y] + gx1 ).
(14.3)
We see from Lemma 3.1 and Theorem 4.4 that [g0 , x] + gx1 = (x g1 )⊥ + gx1 = g1 .
(14.4)
U = {y ∈ gx1 ; [g0 , y] + gx1 = g1 }
(14.5)
Hence, the set is non-empty. The set U is open and Gx -invariant. Furthermore, (14.3) shows that the map (14.1) restricted to G × U is a submersion. The set U satisfies the conditions (1.0), (1.1), (1.2) and (1.5). Suppose we have a non-empty open subset U˜x ⊆ gx1 such that 2
2
(1.4) holds for the supergroup (Gx , gx ) : 2 if g ∈ Gx and y, y ∈ U˜x are such that gy = y , then g ∈ Gx . 2
(14.6)
Since x2 is semisimple, there is an admissible slice Ux2 through x2 in gx0 , with respect to the group G. We may assume that U˜x is contained in the preimage of Ux2 under the map 2 gx1 y → y 2 ∈ gx0 . (14.7)
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Then for y, y ∈ U˜x and g ∈ G such that gy = y we have gy 2 = y 2 . Hence g ∈ Gx . But this together with (14.6) implies the g ∈ Gx . Thus U˜x satisfies (1.3). Hence, if we set Ux = U˜x ∩ U , where U is the set defined in (14.5), then Ux satisfies all the conditions (1.0)-(1.5). Next we shall verify the statement (14.6). The Theorem 4.4 implies that we may 2 2 assume that either x = 0 or x = 0 and that the supergroup (Gx , gx ) corresponds either to the dual pair (Un , Un ) or to (GLn (D), GLn (D)). Since any G-invariant open neighborhood of 0 in g1 is an admissible slice through 0, we may assume that x = 0. We proceed via a case by case analysis performing the computations in terms of matrices. It will be clear from what follows that the sets constructed may be made arbitrarily small and thus form the desired basis for the neighborhoods of x in gx1 . Case (Un , Un ). Set W = Mn (C), V0 = V1 = Cn and τ0 (u0 , v0 ) = v T0 u0 , τ1 (u1 , v1 ) = v T1 iu1
(u0 , v0 ∈ V0 , u1 , v1 ∈ V1 ).
The space W is identified with Hom(V0 , V1 ) by w(v) = wv, w ∈ W , v ∈ V0 . Then w∗ = −iwT
(w ∈ W ).
The restriction of x to Hom(V0 , V1 ) coincides with ξI ∈ W , where ξ ∈ iR \ 0, im(ξ 2 ) < 0. π Thus ξ = re− 4 i , where r ∈ R \ 0. A straightforward calculation using the formula (2.12) shows that under the identification g1 = g1 |V0 = Hom(V0 , V1 ) = W we have x
For 0 < ≤
T
T
g1 = {ξA; A = −A ∈ W } and gx1 = {ξH; H = H ∈ W }.
1 2
(14.8)
let U˜x, be the set of all points w ∈ gx1 such that
λ1 + λ2 = 0 and |λ − ξ| < |ξ| for all eigenvalues λ, λ1 , λ2 of w.
(14.9)
Let be the operator norm on W , viewed as the space of operators on the Hilbert space (V0 , τ0 ). Suppose g, h ∈ Un and ξH ∈ U˜x, are such that ξgHh−1 ∈ U˜x, . Then, by (14.9), H − I < and gHh−1 − I < . Hence H − g −1 h < , and by the triangle inequality g −1 h − I < 2 . Since 2 ≤ 1, we have A = log(g −1 h) ∈ un . Thus g −1 h = exp(A).
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Since ξgHh−1 ∈ U˜x, , we have
T
gHh−1 = gHh−1 . Thus hHg −1 = gHh−1 , or equivalently Hg −1 h = h−1 gH. Therefore H exp(A) = exp(−A)H. Since, by (14.9), H is invertible and since log is injective, this last condition may be expressed as HAH −1 = −A, or equivalently as HA + AH = 0.
(14.10)
Conjugating both sides of (14.10) by an appropriate element of Un we may assume that H is diagonal. Then (14.9) shows that A = 0. Hence g −1 h = I, i.e. g = h. Since the diagonal subgroup {(g, g) ∈ Un × Un } coincides with Gx , we see that the set U˜x, satisfies (1.3). This also shows that the derivative of the map H → H 2 at H is an injective linear map. Furthermore, since H is close to the identity, it is positive definite. Thus H is the unique positive definite square root of H 2 . Hence the map H → H 2 is injective. Case (GLn (D), GLn (D)). For α > 0 let Mn (C)[α] = {A ∈ Mn (C); |Im(a)| < α for all eigenvalues a of A}, and let GLn (C)[α] = exp(Mn (C)[α]). Then, as is well known [11][part. II, p. 17], exp : Mn (C)[π] → GLn (C)[π]
(14.11)
is a bijective analytic diffeomorphism. Moreover, the closure, Cl(GLn (C)[α]) ⊆ GLn (C)[β]
(0 < α < β ≤ π).
(14.12)
Let Mn (C) ⊆ Mn (C) be the set of all matrices A such that the map Mn (C) B → AB + BA ∈ Mn (C)
(14.13)
is surjective. Clearly Mn (C) is a Zariski open, Ad(GLn (C))-invariant neighborhood of the identity I ∈ Mn (C). Lemma 14.1. For any 0 < α < β < π there is an open Ad(GLn (C))-invariant neighborhood of the identity −1 Vα,β = Vα,β ⊆ GLn (C)[α] ∩ Mn (C)a
such that GLn (C)[α] Vα,β ⊆ GLn (C)[β].b
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Proof. The set of eigenvalues of a matrix A ∈ Mn (C) may be viewed as an orbit in Cn under the action of the permutation group. The family of all such orbits has a natural topology. In these terms the set of eigenvalues of a matrix A ∈ Mn (C), is a continuous function. Let S 1 ⊆ Mn (C) denote the unit sphere with respect to the operator norm . Since by Jordan’s Theorem GLn (C)[α] = {A ∈ Mn (C); a = 0, |Arg(a)| < α for all eigenvalues a of A}
(14.14)
the previous paragraph shows that there is an open neighborhood V (α,β) ⊆ GLn (C) of the identity I, such that (S 1 ∩ GLn (C)[α]) V (α,β) ⊆ GLn (C)[β].
(14.15)
Notice that the set (14.14) is closed under the dilations A → tA, t > 0. Hence, (14.15) implies (14.16) GLn (C)[α]V (α,β) ⊆ GLn (C)[β]. Let V α,β = Ad(GLn (C))V (α,β) . As the union of open sets, V α,β is open. Clearly, V α,β is Ad(GLn (C))-invariant and contains the identity. The inclusion (14.16) together with the Ad(GLn (C))-invariance of the sets GLn (C)[γ], γ = α, β, implies (b). Hence the Lemma holds for Vα,β = V α,β ∩ α,β −1 V . Corollary 14.2. Let Vα,β be as in the Lemma 14.1. Suppose A ∈ Vα,β and g, h ∈ GLn (C) are such that hAg −1 = gAh−1 ∈ Vα,β . Then g = h. In particular, if A, B ∈ Vα,β and A2 = B 2 , then A = B. Furthermore, the derivative of the map A → A2 at A ∈ Vα,β is injective. Proof. Set u = g −1 h. Then uA ∈ Vα,β . Since A−1 ∈ Vα,β , Lemma 14.1 implies u = (uA)A−1 ∈ Vα,β Vα,β ⊆ GLn (C)[α] Vα,β ⊆ GLn (C)[β] ⊆ GLn (C)[π]. Hence there is a unique B ∈ Mn (C)[π] such that u = exp(B). Furthermore, uA = g −1 hA = Ah−1 g = Au−1 . Hence exp(B)A = Aexp(−B), or equivalently exp(A−1 BA) = exp(−B).
(14.17)
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Since A−1 BA and −B belong to Mn (C)[π], (14.17) implies A−1 BA = −B, or equivalently AB + BA = 0.
(14.18)
Since A ∈ Mn (C), (14.18) implies B = 0, which means that u = 1. Thus g = h. Let A, B ∈ Vα,β . Then A2 = B 2 is equivalent to AAB −1 = BAA−1 , which implies A = B. The injectivity of the derivative follows from the fact that A ⊆ Mn (C). Let D = R, C or H and let ξ ∈ D \ 0 be such that ξ 2 is in the center of D. Set ⎛ ⎞ ⎜ 0 ξI ⎟ x=⎝ (14.19) ⎠. ξI 0 Under the usual identifications we have
⎛
⎞
⎜ 0 A⎟ g1 = End(Dn ⊕ Dn )1 = {⎝ ⎠ ; A, B ∈ Mn (D)}, B 0 ⎛ ⎞ ⎜g 0 ⎟ G = GL(Dn ⊕ Dn )0 = {⎝ ⎠ ; g, h ∈ GL(Dn )}. 0h A straightforward calculation shows that ⎞ ⎛ −1 ⎜ 0 ξBξ ⎟ gx1 = {⎝ ⎠ ; B ∈ Mn (D)}, B 0 ⎞ ⎛ ⎜g 0 ⎟ n Gx = {⎝ ⎠ ; g ∈ GL(D )}. 0 ξgξ −1
(14.20)
(14.21)
Let Vα,β be the set constructed in Lemma 14.1 for the group GLn (D) if D = C, and for the complexification of GLn (D) if D = C. Set Vα,β (D) = Vα,β ∩ GLn (D).
(14.22)
Then Vα,β (D) is an open Ad(GLn (D))-invariant neighborhood of the identity I ∈ GLn (D). Let ⎞ ⎛ −1
⎜ 0 ξBξ ⎟ Uα,β,ξ (D) ={⎝ ⎠ ; B ∈ ξVα,β (D)} B 0 ⎛ ⎞ ⎜ 0 Aξ ⎟ ={⎝ ⎠ ; A ∈ Vα,β (D)}. ξA 0
(14.23)
T. Przebinda / Central European Journal of Mathematics 4(3) 2006 449–506
503
The set (14.23) is an open Gx -invariant neighborhood of x in gx1 . Lemma 14.3. Suppose y, z ∈ Uα,β,ξ (D) and s ∈ G are such that sys−1 = z. Then s ∈ Gx . Proof. Let
⎛ ⎜g s=⎝ 0
⎞
⎛
⎞
0⎟ ⎠ h
⎜ 0 Aξ ⎟ and y = ⎝ ⎠. ξA 0 ⎞
⎛
Then
⎜ sys−1 = ⎝
0 hξAg −1
−1
gAξh ⎟ ⎠ 0
Since sys−1 ∈ Uα,β,ξ (D), we have gAξh−1 = ξ(hξAg −1 )ξ −1 = ξhξ −1 Ag −1 ξ. Thus (ξhξ −1 )Ag −1 = gA(ξh−1 ξ −1 ).
(14.24)
Furthermore hξAg −1 ∈ ξUα,β,ξ (D). Thus (ξhξ −1 )Ag −1 = ξ −1 hξAg −1 ∈ Uα,β,ξ (D).
(14.25)
By combining (14.24), (14.25) and Corollary 14.2 we see that ξhξ −1 = g, so that h = ξ −1 gξ = ξgξ −1 . The Lemma 14.3 shows that the set (14.23) satisfies the condition (1.3). The rest is also clear from Corollary 14.2.
15
A proof of Theorem 4.7
The ideas presented below originate in [4, sec. 12]. Recall the following Lemma. Lemma 15.1. [12, 8.A.4.5] Let N be a complete metric space and let G be a σ-compact topological group acting on N . Suppose N is the union of a finite number of G-orbits. The we can label the orbits O1 , O2 , ..., Ok , so that for each 1 ≤ j ≤ k, the set Nj =
k
Ol is closed in N.
l=j
We apply the above Lemma to our ordinary classical Lie supergroup (G, g), with N ⊆ g1 equal to the set of nilpotent elements. As is well known, [3], N is the union of a finite number of G-orbits.
504
T. Przebinda / Central European Journal of Mathematics 4(3) 2006 449–506
In particular, for each 1 ≤ j ≤ k there is an open G-invariant set Wj ⊆ g1 such that Wj ∩ Nj = Oj .
(15.1)
Fix 1 ≤ j ≤ k and an element x ∈ Oj . Let U ⊆ g1 be a subspace complementary to the tangent space to the orbit through x. Thus g1 = [g0 , x] ⊕ U.
(15.2)
For each z ∈ U we have a linear map Tz : g0 ⊕ U (y, z ) → [y, x + z] + z ∈ g1 .
(15.3)
Notice that, by (15.2), T0 is surjective. Further, the map U z → Tz ∈ Hom(g0 ⊕ U, g1 ) is affine and hence continuous. Therefore the set of all z ∈ U such that rank(Tz ) ≥ rank(T0 ) (= dim g1 )
(15.4)
is an open neighborhood of 0 in U . Let us denote this neighborhood by U1 . Let Φ : G × U1 (g, z) → g(x + z)g −1 ∈ g1 .
(15.5)
The derivative of Φ at (g, z) coincides with the following linear map g0 ⊕ U (y, z ) → g([y, x + z] + z )g −1 ∈ g1 .
(15.6)
By (15.4), the map (15.6) is surjective. Thus Φ is a submersion. Recall the set Wj , (15.1). Let U2 = {z ∈ U1 ; x + z ∈ Wj }.
(15.7)
Φ(G × U2 ) ⊆ Wj .
(15.8)
Then Let W ⊆ g0 be a subspace complementary to the kernel of the map ad(x) : g0 y → [y, x] ∈ g1
(15.9)
W y → [y, x] ∈ [g0 , x]
(15.10)
so that the map is a linear bijection. By (15.6), the derivative of the map W × U2 (y, z) → Φ(exp(y), z) = exp(y)(x + z) exp(−y) ∈ g1
(15.11)
at (0, 0) coincides with the following linear map W ⊕ U (y, z ) → [y, x] + z ∈ g1 ,
(15.12)
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505
which, by (15.2) and (15.10), is a linear bijection. Hence, there is an open neighborhood W1 of 0 in W and an open neighborhood U3 of 0 in U2 , such that the map W1 × U3 (y, z) → exp(y)(x + z) exp(−y) ∈ g1
(15.13)
is an diffeomorphism onto an open neighborhood of x in g1 . Let W2 ⊆ W1 be an open neighborhood of 0 such that exp(ad(W2 ))x ⊆ Nj is an open neighborhood of x in Nj . This is compatible with (15.8). Choose an open neighborhood W0 ⊆ W2 of 0 and an open neighborhood U0 ⊆ U3 of 0 such that Φ(exp(W0 ), U0 ) ∩ Nj ⊆ exp(ad(W2 ))x ∩ Nj .
(15.14)
Suppose z ∈ U0 is such that x + z ∈ Oj .
(15.15)
Since x + z = Φ(exp(0), z), (15.14) implies that there is y ∈ W2 such that x + z = exp(y) x exp(−y).
(15.16)
Φ(exp(0), z) = Φ(exp(y), 0).
(15.17)
Thus But then (15.13) implies that y = 0 and z = 0. Thus (x + U0 ) ∩ Oj = {x}.
(15.18)
References [1] N. Burgoyne and R. Cushman: “Conjugacy Classes in Linear Groups”, J. Algebra, Vol. 44, (1975), pp. 339–362. [2] D. Collingwood and W. McGovern: Nilpotent orbits in complex semisimple Lie algebras, Reinhold, Van Nostrand, New York, 1993. [3] A. Daszkiewicz, W. Kra´skiewicz and T. Przebinda: “Dual Pairs and KostantSekiguchi Correspondence. II. Classification of Nilpotent Elements”, Centr. Eur. J. Math., Vol. 3, (2005), pp. 430-464. [4] Harish-Chandra: “Invariant Distributions on Lie algebras”, Amer. J. of Math., Vol. 86, (1964), pp. 271–309. [5] R. Howe: Analytic Preliminaries, preprint. [6] R. Howe: “Remarks on classical invariant theory”, Trans. Amer. Math. Soc., Vol. 313, (1989), pp. 539–570, [7] R. Howe: “Transcending Classical Invariant Theory”, J. Amer. Math. Soc., Vol. 2, (1989), pp. 535–552. [8] V. Kac: “Lie superalgebras”, Adv. Math., Vol. 26, (1977), pp. 8–96.
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[9] B. Kostant: Graded manifolds, graded Lie theory, and prequantization, Lecture Notes in Math., Vol. 570, Springer-Verlag, Berlin-New York, 1977, pp. 177–306. [10] M. Spivak: A comprehensive introduction to differential geometry, Brandeis University, Waltham, Massachusetts, 1970. [11] V.S. Varadarajan: Harmonic Analysis on Real Reductive Groups I and II,Lecture Notes in Math., Vol. 576, Springer Verlag, 1977. [12] N. Wallach: Real Reductive Groups I, Academic Press, INC, 1988.
DOI: 10.2478/s11533-006-0015-8 Research article CEJM 4(3) 2006 507–524
Duality triads of higher rank: Further properties and some examples Matthias Schork∗ Alexanderstr. 76, 60489 Frankfurt, Germany
Received 6 February 2006; accepted 27 February 2006 Abstract: It is shown that duality triads of higher rank are closely related to orthogonal matrix polynomials on the real line. Furthermore, some examples of duality triads of higher rank are discussed. In particular, it is shown that the generalized Stirling numbers of rank r give rise to a duality triad of rank r. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Duality triad, recurrence relation, orthogonal polynomials MSC (2000): 05Axx, 11B37, 11B83
1
Introduction
In [10, 11] the notion of duality triad was introduced. By this the following system is meant. Let three sequences i = {ik }k≥0 , q = {qk }k≥0 , and d = {dk }k≥0 of complex numbers with ik = 0 be given. For such a triple (i, q, d) we introduce the following “dynamical system”. The discrete time steps n start with 0 and at every time n we consider (i,q,d) the sequence cn = {cn,k }k≥0 ≡ {cn,k }k≥0 of complex numbers satisfying c0,k = δ0,k and the recursion relation cn+1,k = ik−1 cn,k−1 + qk cn,k + dk+1 cn,k+1 .
(1)
Given the initial values and the triple of sequences, the coefficients cn,k are uniquely (i,q,d) determined. Let us consider the polynomial sequence {Φn (x)}n≥0 ≡ {Φn (x)}n≥0 (i.e., deg Φn (x) = n) satisfying Φ0 (x) = 1 and the recursion relation xΦn (x) = dn Φn−1 (x) + qn Φn (x) + in Φn+1 (x). ∗
E-mail: [email protected]
(2)
M. Schork / Central European Journal of Mathematics
This recursion relation for the sequence of triad polynomials Φn (x) is dual to the relation (1) in the sense that the following inversion relation holds [10, 11]: xn = cn,k Φk (x). (3) k≥0
This system of numbers cn,k and polynomials Φn (x) satisfying (1), (2), and (3) for the given triple of sequences (i, q, d) is called duality triad (associated to (i, q, d)) [10, 11]. As stressed in [10], these duality triads are dual recurrences satisfying (1) and (2) as used in dynamical data structure theory [8, 9] which furthermore satisfy a third relation, the inversion relation (3). In general, the coefficients cn,k are thus the expansion coefficients (or connection coefficients) of the monomials xn in the basis of polynomials {Φk (x)}k≥0 . In [4, 10–12, 18] these triads were studied and several well-known sequences of combinatorial numbers cn,k and polynomials Φn (x) were shown to be special cases. Due to (2) the triad polynomials are very closely related to orthogonal polynomials on the real line. In [18] a generalization of duality triads to higher rank (where the rank one case corresponds to the usual duality triads) was suggested and it was conjectured that the Stirling numbers of higher rank introduced in [2, 3] should yield an interesting example of such a generalized duality triad. A precise definition of duality triads of higher rank was given in [19] and first properties were derived. However, only toy examples were given in [19]. It is the aim of the present paper to present some concrete examples of duality triads of higher rank. In particular, it is shown that the generalized Stirling numbers of higher rank (and their q-deformed analogues introduced in [17]) give indeed rise to duality triads of higher rank. The main structural result of this paper is a description of the intimate connection between duality triads of higher rank and orthogonal matrix polynomials on the real line, generalizing the above-mentioned connection of the rank one case. The structure of the paper is as follows. In Section 2 some examples of duality triads (of rank one) and their relation to dynamical data structures and orthogonal polynomials are discussed. In Section 3 we recall the definition of duality triads of higher rank, introduce the important class of Hermitian duality triads and give a possible interpretation of the defining recursion relation in terms of dynamical data structures. In Section 4 some properties of orthogonal matrix polynomials are recalled and the connection between Hermitian duality triads of rank r and orthogonal matrix polynomials of rank r on the real line is established. Some explicit examples of duality triads of higher rank are discussed in Section 5. Finally, in Section 6 some conclusions are presented.
2
Duality triads: Some examples and properties
In this section we describe a few examples of duality triads which will play a prominent role later on. Many more examples can be found in [4, 10–12, 18]. We also make some remarks concerning the relationship between duality triads and dynamical data structures as discussed in [8] and orthogonal polynomials on the real line.
M. Schork / Central European Journal of Mathematics
Example 2.1. (Pascal triad) ik = 1, qk = 1, dk = 0. The corresponding cn,k satisfy the recursion relation cn+1,k = cn,k−1 + cn,k and are given by the binomial coefficients, i.e., cn,k = nk . The triad polynomials are given by Φk (x) = (x − 1)k so that the inversion relation (3) becomes in this case n
x =
n n
k
k=0
(x − 1)k .
(4)
Example 2.2. (Stirling triad) ik = 1, qk = k, dk = 0. The corresponding cn,k satisfy cn+1,k = cn,k−1 + kcn,k and are given by the Stirling numbers of second kind, i.e., cn,k = S(n, k) where (see [21]) k (−1)k p k S(n, k) = pn . (−1) p k! p=0
(5)
The triad polynomials are given by Φk (x) = x(x − 1) · · · (x − k + 1) =: xk and (3) becomes in this case n n S(n, k)xk . (6) x = k=0
Example 2.3. (q-deformed Stirling triad) ik = q k , qk = kq , dk = 0, where the basic n k nq ! q-numbers are given by kq = 1−q , mq ! = m k=1 kq and k q = kq !(n−k)q ! [1]. The corre1−q sponding cn,k satisfy cn+1,k = q k−1 cn,k−1 + kq cn,k and are given by the q-deformed Stirling numbers of second kind, i.e., cn,k = S(n, k|q) where (see [17] and the references therein) k (−1)k k p (k−p ) 2 S(n, k|q) = (−1) q pn . p q q kq ! p=0
(7)
As a reference for later discussions we denote the recursion relation explicitly, S(n + 1, k|q) = q k−1 S(n, k − 1|q) + kq S(n, k|q).
(8)
Introducing χk (x) := x(x − 1q ) · · · (x − (k − 1)q ), the triad polynomials are given by k Φk (x) = q −(2) χk (x) so that (3) becomes xn =
n
k
q −(2) S(n, k|q)χk (x).
(9)
k=0
Introducing xkq := xq (x − 1)q · · · (x − k + 1)q , a more natural relation generalizing (6) is xnq
=
n
S(n, k|q)xkq .
(10)
k=0
Choosing the special value q = 1 leads to Example 2.2. Note, however, that (10) is not an example of the inversion relation (3) in the strict sense since the occurring functions are
M. Schork / Central European Journal of Mathematics
polynomials in xq but not in x. For further discussions see Remark 5.8. As discussed in [18] it is possible to generalize this example to various different generalizations of Stirling numbers, e.g., to the Stirling numbers occurring in “ψ-extended umbral calculus” (see [13] for an up-to-date account). Remark 2.4. (Dynamical data structures) Let us discuss briefly the relation between duality triads and dynamical data structures as discussed in [8]. As a concrete example of a data structure we think of a file containing some items (representing, e.g., a list or a stack). Let us denote by cn,k the number of possible paths, i.e., histories, of the given file starting from height 0 (i.e., the empty file) that are at height (= file size) k at time n. We think of the the height as the location of a particle on the line. It jumps left or right or sits at each time step according to whether the operation Insertion I, Deletion D or a Query Q is applied to the file (here we assume that that only these operations are permitted for the corresponding data structure). Let us denote by N pos(O, k) the number of possibilities for performing the operation O (where O ∈ {I, D, Q}) on the file of size k. Abbreviating ik := N pos(I, k), dk := N pos(D, k), qk := N pos(Q, k), it is then clear that cn,k satisfies the recursion relation (1). In order to interpret (1) in this setting the coefficients ik , dk , qk should be non-negative integers. Remark 2.5. (Orthogonal polynomials on the real line) We briefly review some basic facts about orthogonal polynomials following the standard references [5, 22] (see also the recent survey [23] discussing many new developments). Let μ be a positive Borel measure (with an infinite number of points in its support) with support on the real line such that all moments μn := R xn dμ(x) exist. Then there exist unique polynomials pn (x) = κn xn + · · · with κn > 0 that form an orthonormal system in L2 (R, μ), i.e., pm (x) dμ(x) = δn,m . The pn ’s are called orthonormal polynomials (pn , pm ) := R pn (x)¯ corresponding to μ. One of the most remarkable consequences of the fact that μ is supported on the real line is that the pn ’s obey a three-term recurrence relation xpn (x) = an+1 pn+1 (x) + bn pn (x) + an pn−1 (x)
(11)
n where the coefficients can be determined by an = κκn−1 > 0 and bn = R xp2n (x) dμ(x). Conversely, any system of polynomials satisfying (11) with real an > 0, bn is an orthonormal system with respect to a (not necessarily unique) measure μ on the real line (Favard’s theorem). Comparing (11) with (2) shows the close connection of the triad polynomials Φn (x) to orthogonal polynomials. The main difference is that the tridiagonal transfer (l) matrix T of a duality triad (15) is not necessarily symmetric (depending on μn ), see Theorem 3.5.
3
Duality triads of higher rank
Let us introduce a concise notation following [18, 19]. Thus, (−1)
μk
≡ ik ,
(0)
μk ≡ qk ,
(1)
μk ≡ dk .
(12)
M. Schork / Central European Journal of Mathematics
Remark 3.1. With this notation the basic sequences of the Stirling triad of Example (l) (l) 2.2 can be written as μk = k 1+l if l ≤ 0 and μk = 0 if l > 0. The q-deformed versions (l) (l) of Example 2.3 are given by μk = q −lk kq1+l if l ≤ 0 and μk = 0 if l > 0. The basic recursion relation (1) can then be written as cn+1,k =
1
(l)
μk+l cn,k+l
(13)
l=−1
and the dual recursion relation (2) is given by xΦn (x) =
1
μ(−l) n Φn+l (x).
(14)
l=−1
Defining the tridiagonal transfer matrix T by Tkl :=
1
(σ)
μk+σ δk+σ,l ,
(15)
σ=−1
one can write (13) as cn+1 = T cn with the interpretation as a dynamical system on the space of sequences. The main idea of a duality triad of rank r ≥ 1 consists in generalizing the basic three-term recursion (13) to a (2r + 1)-term recursion relation for the cn,k . Now, let r be a natural number and assume that we are given a (2r + 1)-tuple of sequences μ = (μ(−r) , . . . , μ(r) ). Then the straightforward generalization of (13) is the (2r + 1)-term recursion relation r (l) cn+1,k = μk+l cn,k+l (16) l=−r
with initial condition c0,k = δ0,k . Defining the linear spaces t C∞ f in := {c = (c0 , c1 , . . .) | ck ∈ C, only finitely many ck = 0}, k akj xj , akj ∈ C, akk = 0}, PC := {Ψ(x) = (Ψ0 (x), Ψ1 (x), . . .) | Ψk (x) = j=0
and the bilinear pairing ·|· : PC × C∞ f in → C[x], given by Ψ(x)|c := may now recall the definition of duality triads of rank r given in [19].
∞
k=0 ck Ψk (x),
we
Definition 3.2. (Duality triad of rank r) Let a (2r + 1)-tuple of sequences μ = (−r) (μ(−r) , . . . , μ(r) ) with μk = 0 for all k and a polynomial pr (x) of degree r be given. Furthermore, let an r-tuple Ψ(x) = (Ψ0 (x), . . . , Ψr−1 (x)) of polynomials satisfying Ψ0 (x) = 1 and deg Ψk (x) = k be given. The associated duality triad or rank r is defined by the trans (σ) fer matrix T ≡ T (μ) with Tkl := rσ=−r μk+σ δk+σ,l , a sequence {cn }n≥0 with cn ∈ C∞ f in , and a Φ(x) ∈ PC such that: (i) cn+1 = T cn (together with c0,k = δ0,k ), (ii) pr (x)Φ(x) = Φ(x)T (together with Φk (x) = Ψk (x) for 0 ≤ k ≤ r − 1), and
M. Schork / Central European Journal of Mathematics
(iii) Φ(x)T |cn = Φ(x)|T cn for every n ≥ 0. It was shown in [19] that every duality triad of rank one with p1 (x) = x is a duality triad in the original sense of [10, 11] and vice versa. To be more explicit, (i) reproduces the recursion relation (16) whereas (ii) yields the dual recursion relation for the triad polynomials r pr (x)Φn (x) = μ(−l) (17) n Φn+l (x) l=−r
and is the generalization of (14). The inversion relation (iii) is given explicitly by n
[pr (x)] =
rn
cn,k Φk (x)
(18)
k=0
and is the generalization of (3). The transfer matrix T = (Tkl )k,l∈N is a (2r + 1)-diagonal (0) matrix and has around the element Tkk = μk the structure ⎞ ⎛ .. .. .. . . . ⎟ ⎜ ⎟ ⎜ ⎜ . . . μ(0) μ(1) μ(2) · · · ⎟ ⎟ ⎜ k−1 k k+1 ⎟ ⎜ ⎟ ⎜ (0) (1) T = ⎜ · · · μ(−1) . (19) μk+1 · · · ⎟ k−1 μk ⎟ ⎜ ⎟ ⎜ ⎜ · · · μ(−2) μ(−1) μ(0) · · · ⎟ k−1 k k+1 ⎟ ⎜ ⎠ ⎝ .. .. .. . . . Definition 3.3. (Hermitian duality triad of rank r) A duality triad of rank r associated to the (2r + 1)-tuple of sequences μ = (μ(−r) , . . . , μ(r) ) is called Hermitian if the (σ) associated transfer matrix T with Tkl := rσ=−r μk+σ δk+σ,l is Hermitian. Proposition 3.4. For an Hermitian duality triad of rank r the sequence μ(0) is real and the sequences μ(−l) with 1 ≤ l ≤ r can be expressed through the sequences μ(l) with positive index by (−l) (l) μk = μk+l . (20) Thus, only the (r + 1) sequences μ(l) with 0 ≤ l ≤ r are independent. Proof. This follows immediately from (19).
Note that the rather innocent looking property of being Hermitian is in fact a rather strong restriction on a duality triad of rank r. In particular, the explicit examples of duality triads of rank r discussed in Section 5 are not Hermitian. In the rank one case (with notations from the introduction) being Hermitian means that the qk are real and that one has the relation ik−1 = dk . Assuming in particular that the sequence d = (dk )k∈N is real and positive, the recursion relation (2) of the triad polynomials Φn (x) becomes xΦn (x) = dn Φn−1 (x) + qn Φn (x) + dn+1 Φn+1 (x)
(21)
M. Schork / Central European Journal of Mathematics
which is (upon identifying the sequences an ≡ dn and bn ≡ qn ) exactly the recursion relation (11) of orthogonal polynomials on the real line. Let us collect the above observations in the following theorem. Theorem 3.5. Let an Hermitian duality triad of rank one (with polynomial p1 (x) = x) associated to the pair of real sequences d ≡ μ(1) and q = μ(0) with dk > 0 be given. Then the triad polynomials Φn (x) are orthogonal polynomials on the real line in the sense of (11) and determine a positive measure on the real line (Favard’s theorem). Conversely, given a sequence of orthogonal polynomials Φn (x) on the real line satisfying (11) there exists an Hermitian duality triad of rank one (with polynomial p1 (x) = x) associated to (1) (0) (−1) the sequences μn ≡ an , μn ≡ bn and μn ≡ an+1 such that the triad polynomials are equal to the Φn (x). The generalization of this result to duality triads of higher rank will be discussed in the next section (see Theorem 4.2). Remark 3.6. (Dynamical data structures revisited) Let us try to connect duality triads of rank r ≥ 1 to dynamical data structures which were briefly recalled in Remark 2.4 with their connection to duality triads (of rank one). The coefficient cn,k has an interpretation as the number of possible histories a file (empty at the beginning) with k items has after n time steps where at each time step one of the operations I, D or Q is performed and where exist, e.g., ik possibilities to perform an insertion I if the file contains k items. This interpretation led directly to (1). Having this interpretation in mind, one should write the recursion relation (16) of a duality triad of rank r as (r)
(1)
(1)
(r)
cn+1,k = ik−r cn,k−r + · · · + ik−1 cn,k−1 + qk cn,k + dk+1 cn,k+1 + · · · + dk+r cn,k+r .
(22)
(σ)
The interpretation is then that one has ik−σ (with 1 ≤ σ ≤ r) possibilities of inserting (τ ) σ items simultaneously in the file containing k − σ items and having dk+τ (with 1 ≤ τ ≤ r) possibilities of deleting τ items simultaneously in the file containing k + τ items (σ) (τ ) (of course, for this interpretation all coefficients ik−σ , qk , dk+τ have to be non-negative integers). Thus, if the data structure is accessed sequentially (i.e., one item after another), (σ) (τ ) the coefficients ik−σ with σ > 1 as well as dk+τ with τ > 1 should vanish. In other words, the case r > 1 is not a good model. However, if one imagines that the file can be accessed in a parallel fashion (e.g., by r independent “operators”) then a possible application of duality triads of higher rank to the study of dynamical data structures seems not to be too far fetched.
4
Duality triads of higher rank and orthogonal matrix polynomials on the real line
In this section we draw a connection between duality triads of rank r and orthogonal matrix polynomials of rank r on the real line. This generalizes the connection between
M. Schork / Central European Journal of Mathematics
duality triads of rank one and scalar orthogonal polynomials discussed in Theorem 3.5. Before we do this we first recall some basic facts about orthogonal matrix polynomials (following [7, 15, 20]) and their relation to higher order scalar recurrence relations [7]. An r × r matrix P (x) = (pij (x))1≤i,j≤r with polynomial entries pij (x) of degree at most n is called a matrix polynomial of degree at most n. Alternatively, one can write P (x) = Cn xn + · · · + C0 with some numerical matrices Ck of size r × r. A matrix μ(r) = (μij )1≤i,j≤r of complex Borel measures defined on the real line is positive definite if for any Borel set E the matrix μ(r) (E) is positive semidefinite. We assume that all moments of μ(r) are finite. With such a matrix μ(r) one can define a matrix inner product on the space of r×r matrix polynomials via (P, Q) := R P (x) dμ(r) (x) Q∗ (x), and if (P, P ) is nonsingular for any P with nonsingular leading coefficient, then just as in the scalar case one can generate a sequence {Pn }n≥0 of matrix polynomials which is orthonormal with respect to μ(r) , i.e., (Pn , Pm ) = R Pn (x) dμ(r) (x) Pm∗ (x) = δn,m Ir . The sequence {Pn }n≥0 is determined only up to left multiplication by unitary matrices, i.e., if Un are unitary matrices then {Un Pn }n≥0 also forms an orthonormal system with respect to μ(r) . Just as in the scalar case the orthogonal matrix polynomials satisfy a three-term recurrence relation xPn (x) = An+1 Pn+1 (x) + Bn Pn (x) + A∗n Pn−1 (x) (23) where An are nonsingular and Bn Hermitian. Conversely, the analogue of Favard’s theorem is also true: If a sequence of matrix polynomials {Pn }n≥0 satisfies (23) with nonsingular An and Hermitian Bn then there exists a positive definite measure matrix μ(r) such that Pn are orthogonal with respect to μ(r) . Orthogonal matrix polynomials of rank r are closely related to (2r + 1)-term recurrence relations for scalar polynomials. Let a polynomial h(x) of degree r be given. Instead of using the basis of monomials {1, x, x2 , . . .} to span the linear space of polynomials, one can use the basis {xj hi (x) | 0 ≤ j < r, i = 0, 1, . . .}, i.e., {1, x, . . . , xr−1 , h(x), xh(x), . . . , xr−1 h(x), h2 (x), xh2 (x), . . .}. A polynomial p(x) of degree nr + m (0 ≤ m < r) can then expanded in this basis as p(x) =
n r−1
ai,j xj hi (x).
(24)
i=0 j=0
Let us define operators Rh,r,j (for 0 ≤ j ≤ r − 1) which take from p just those terms of the form ai,j xj hi (x) and then remove the common factor xj and change h(x) to x, i.e., Rh,r,j [p](x) =
n
ai,j xi .
(25)
i=0
Now, we can state the following theorem due to Duran and Van Assche [7]. Theorem 4.1 (Duran, Van Assche). Suppose {pn (x)}n≥0 is a sequence of polynomials satisfying the following (2r + 1)-term recurrence relation r d¯n,k pn−k (x) + dn+k,k pn+k (x) h(x)pn (x) = dn,0 pn (x) + k=1
(26)
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where dn,0 (n = 0, 1, . . .) is a real sequence and dn,k (n = 0, 1, 2, . . .) are complex sequences for k = 1, . . . , r with dn,r = 0 for every n and with the initial conditions pk (x) = 0 for k < 0 and pk given polynomials of degree k, for k = 0, . . . , r − 1. Define the sequence of matrix polynomials {Pn (x)}n≥0 of rank r by ⎛ ⎞ Rh,r,r−1 [pnr ](x) Rh,r,0 [pnr ](x) · · · ⎜ ⎟ ⎜ ⎟ ⎜ Rh,r,0 [pnr+1 ](x) · · · Rh,r,r−1 [pnr+1 ](x) ⎟ ⎜ ⎟ Pn (x) := ⎜ (27) ⎟. . . ⎜ ⎟ .. .. ⎜ ⎟ ⎝ ⎠ Rh,r,0 [pnr+r−1 ](x) · · · Rh,r,r−1 [pnr+r−1 ](x) Then this sequence of matrix polynomials is orthonormal on the real line with respect to a positive definite matrix measure and satisfies a three-term recurrence relation of the form (23) where the coefficients An , Bn can be described explicitly in terms of dn,k . Conversely, given a sequence Pn (x) = (Pn;k,l (x))1≤k,l≤r of orthonormal matrix polynomials of rank r on the real line (satisfying a three-term recurrence (23)), the sequence pn (x) of scalar polynomials defined for n ∈ N and 0 ≤ m ≤ r − 1 by pnr+m (x) :=
r
xj Pn;m,j (h(x))
(28)
j=0
satisfies a (2r + 1)-term recursion relation of the form (26). We now come to the analogue of Theorem 3.5 and show the intimate connection between duality triads of higher rank and orthogonal matrix polynomials. Theorem 4.2. Let an Hermitian duality triad of rank r associated to the polynomial pr (x) and the (r + 1) sequences μ(l) with 0 ≤ l ≤ r be given (the r polynomials Ψk (x) with 0 ≤ k ≤ r − 1 have also to be fixed). Then there exists a sequence of matrix polynomials Pn (x) (determined up to multiplication by unitary matrices Un ) which satisfy a three-term recursion of the form (23) - where the coefficients An , Bn can be described explicitly in (l) terms of the μk - and are orthogonal with respect to a positive definite matrix measure (Favard’s theorem for matrix polynomials). Conversely, given a sequence Pn (x) of orthogonal matrix polynomials of rank r on the real line and a polynomial h(x) of degree r, there exists an associated Hermitian duality triad of rank r (with polynomial pr (x) = h(x)) whose triad polynomials are completely determined by the entries of the Pn (x). Proof. Let an Hermitian duality triad of rank r be given. The triad polynomials Φn (x) satisfy the recursion relation (17). Writing this more explicitly and using the relation (20) valid for an Hermitian duality triad, we obtain pr (x)Φn (x) =
μ(0) n Φn (x)
+
r
(k) μ(k) Φ (x) + μ Φ (x) . n−k n n+k n+k
k=1
(29)
M. Schork / Central European Journal of Mathematics (0)
(k)
Defining the polynomial h(x) := pr (x) and the sequences dn,0 := μn and dn,k := μn for 1 ≤ k ≤ r, this equation becomes r h(x)Φn (x) = dn,0 Φn (x) + dn,k Φn−k (x) + dn+k,k Φn+k (x) .
(30)
k=1
Thus, the triad polynomials Φn (x) satisfy (26) where the sequence dn,0 is real and where (r)
(−r)
dn,r ≡ μn = μn−r = 0 by definition of the duality triad of rank r (and the first r polynomials are given explicitly by Φk (x) = Ψk (x) for 0 ≤ k ≤ r − 1). Thus, all the assumptions of Theorem 4.1 are satisfied. Defining Rh,r,j [Φn ](x) as in (25) and with their help the corresponding matrix polynomials Pn (x) as in (27), the theorem of Duran and Van Assche (Theorem 4.1) assures us that the Pn (x) satisfy the asserted three-term recursion relation and are orthogonal with respect to a positive definite measure matrix on the real line, thereby showing the first assertion. Now, let us show the converse, i.e., the polynomial h(x) and a sequence Pn (x) is given. The second part of the theorem of Duran and van Assche (Theorem 4.1) assures us that the polynomials pn (x) defined by (28) satisfy a (2r + 1)-term recursion relation of the form (26) with certain coefficients (k) dn,k . Defining the sequences μn := d¯n,k for 0 ≤ k ≤ r yields the basic input of an Hermitian duality triad of rank r. Example 4.3. (Discrete Sobolev orthogonal polynomials) An important class of polynomials satisfying a higher order recurrence relation is obtained by taking polynomials orthogonal with respect to an inner product of (discrete) Sobolev type, (p, q) =
p(x)q(x) dμ(x) + R
Mi M
λi,j p(j) (ci )q (j) (ci ),
(31)
i=1 j=0
where p, q are real polynomials on the real line and λi,j ≥ 0. Here derivatives are taken at M points ci ∈ R, and at the point ci the highest derivative is of order Mi [7]. Introducing Mi +1 of degree M + M the polynomial h(x) := M i=1 (x − ci ) i=1 Mi =: r, it can be shown that the corresponding orthogonal polynomials pn with (pn , pm ) = δn,m satisfy a recursion relation of the form h(x)pn (x) = rk=−r an,k pn+k (x), i.e., a (2r+1)-term recursion relation. It is clear that by defining the sequences μ(l) appropriately in terms of the an,k one obtains a duality triad of rank r. Considering the particular case M = 1 and denoting the highest derivative M1 ≡ s, one finds r = s + 1 and, therefore, a (2s + 3)-term recursion relation for the polynomials. This is the original observation of Marcell´an and Ronveaux [14]. Before closing this section let us mention that there exists a close connection between orthogonal polynomials on an algebraic harmonic curve and orthogonal matrix polynomials on the real line (due to Marcell´an and Sansigre [15]). The orthogonal polynomials on the curve satisfy a (2r + 1)-term recursion relation where r is given by the degree of the polynomial h(z) defining the algebraic curve.
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5
Examples of duality triads of higher rank
In this section some examples of duality triads of rank r ≥ 1 generalizing the duality triads (of rank one) considered in Section 2 will be discussed. Note that none of the examples considered here is Hermitian (and, therefore, not corresponding to orthogonal matrix polynomials by Theorem 4.2). Example 5.1. (“Embedded” Pascal triad) This is an example which was treated briefly in [19]. Let the (2r + 1)-tuple μ = (μ(−r) , 0, . . . , 0, μ(0) , 0, . . . , 0) of sequences be given (−r) (0) := 1 and μk := 1. Furthermore, a polynomial pr (x) and an r-tuple of where μk polynomials Ψ(x) = (Ψ0 (x), . . . , Ψr−1 (x)) has to be given. If we write n = r nr + n∗ with 0 ≤ n∗ ≤ r − 1 then the triad polynomials Φn are given for n ≥ r by n Φn (x) = (pr (x) − 1)[ r ] Ψn∗ (x).
(32)
Let us turn to the coefficients cn,k satisfying cn+1,k = cn,k +cn,k−r . It is clear that cn,k = 0 if n ¯ and obtain cn,kr k is not a multiple of r. If k is a multiple of r we may write k = kr ¯ = k ¯ . Drawing a table of the coefficients cn,k shows why we have called this example “embedded” ∗ Pascal triad (see Example 2.1). Choosing Ψn∗ (x) = xn (for 0 ≤ n∗ ≤ r − 1) it follows ¯ k that Φkr ¯ (x) = (pr (x) − 1) . Since cn,k vanishes if k is not a multiple of r the inversion relation (18) is in this case given by n n ¯ (pr (x) − 1)k . [pr (x)] = ¯ k n
(33)
¯ k=0
By choosing different pr (x) (e.g., pr (x) = xr or pr (x) = xr ) one obtains different generalizations of (4). (l)
(l)
Example 5.2. (Pascal triad of rank r) Let μk := 1 if l ≤ 0 and μk := 0 if l > 0. Thus, the coefficients cn,k satisfy the recursion relation cn+1,k = cn,k + cn,k−1 + · · · + cn,k−r .
(34)
Clearly, the case r = 1 corresponds to the usual Pascal triad treated in Example 2.1. A small induction shows that the coefficients (for fixed n) are symmetric around the one in the middle, i.e., cn,rn−k = cn,k . Following [6], p. 77 (see also [16], p. 167) we introduce (r) in [6]) by the identity the polynomial coefficients An,k (denoted by n,r+1 k (1 + x + · · · + x ) = r n
rn
(r)
An,k xk .
(35)
k=0 (r)
From this it follows immediately that A0,k = δk,0 as well as (r)
(r)
(r)
(r)
An+1,k = An,k + An,k−1 + · · · + An,k−r
(36)
M. Schork / Central European Journal of Mathematics (r)
so that An,k ≡ cn,k from (34). The case r = 1 leads via the binomial formula directly to (1) the binomial coefficient An,k = nk , see Example 2.1. In the general case one may use the multinomial formula for the left-hand side of (35) to obtain n! (r) An,k = . (37) (n − l1 − · · · − lr )!l1 ! · · · lr ! 0≤l ,...,lr ≤n 1 l1 +2l2 +···+rlr =k
(2)
(2)
Specializing (37) to the case r = 2 one obtains for 0 ≤ k ≤ n (recall that An,2n−k = An,k ) the following explicit values of the trinomial coefficients k
(2) An,k
=
2 l=0
k2 n k−l n! = k−l l (n − k + l)!(k − 2l)!l! l=0
(38)
(where we have denoted by x the greatest integer less than or equal to x). The maximum n2 nn−l (2) (2) value of the An,k for fixed n is attained if k = n and equals An,n = l=0 . This l l particular value for the middle trinomial coefficient is also mentioned in [16]. A table (2) for the first few trinomial coefficients An,k ≡ n,3 as well as quadrinomial coefficients k n,4 (3) A ≡ k can be found on p. 78 in [6]. Note that taking x → 1 in (35) shows that n,k (r) rn n k=0 An,k = (1 + r) . Now, let us determine the remaining properties of a duality triad of rank r. Defining pr (x) := 1 + x + · · · + xr as well as Φk (x) := xk for k ≥ r (together with Ψk (x) := xk for 0 ≤ k ≤ r − 1) shows that (35) is the inversion relation (18) in this case. What remains to be checked is the dual recursion relation (17) for the triad polynomials Φk (x) = xk which is given in this case by (1 + x + · · · + x )x = r
n
r
xn+l
(39)
l=0
and which evidently holds. It is interesting to note that in the case r = 1 one obtains p1 (x) = 1 + x and Φk (x) = xk , thus reproducing Example 2.1 only up to a shift x x − 1 (in Example 2.1 one has p1 (x) = x and Φk (x) = (x − 1)k ), suggesting that the definition of the Pascal triad in the rank one case is not the natural choice. However, we may now summarize the above observations in the following theorem. Theorem 5.3. (Pascal triad of rank r) For a given r ≥ 1 let pr (x) := 1 + x + · · · + xr , (l) Ψk (x) := xk for k = 0, . . . , r − 1 and the (2r + 1) sequences μ(l) defined by μk := 1 if (l) l ≤ 0 and μk := 0 if l > 0 be given. Then the associated duality triad of rank r consists (r) of the coefficients cn,k ≡ An,k of (37) and the triad polynomials Φk (x) ≡ xk satisfying the recursion relation (36), the dual recursion relation (39) and the inversion relation (35). Choosing r = 1 yields the Pascal triad treated in Example 2.1 (up to a shift x x − 1). Example 5.4. (Generalized Pascal triad of rank r) We will now generalize (35) slightly (r) by introducing a vector λ = (λ1 , . . . , λr ) ∈ Zr and defining coefficients An,k (λ) by (1 + λ1 x + · · · + λr x ) = r n
rn k=0
(r)
An,k (λ)xk ;
(40)
M. Schork / Central European Journal of Mathematics (r)
(r)
(r)
if λ = (1, . . . , 1) then An,k (λ) = An,k from above. The definition implies A0,k (λ) = δk,0 and the recursion relation (r)
(r)
(r)
(r)
An+1,k (λ) = An,k (λ) + λ1 An,k−1 (λ) + · · · + λr An,k−r (λ).
(41) (l)
Thus, the sequences μ(l) of the corresponding duality triad of rank r are given by μk = λ−l (0) (l) if l < 0, μk = 1 and μk = 0 if l > 0. In view of (40) one defines pr (x) := 1 + λ1 x + · · · + λr xr , Φk (x) := xk as well as Ψk (x) := xk so that (40) is already the inversion relation. The (r) data {μ(l) , pr (x), Ψk (x), Φk (x), cn,k ≡ An,k (λ)} comprise the generalized Pascal triad of rank r (the dual recursion relation for the triad polynomials remains to be checked). Using the multinomial formula for the left-hand side of (40) yields a formula for the coefficients (r) An,k (λ) similar to (37). Note that by choosing the particular vector λ = (0, . . . , 0, 1) (r)
(r)
(r)
one obtains the recursion relation An+1,k (λ) = An,k (λ) + An,k−r (λ) of Example 5.1. As another example, let us consider the case r = 2 with the vector λ = (−1, 1). The (2) (2) (2) coefficients An,k (−1, 1) ≡ An,k ((−1, 1)) satisfy the recursion relation An+1,k (−1, 1) = (2)
(2)
(2)
An,k (−1, 1) − An,k−1 (−1, 1) + An,k−r (−1, 1) and are related to the trinomial coefficients (2)
(2)
(2)
An,k corresponding to the vector λ = (1, 1) by An,k (−1, 1) = (−1)k An,k . Remark 5.5. It is tempting to consider r → ∞ in (35) (see also [16], p. 189, for a related discussion). Summing the geometric series yields as definition of the cor ∞ (∞) k responding coefficients (1 − x)−n = k=0 An,k x . Thus, one has the explicit values (∞) (∞) (∞) (∞) (∞) = n+k−1 and the recursion relation An+1,k = An,k +An,k−1 +· · ·+An,0 An,k = (−1)k −n k k which one may consider as the limit r → ∞ of (36). Note, however, that this will not give rise to a duality triad of higher rank since the order of the recursion relation is not fixed, but depends on the second index. Example 5.6. (Stirling triad of rank r) The generalized Stirling numbers of second kind Sr,s (n, k) were introduced in [2, 3]. In the special case r = s = 1 they reduce to the conventional Stirling numbers of Example 2.2, i.e., S1,1 (n, k) ≡ S(n, k), and in the case r = 2, s = 1 they reduce to Lah numbers. In the most important special case s = r (which we will call Stirling numbers of rank r) it was shown in [2, 3] that k (−1)k p k Sr,r (n, k) = [pr ]n (−1) (42) p k! p=r for r ≤ k ≤ rn (and Sr,r (n, k) = 0 otherwise) and that the recursion relation is given written in a form particularly suited for our purpose - by 0 r (k + l)! Sr,r (n + 1, k) = (43) Sr,r (n, k + l) −l (k − r)! l=−r (the recursion relation for the general case Sr,s (n, k) can be found in [17]). Furthermore, these Stirling numbers of rank r were also shown to be connection coefficients, i.e., rn r n [x ] = Sr,r (n, k)xk . (44) k=r
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Now, we want to show that the Sr,r (n, k) lead to a duality triad of rank r which generalizes Example 2.2 in a beautiful way. Let us, therefore, define the (2r + 1) sequences μ(l) by ⎧ ⎪ ⎨ 0 if 0 < l ≤ r, (l) μk := (45) ⎪ ⎩ r k r+l if −r ≤ l ≤ 0. −l (−r)
Note that μk = 1 = 0. Furthermore, let pr (x) := xr and Ψk (x) := xk for k = 0, . . . , r−1. The coefficients cn,k satisfy c0,k = δ0,k as well as the recursion relation (16) which is given with the above choice (45) by cn+1,k
0 0 r r (k + l)! r+l (k + l) cn,k+l = = cn,k+l . −l −l (k − r)! l=−r l=−r
(46)
Comparing this with (43) shows that the cn,k are given by the Stirling numbers of rank r, i.e., cn,k = Sr,r (n, k). Furthermore, comparing the inversion relation (18) with (44) (and recalling pr (x) = xr ) shows that the triad polynomials are given by Φk (x) = xk . It remains to be shown that these triad polynomials satisfy the dual recursion relation (17) (−l) which is in this case given by xr Φn (x) = rl=−r μn Φn+l (x), or, more explicitly, by r n
xx =
r r l=0
l
nr−l xn+l .
(47)
We will not check this explicitly but just remark that the identity xn+1 = xn (x − n) (and its iterations xn+l = xn lj=1 (x − (n + l) + j)) is particularly helpful. Let us summarize the above observations in the following theorem. Theorem 5.7. (Stirling triad of rank r) For a given r ≥ 1 let pr (x) := xr , Ψk (x) := xk for k = 0, . . . , r − 1 and the (2r + 1) sequences μ(l) defined by (45) be given. Then the associated duality triad of rank r consists of the coefficients cn,k ≡ Sr,r (n, k) of (42) and the triad polynomials Φk (x) ≡ xk satisfying the recursion relation (43), the dual recursion relation (47) and the inversion relation (44). Choosing r = 1 yields the Stirling triad treated in Example 2.2. Remark 5.8. Before we discuss the next example we want to give some comments concerning the q-deformed situation. As already mentioned at the end of Example 2.3 the inversion relation (9) does not look natural while the much more natural looking equation (10) is not an example of the inversion relation (3) since the occurring functions are polynomials in xq but not in x. To remedy this situation one has the following way out: One should make the domain of the “variable” explicit in Definition 3.2. This means that one should call the “duality triad of rank r” according to this definition more precisely a “duality triad of rank r over C[x]”. In the q-deformed situation one should replace the ring C[x] by C[xq ] where xq is the new variable. Thus, one should replace PC from above by PCq := {Ψ(xq ) = (Ψ0 (xq ), Ψ1 (xq ), . . .) | Ψk (xq ) = kj=0 akj xjq , akj ∈ C, akk = 0} and
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the bilinear pairing ·|· : PC × C∞ the corresponding q-deformed version f in → C[x] by q ∞ ·|· q : PC × Cf in → C[xq ] given by Ψ(xq )|c q := ∞ k=0 ck Ψk (xq ). Then one should give a definition of “duality triad of rank r over C[xq ]” in complete analogy to Definition 3.2 but where now the variable xq replaces x (and pr (xq ) is a polynomial of degree r in xq , etc.). Since all the manipulations are only algebraic (no considerations of convergence, limits, etc.) one is tempted to consider even more general situations where more or less arbitrary “variables” instead of x or xq are allowed. However, we will not do this but consider the q-deformed situation in a more formal way, mimicking as close as possible the undeformed case. Example 5.9. (q-deformed Stirling triad of rank r) In this example we consider the q-deformed version of Example 5.6. It is the common generalization of Example 5.6 (to the case where q = 1) and Example 2.3 (to the case r ≥ 1). Let us first recall some properties of the q-deformed Stirling numbers of rank r Sr,r (n, k|q); these generalized Stirling numbers were introduced in [17] and all the properties mentioned can be found therein. First of all, they are explicitly given by k k−p (−1)k p ( 2 ) k Sr,r (n, k|q) = (−1) q (pr )n p q q kq ! p=r
(48)
(where r ≤ k ≤ rn) and are the obvious common generalization of (42) and (7). They satisfy the recursion relation Sr,r (n + 1, k|q) =
0
q
−l(k−r)
l=−r
r −l
(k + l)q ! Sr,r (n, k + l|q) q (k − r)q !
(49)
(which is the common generalization of (43) and (8)) and are also connection coefficients [xrq ]n
=
rn
Sr,r (n, k|q)xkq
(50)
k=r
(which is the common generalization of (44) and (10)). Since the procedure is very similar to the one given in Example 5.6 we will be brief. The decisive step is to define the sequences μ(l) correctly. In analogy to (45) we define ⎧ ⎪ ⎨ 0 if 0 < l ≤ r, (l) (51) μk := ⎪ ⎩ q −l(k−l−r) r kqr+l if −r ≤ l ≤ 0. −l q
Since (l) μk+l
=q
−l(k−r)
r −l
(k +
q
r+l l)q
=q
−l(k−r)
r −l
(k + l)q ! q (k − r)q !
(52)
one sees immediately that the recursion relation (16) for the cn,k has the same form as (49) so that the coefficients cn,k are given by the q-deformed Stirling numbers of rank r, i.e., cn,k = Sr,r (n, k|q). The inversion relation (50) shows that one has to choose pr (xq ) := xrq
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and that the triad polynomials are given by Φk (xq ) := xkq . The dual recursion relation (17) for the triad polynomials is in this case given by r r−l n+l r n l(n+l−r) r xq x q = q nq x q . (53) l q l=0 As above, we will not check this identity but immediately summarize the above observations in the following theorem. Theorem 5.10. (q-deformed Stirling triad of rank r) For a given r ≥ 1 let pr (xq ) := xrq , Ψk (xq ) := xkq for k = 0, . . . , r−1 and the (2r+1) sequences μ(l) defined by (51) be given. Then the associated duality triad of rank r consists of the coefficients cn,k ≡ Sr,r (n, k|q) of (48) and the triad polynomials Φk (xq ) ≡ xkq satisfying the recursion relation (49), the dual recursion relation (53) and the inversion relation (50). Choosing q = 1 yields the Stirling triad of rank r treated in Example 5.6 while choosing r = 1 yields the q-deformed Stirling triad treated in Example 2.3. In (54) the connections between the sequences μ(l) (with −r ≤ l ≤ 0) of the various (generalized) Stirling triads are sketched. In the upper left corner one has the usual (l) Stirling triad with μk = k 1+l (see Remark 3.1). The arrow down means going to the q-deformed situation so that one has in the lower left corner the q-deformed Stirling triad (see Remark 3.1). The arrow to the right means going from r = 1 to arbitrary r ≥ 1. Therefore, one has in the upper right corner the Stirling triad of rank r (Example 5.6) and in the lower right corner the q-deformed Stirling triad of rank r (Example 5.9). k 1+l ⏐ ⏐ ⏐ ⏐
−→
q −lk kq1+l −→ q
r r+l k −l ⏐ ⏐ ⏐ ⏐ −l(k−l−r) r
(54)
r+l k . −l q q
Recall that in the rank one case it is possible to construct duality triads associated to various extensions of the conventional Stirling numbers (of which the q-deformed versions are only a particular example), see Example 2.3. Here one should have in mind the Stirling numbers S(n, k|ψ) of ψ-extended umbral calculus [13]. It seems to be interesting to find out whether there exists a natural extension of this ψ-extended Stirling triad to higher rank r ≥ 1.
6
Conclusions
In this paper we have discussed some properties of duality triads of rank r ≥ 1, in particular the connection to orthogonal polynomials. It was shown that a Hermitian duality triad of rank r gives rise to a sequence of orthogonal matrix polynomials (with matrices of size r × r), generalizing the scalar rank one case. Conversely, given a (2r + 1)-term recurrence
M. Schork / Central European Journal of Mathematics
relation for a sequence of polynomials, one obtains an associated duality triad of rank r by choosing the sequences μ(l) defining the duality triad appropriately. This shows that one may associate a duality triad of higher rank to sequences of polynomials which satisfy a higher order recurrence relation. As examples for such sequences discrete Sobolev orthogonal polynomials and orthogonal polynomials on algebraic curves were mentioned. Some concrete examples of duality triads of higher rank were discussed explicitly. In particular, it was shown that the generalized Stirling numbers of higher rank give rise to a duality triad of higher rank. Its q-deformed analogue was also discussed in a slightly formal way and it was stressed that one could make this discussion completely rigorous by extending the notion of duality triad (of higher rank) slightly. Another example of a duality triad of higher rank associated to the polynomial coefficients was discussed and it was shown that this represents a natural generalization of the Pascal triad. Turning to the relation between duality triads and dynamical data structures, it was shown that the defining recursion relation of a duality triad of rank r can be interpreted in terms of histories of a file where up to r items can be inserted or deleted simultaneously (i.e., in one time step). Thus, one may hope that duality triads of higher rank might prove to be useful for the study of dynamical data structures where parallel access is allowed.
References [1] G.E. Andrews: The Theory of Partitions, Addison Wesley, Reading, 1976. [2] P. Blasiak, K.A. Penson and A.I. Solomon: “The Boson Normal Ordering Problem and Generalized Bell Numbers”, Ann. Comb., Vol. 7, (2003), pp. 127–139. [3] P. Blasiak, K.A. Penson and A.I. Solomon: “The general boson normal ordering problem”, Phys. Lett. A, Vol. 309, (2003), pp. 198–205. [4] E. Borak: “A note on special duality triads and their operator valued counterparts”, Preprint arXiv:math.CO/0411041. [5] T.S. Chihara: An Introduction to Orthogonal Polynomials, Gordon & Breach, New York, 1978. [6] L. Comtet: Advanced Combinatorics, Reidel, Dordrecht, 1974. [7] A.J. Duran and W. Van Assche: “Orthogonal matrix polynomials and higher order recurrence relations”, Linear Algebra Appl., Vol. 219, (1995), pp. 261–280. [8] P. Feinsilver and R. Schott: Algebraic structures and operator calculus. Vol. II: Special functions and computer science, Kluwer Academic Publishers, Dordrecht, 1994. [9] I. Jaroszewski and A.K. Kwa´sniewski: “On the principal recurrence of data structures organization and orthogonal polynomials”, Integral Transforms Spec. Funct., Vol. 11, (2001), pp. 1–12. [10] A.K. Kwa´sniewski: “On duality triads”, Bull. Soc. Sci. Lettres L ´od´z, Vol. A 53, Ser. Rech. D´eform. 42, (2003), pp. 11–25. [11] A.K. Kwa´sniewski: “On Fibonomial and other triangles versus duality triads”, Bull. Soc. Sci. Lettres L ´od´z, Vol. A 53, Ser. Rech. D´eform. 42, (2003), pp. 27–37. [12] A.K. Kwa´sniewski: “Fibonomial Cumulative Connection Constants”, Bulletin of the
M. Schork / Central European Journal of Mathematics
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
ICA, Vol. 44, (2005), pp. 81–92. A.K. Kwa´sniewski: “On umbral extensions of Stirling numbers and Dobinski-like formulas”, Adv. Stud. Contemp. Math., Vol. 12, (2006), pp. 73–100. F. Marcell´an and A. Ronveaux: “On a class of polynomials orthogonal with respect to a discrete Sobolev inner product”, Indag. Math., Vol. 1, (1990), pp. 451–464. F. Marcell´an and G. Sansigre: “On a Class of Matrix Orthogonal Polynomials on the Real Line”, Linear Algebra Appl., Vol. 181, (1993), pp. 97–109. J. Riordan: Combinatorial Identities, Wiley, New York, 1968. M. Schork: “On the combinatorics of normal-ordering bosonic operators and deformations of it”, J. Phys. A: Math. Gen., Vol. 36, (2003), pp. 4651–4665. M. Schork: “Some remarks on duality triads”, Adv. Stud. Contemp. Math., Vol. 12, (2006), pp. 101–110. M. Schork: “On a generalization of duality triads”, Cent. Eur. J. Math., Vol 4(2), (2006), pp. 304–318. A. Sinap and W. Van Assche: “Orthogonal matrix polynomials and applications”, J. Comput. Appl. Math., Vol. 66, (1996), pp. 27–52. R.P. Stanley: Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, 1999. G. Szeg¨o: Orthogonal Polynomials, American Mathematical Society, 1948. V. Totik: “Orthogonal Polynomials”, Surv. Approximation Theory, Vol. 1, (2005), pp. 70–125.
DOI: 10.2478/s11533-006-0012-y Research article CEJM 4(3) 2006 525–530
Remarks on affine complete distributive lattices Dominic van der Zypen∗ Allianz Suisse, CH-3001 Berne, Switzerland
Received 20 September 2005; accepted 14 MArch 2006 Abstract: We characterise the Priestley spaces corresponding to affine complete bounded distributive lattices. Moreover we prove that the class of affine complete bounded distributive lattices is closed under products and free products. We show that every (not necessarily bounded) distributive lattice can be embedded in an affine complete one and that Q ∩ [0, 1] is initial in the class of affine complete lattices. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Distributive lattice, affine complete, Priestley spaces MSC (2000): 06D50, 06D99
1
Affine complete lattices
A k-ary function f on a bounded distributive lattice L is called compatible if for any congruence θ on L and (ai , bi ) ∈ θ, (i = 1, ..., k) we always have (f (a1 , ..., ak ), f (b1 , ....bk )) ∈ θ. It is easy to see that the projections pri : Lk → L are compatible. With induction on polynomial complexity one shows that every polynomial function is compatible (see [4]). A lattice L is called affine complete, if conversely every compatible function on L is a polynomial and if it is bounded and distributive. G. Gr¨atzer [2] gave an intrinsic characterization of bounded distributive lattices that are affine complete: Theorem 1.1. [2] A bounded distributive lattice is affine complete if and only if it does not contain a proper interval that is a Boolean lattice in the induced order. Note that in particular, no finite bounded distributive lattice L is affine complete: ∗
E-mail: [email protected]
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Let x ∈ L be an element distinct from 1. Then x has an upper neighbor, ie, there exists y ∈ L such that [x, y] = {x, y} which is isomorphic to the 2-element Boolean lattice. Example 1.2. The bounded distributive lattices [0, 1] and [0, 1]×[0, 1] are affine complete. NOTE: From now on, all lattices considered are assumed to be bounded and distributive, unless otherwise stated.
2
Priestley duality
In [5], Priestley proved that the category D of bounded distributive lattices with (0, 1)preserving lattice homomorphisms and the category P of compact totally order-disconnected spaces (henceforth referred to as Priestley spaces) with order-preserving continuous maps are dually equivalent. A compact totally order-disconnected space (X; τ, ) is a poset (X; ) endowed with a compact topology τ such that, for x, y ∈ X, whenever x y, then there exists a clopen decreasing set U such that x ∈ U and y ∈ U . (A decreasing set or a down-set is a subset D of a partially ordered set P such that x ≤ y in P and y ∈ D imply x ∈ D.) The functor D : D → P assigns to each object L of D a Priestley space (D(L); τ (L), ⊆), where D(L) is the set of all prime ideals of L and τ (L) is a suitably defined topology (the details of which will not be required here). The functor E : P → D assigns to each Priestley space X the lattice (E(X); ∪, ∩, ∅, X), where E(X) is the set of all clopen decreasing sets of X. Priestley duality therefore provides us with a“dictionary”between the world of bounded distributive lattices and a certain category of ordered topological spaces. This is interesting in particular because free products of lattices are “translated” into products of Priestley spaces. We will use this fact for showing that the class of affine complete bounded distributive lattices is closed under free products.
3
Affine complete Priestley spaces
The aim of this section is to characterize the Priestley spaces corresponding to affine complete distributive (0,1)-lattices. Such spaces will be called affine complete Priestley spaces. In other words, a Priestley space X is affine complete iff E(X) is affine complete. The following theorem provides a rather straightforward translation of the algebraic concept of affine completeness in order-topological terms. Theorem 3.1. Let X be a Priestley space. Then the following statements are equivalent: (1) E(X) is affine complete. (2) If U ⊆ V are clopen down-sets and U = V , then the subposet V \ U of X is not an antichain, i.e. V \ U contains a pair of distinct comparable elements. Proof. (1) =⇒ (2). Suppose V \U is an antichain. Let C ∈ [U, V ] ⊆ E(X). Take
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C = U ∪ (V \C). Claim: C is a clopen down-set of X. It is clear that C is a clopen subset of X since V \ C = V ∩ (X \ C). Now, let c ∈ C and assume x < c. Then if c ∈ U , we are done, since U is a down-set. Assume c ∈ V \ U . Since V is a down-set, we get x ∈ V , and the fact that V \ U is an antichain tells us that x cannot be a member of V \ U . Therefore x ∈ U ⊆ C which proves that C is indeed a (clopen) down-set. Moreover, C is the complement of C in [U, V ], i.e. C ∩ C = U and C ∪ C = V . Because C was arbitrary, we see that [U, V ] is a proper Boolean interval of E(X), whence E(X) is not affine complete. (2) =⇒ (1). Suppose U ⊆ V are distinct clopen down-sets. By assumption, there are elements x, y ∈ V \U such that x < y. There is a clopen down-set A with x ∈ A and y∈ / A. Consider B = (A ∩ V ) ∪ U . So B ∈ [U, V ] and y ∈ / B. Now we show that B has no complement in [U, V ]: Take any C ∈ [U, V ] with C ∪ B = V . Then y ∈ C, but since C is a down-set, we have x ∈ C, thus x ∈ (B ∩ C)\U and B ∩ C = U . So whatever C we pick, C is no complement for B, i.e. B is not complemented, and consequently [U, V ] is not Boolean. It follows that no proper interval of E(X) is Boolean. We can formulate the above result in a more concise way: Corollary 3.2. A Priestley space X is affine complete if and only if each nonempty open set contains two distinct comparable points. Proof. It follows directly from theorem 3.1 that if each nonempty open set contains two distinct points that are comparable, then X is affine complete. Conversely, suppose that U is a nonempty open set which is an antichain. Recall that the clopen decreasing sets and their complements (clopen increasing sets) form a base of any Priestley space. So there exist open down-sets C1 , C2 such that ∅ = C1 ∩(X\C2 ) ⊆ U . Then [C1 ∩C2 , C1 ] is a proper interval such that C1 \(C1 ∩C2 ) = C1 ∩(X\C2 ) is an antichain (as a subset of the antichain U ). Thus theorem 3.1 implies that X is not affine complete. Note that the proof works exactly the same way if each occurrence of “open” is replaced by “clopen” (basically because each Priestley space is zero-dimensional). So we can state as well: A Priestley space X is affine complete if and only if each nonempty clopen set contains two distinct comparable points.
4
Products of affine complete lattices
We prove in this section that arbitrary products of affine complete lattices are affine complete. We don’t need Priestley duality to do this. Priestley duals of affine com-
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plete lattices, i.e. affine complete Priestley spaces, will come into play when we consider coproducts of affine complete lattices. Theorem 4.1. If (Li )i∈I is a family of (bounded distributive) affine complete lattices, then Πi∈I Li is affine complete. Proof. We prove the contrapositive of the theorem. Suppose that Πi∈I Li is not affine complete. Then it contains a proper interval [ξ, η] that is Boolean. There exists some k ∈ I such that ξ(k) < η(k). We claim that [ξ(k), η(k)] ⊆ Lk is a Boolean interval. Set x = ξ(k), y = η(k). Suppose l ∈ [x, y] and define λ ∈ Πi∈I Li by λ(k) = l and λ(i) = ξ(i) if i = k Because [ξ, η] is Boolean, there exists λ ∈ Πi∈I Li such that λ ∧ λ = ξ and λ ∨ λ = η. Thus it is easy to see that l := λ (k) is the complement of l ∈ [x, y]. Therefore, [x, y] is a proper Boolean interval of Lk and whence Lk is not affine complete. Example 4.2. Theorem 4.1 implies that [0, 1]N is affine complete.
5
Free products of affine complete lattices
Now we turn our attention to free products of affine complete bounded distributive lattices; we prove they are complete. A convenient way to obtain this result is to dualise the problem into the category of Priestley spaces. Recall that free products in D are categorically speaking coproducts in D. Since Priestley duality is a pair of contravariant functors, coproducts correspond to products in P and vice versa; this is stated in the following proposition in a more general way. Proposition 5.1. [3] Let A and B be categories, and assume that F : A → B and G : B → A are contravariant functors that form a dual equivalence. Then: (1) If A is a product of a family of objects (Ai )i∈I of A, then F(A) is a coproduct of (F(Ai ))i∈I . (2) If A is a coproduct of a family of objects (Ai )i∈I of A, then F(A) is a product of (F(Ai ))i∈I . We recall that a Priestley space was shown to be affine complete if and only if each non-empty open subset contains two distinct comparable points. Theorem 5.2. If (Xi )i∈I is a family of affine complete Priestley spaces, then Πi∈I Xi is affine complete.
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Proof. Suppose that Xi is affine complete for every i ∈ I. It suffices to show that every nonempty subset V of Πi∈I Xi of the form (U1 ) ∩ ... ∩ πi−1 (Ur ) V = πi−1 r 1 contains two distinct comparable elements (where Uk ⊆ Xik open, nonempty). Take U1 . It contains elements a < b, because Xi1 is affine complete. Now pick ξ ∈ V . Define ξ1 , ξ2 ∈ V by ξ1 (i1 ) = a and ξ1 (i) = ξ(i) if i = i1 and ξ2 (i1 ) = b and ξ2 (i) = ξ(i) if i = i1 . Clearly, ξ1 , ξ2 are distinct comparable elements of V .
Applying the Priestley duality now yields: Corollary 5.3. The class of (bounded distributive) affine complete lattices is closed under free products.
6
Embedding lattices in affine complete lattices
First we will stay away from affine completeness in the worst possible way: we will embed each L into a powerset of some set, which, being Boolean, is as affine incomplete as it gets. The following fact is well-known: Lemma 6.1. Let L be a distributive lattice (L need not be bounded). There is a set X and a lattice embedding j : L → P(X) where P(X) is the powerset of the set X. Next, we will embed that powerset in an affine complete lattice. Lemma 6.2. Let X be a set and let Q = {q ∈ Q; 0 q 1}. Then there is a lattice embedding j : P(X) → QX . Moreover, Q is affine complete. Proof. Set j : S → χS ∈ QX for every S ⊆ X, where χS is defined by / S. χS (x) = 1 if x ∈ S and χS (x) = 0 if x ∈ It is easy to see that j is a lattice embedding. Next, we claim that Q is affine complete. ∈ [x, y] has no complement a in [x, y]: Take any x < y in Q. Then the element a = x+y 2 Otherwise we would have a ∧ a = x which would imply a = x, but then a ∨ a = a = y.
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So [x, y] is not Boolean, whence Q has no proper Boolean interval. Therefore, Q is affine complete. Moreover, by 4.1, QX is affine complete which concludes the proof. Lemmas 6.1 and 6.2 now imply: Corollary 6.3. Every distributive lattice (not necessarily bounded) can be embedded in a bounded affine complete lattice. Admittedly, the construction provided by 6.1 and 6.2 is highly non-unique and has no minimality properties.
References [1] B.A. Davey and H.A. Priestley: Lattices and Order, Cambridge University Press, 1990. [2] G. Gr¨atzer: “Boolean functions on distributive lattices”, Acta Math. Acad. Sci. Hung., Vol. 15, (1964), pp. 195–201. [3] S. MacLane: Categories for the working mathematician, 2nd ed., Springer Verlag, (1998). [4] M. Ploˇsˇcica: “Affine Complete Distributive Lattices”, Order, Vol. 11, (1994), pp. 385–390. [5] H.A. Priestley: “Representation of distributive lattices by means of ordered Stone spaces”, Bull. London Math. Soc., Vol. 2, (1970), pp. 186–190. [6] H.A. Priestley: “Ordered topological spaces and the representation of distributive lattices”, Proc. London Math. Soc., Vol. 3(24), (1972), pp. 507–530. [7] D. van der Zypen: Aspects of Priestley Duality, Thesis (PhD), University of Bern, 2004.
DOI: 10.2478/s11533-006-0022-9 Research article CEJM 4(3) 2006 531–546
Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences Roman Witula, Damian Slota∗ Institute of Mathematics, Silesian University of Technology, 44-100 Gliwice, Poland
Received 8 March 2006; accepted 2 June 2006 Abstract: In this paper some decompositions of Cauchy polynomials, Ferrers-Jackson polynomials and polynomials of the form x2n + y 2n , n ∈ N, are studied. These decompositions are used to generate the identities for powers of Fibonacci and Lucas numbers as well as for powers of the so called conjugate recurrence sequences. Also, some new identities for Chebyshev polynomials of the first kind are presented here. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Cauchy polynomials, Ferrers-Jackson polynomials, Chebyshev polynomials, Fibonacci and Lucas numbers, recurrence sequences MSC (2000): 11B83, 26C99, 11B39
1
A brief exposition of the content of the paper
In Section 2 the following decompositions of Cauchy polynomials: pn (x, y) := (x + y)2n+1 − x2n+1 − y 2n+1 ,
n ∈ N,
as well as Ferrers-Jackson polynomials qn (x, y) := (x + y)2n + x2n + y 2n , ∗
[email protected]
n ∈ N,
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are discussed (see Lemma 2.2): 2k+1 2 2n + 1 n − k (x + x y + y 2 )n−1−3k x y (x + y) n − k 2k + 1
(n−1)/3
pn (x, y) =
k=0
and
n/3
qn (x, y) =
k=0
2k 2n n − k x y (x + y) (x2 + x y + y 2 )n−3k . n−k 2k
(1)
(2)
These decompositions are differ from those already published – see especially [13]. Also, presented here proof of decompositions (1) and (2) by induction is based on a simple recurrence dependence between polynomials pn (x, y) and qn (x, y) seems to be new. In this paper, decompositions (1) and (2) are applied to generating the identities for powers of Fibonacci and Lucas numbers (some of them – the simplest ones – are identical with the identities discussed in [1, 6, 7]; also in [4] some of these simplest identities are presented). In Sections 4 and 5 the notion of pairs and triples of conjugate recurrence sequences is introduced. The identities for the powers of elements of such sequences are also derived. In the last section of the paper, the following decomposition of polynomials: x2n +y 2n , n ∈ N is discussed: n x2n + y 2n = ωr,n · (x y)r · (x2 + x y + y 2 )n−r . (3) r=0
The combinatorial and analytical descriptions of coefficients ωr,n are provided, in particular the analytical description where Chebyshev polynomials of the first kind are used. Formula (3) is applied to generate some identities for the powers of Fibonacci and Lucas numbers. Also, some new combinatorial identities for the Chebyshev polynomials of the first kind are presented. Publications concerning the identities of the sums of the powers of elements of recurrence sequences, (especially second order), are fairly recent [1–3, 5–11, 14]. The results presented in the publications generally indicate two directions: generating formulas with a bounded number of elements (our paper falls into this category) and generating formulas with an increasing number of elements [6–9]. Some papers are focused on formulas for generating functions of sequences of the powers of elements of recurrence sequences [3, 5, 14], as well as generating the identities by means of such functions. Other papers [10, 11] discuss generalized forms of certain known identities (especially for Fibonacci and Lucas numbers).
2
Decompositions of Cauchy and Ferrers-Jackson polynomials
Lemma 2.1. The following recurrence relations are satisfied: pn+1 (x, y) = u pn (x, y) + t qn (x, y)
(4)
qn+1 (x, y) = u qn (x, y) + t pn−1 (x, y),
(5)
and
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for every n = 1, 2, . . ., where p0 (x, y) ≡ 0, t := x y (x + y) and u := x2 + x y + y 2 . For example, we have 2 2 q2 (x, y) = q1 (x, y) , 2 q3 (x, y) = q1 (x, y) q2 (x, y) + 6 t2 , p1 (x, y)q1 (x, y) = 6 t u, etc. Proof. Only the proof of relation (4) is given here: q1 (x, y) pn (x, y) = pn+1 (x, y) + (x2 + y 2 )(x + y)2n+1 − − (x + y)2 + y 2 x2n+1 − (x + y)2 + x2 y 2n+1 = 2 pn+1 (x, y) − 2 x y (x + y)2n+1 − − (2 x y + 2 y 2 ) x2n+1 − (2 x y + 2 x2 ) y 2n+1 = 2 pn+1 (x, y) − 2 t qn (x, y). Lemma 2.2. The following identities hold 2k+1 2 2n + 1 n − k pn (x, y) = (x + x y + y 2 )n−1−3k x y (x + y) n − k 2k + 1 k=0 (n−1)/3 2n + 1 n − k t2k+1 un−1−3k = n − k 2k + 1 k=0 (n−1)/3
:= Vn (t, u),
n = 1, 2, . . .
(6)
2k 2n n − k qn (x, y) = x y (x + y) (x2 + x y + y 2 )n−3k n−k 2k k=0 n/3 2n n − k t2k un−3k = n−k 2k k=0 n/3
:= Wn (t, u),
n = 1, 2, . . .
where t := x y (x + y) and u := x2 + x y + y 2 .
(7)
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Proof. The proof of Lemma (2.2) is followed by induction on n and by applied Lemma (2.1). For example, we have
2n + 1 n − k 2n n − k + t2k+1 un−3k + n − k 2k + 1 n − k 2k k=0 2n n − k 2k+1 n−3k + u = t n−k 2k k=n/3
(n−1)/3
Vn+1 = u Vn + t Wn =
(only if 3|n)
(n−1)/3
=
k=0
(n−1)/3
=
k=0
n/3
=
k=0
(n − k)!(2n + 3) 2k+1 n−3k t u + (2k + 1)!(n − 3k)!
3t2k+1 un−3k =
k=n/3 (only if 3|n)
2n + 3 n − k + 1 2k+1 n−3k u + t n − k + 1 2k + 1 2n + 3 n − k + 1 2k+1 n−3k + u = t n − k + 1 2k + 1 k=n/3
(only if 3|n)
2n + 3 n − k + 1 2k+1 n−3k u . t n − k + 1 2k + 1
The first six polynomials Vn (t, u) and Wn (t, u) are presented below: n Vn (t, u)
Wn (t, u)
1
3t
2u
2
5tu
2 u2
3
7 t u2
2 u3 + 3 t2
4
3 t (3 u3 + t2 )
2 u4 + 8 t2 u
5
11 t u (u3 + t2 )
2 u5 + 15 t2 u2
6
13 t u2 (u3 + 2 t2 ) 2 u6 + 24 t2 u3 + 3 t4
Remark 2.3. If n ∈ N and t, u ∈ Z then the following relations hold true: i) if 2n + 1 is a prime number, then (2n + 1) | Vn (t, u); ii) n ≡ 1 (mod 3) =⇒ t | Vn (t, u) and u | Wn (t, u); iii) n ≡ 2 (mod 3) =⇒ t u | Vn (t, u) and u2 | Wn (t, u); iv) n ≡ 0 (mod 3) =⇒ t u2 | Vn (t, u).
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3
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Identities for the powers of Fibonacci and Lucas numbers
In this section, the polynomials Vn (t, u) and Wn (t, u) determined in Section 2 are used to generate two separate sets of identities for the powers of Fibonacci and Lucas numbers. Below we presented special cases of the following identities: pk (x, y) = Vn (t, u),
k∈N
(8)
qk (x, y) = Wn (t, u),
k ∈ N,
(9)
and for the values x, y which are Fibonacci and Lucas numbers with appropriately selected indices. For example, we obtain the following formulas: 11 11 n 3 Ln+2 − L11 × n − Ln+4 = 33 Ln Ln+2 Ln+4 8 L2n+4 + 11 (−1) 2
n 3 × 8 L2n+4 + 11 (−1) + 3 Ln Ln+2 Ln+4 , and 13 13 13 Fn+2m − F2m Fn+1 − F2m−1 Fn =
2 2 = 13 F2m−1 F2m Fn Fn+1 Fn+2m Fn+2m + F2m−1 F2m Fn Fn+1 × 3 2
2 + F2m−1 F2m Fn Fn+1 + 2 F2m−1 F2m Fn Fn+1 Fn+2m . × Fn+2m
We will also consider three variations of (8) for k = 7: p7 (x, y) − 3 t5 = 5 t u3 3 u3 + 10 t2 , p7 (x, y) − 36 u3 t3 = t 3 u3 + t2 5 u3 + 3 t2 , p7 (x, y) − t3 5 u3 + 3 t2 = 15 t u3 u3 + 3 t2 . The above mentioned identities, are discussed for the following values of arguments (three nontrivial examples are given below): a) for x = F2m Ln+4m+1 and y = F2m+1 Ln (n, m ∈ N0 ): t = F4m+1 F2m F2m+1 Ln Ln+2m Ln+4m+1 , x + y = F4m+1 Ln+2m ,
2 u = F4m+1 Ln+2m − F2m F2m+1 Ln Ln+4m+1 2 2 = F4m+1 L2n+4m − F2m F2m+1 L2n+4m+1 + 2 F4m+1 − F2m F2m+1 L4m (−1)n ; b) for x = Fn and y = Fn+4m+2 (n, m ∈ N0 ): x + y = F2m+1 Ln+2m+1 , t = F2m+1 Fn Ln+2m+1 Fn+4m+2 , 2 u = F2m+1 Ln+2m+1 −Fn Fn+4m+2 = 15 ((L4m+2 −3)L2n+4m+2 + (3L4m+2 −4)(−1)n ); and for x = Ln and y = Ln+4m+2 (n, m ∈ N0 ): x + y = 5 F2m+1 Fn+2m+1 , t = 5 F2m+1 Fn+2m+1 Ln Ln+4m+2 , 2 u = 5F2m+1 Fn+2m+1 − Ln Ln+4m+2 = (L4m+2 + 1)L2n+4m+2 − (3L4m+2 + 4)(−1)n ;
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c) for x = −F2m−1 Fn and y = Fn+2m (or x = −F2m−1 Fn , y = −F−2m Fn+1 , n, m ∈ N0 ): t = −F2m−1 F2m Fn Fn+1 Fn+2m , x + y = F2m Fn+1 , 2 2 u = F2m Fn+1 + F2m−1 Fn Fn+2m = Fn+2m − F2m−1 F2m Fn Fn+1 ; and for x = −F2m−1 Ln and y = Ln+2m (or x = −F2m−1 Ln and y = −F2m Ln+1 , n, m ∈ N0 ): t = −F2m−1 F2m Ln Ln+1 Ln+2m , x + y = F2m Ln+1 , 2 u = F2m Ln+1 + F2m−1 Ln Ln+2m = L2n+2m − F2m−1 F2m Ln Ln+1 .
4
Identities for the powers of elements of conjugate recurrence sequences
The identities of the Fibonacci and Lucas numbers discussed in the previous Section may be generalized to certain pairs of recurrence sequences {xn } and {yn } satisfying the same recurrence equation, yet with different initial conditions, i.e. the sequences described in the following lemma: Lemma 4.1. Let a, b, c, d, x1 , x2 , y1 , y2 ∈ C, a b c = 0. We assume that the following identities hold xn+2 = a xn+1 + b xn , yn+2 = a yn+1 + b yn (10) xn+2 + c xn = yn+1
(11)
yn+2 + c yn = d xn+1 .
(12)
and Moreover, we suppose that the following condition is satisfied: if A, B ∈ C and A xn+1 + B xn = 0 for sufficiently large n ∈ N then A = B = 0. Then we have either c = b and d = a2 + 4 b or yn+1 ≡ a xn+1 . Proof. In turn, from (12), (11) and (10), we have: xn+3 + 2 c xn+1 + c2 xn−1 = d xn+1 , a xn+2 + (b + 2 c − d) xn+1 + c2 xn−1 = 0, (a2 + b + 2 c − d) xn+1 + a b xn + c2 xn−1 = 0, Hence, we obtain:
⎧ ⎪ ⎨ a b = −a (a2 + b + 2 c − d) ⎪ ⎩ c2 = −b (a2 + b + 2 c − d),
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or equivalently:
537
⎧ ⎪ ⎪ a2 + b + 2 c − d = 0 ⎪ ⎪ ⎨ b2 = c2 ⇔ c = ±b ⎪ ⎪ ⎪ ⎪ ⎩ d = a2 + 2 b + 2 c.
If c = −b then yn+1 = a xn+1 , n ∈ N. If c = b then d = a2 + 4 b.
Now, let us suppose that the elements of recurrence sequences {xn } and {yn } satisfy the conditions (10)–(12) for c = b and d = a2 + 4 b. Then the following eight identities could be derived (more precisely 3 × 8 = 24 new identities if we count the additional identities for t and u): a) for x = xn+2 and y = b xn : 2k+1 2k+1 yn+1 − xn+2 − (b xn )2k+1 = Vk (t, u), 2 where t = b xn+2 yn+1 xn and u = yn+1 − b xn xn+2 ; b) for x = yn+2 and y = b yn : 2 2k+1 2k+1 (a + 4b) xn+1 − yn+2 − (b yn )2k+1 = Vk (t, u),
where t = b (a2 + 4 b) yn xn+1 yn+2 and u = (a2 + 4 b)2 x2n+1 − b yn yn+2 ; c) for x = (a2 + 4 b) xn and y = a yn : (2 yn+1 )2k+1 − (a yn )2k+1 − ((a2 + 4 b) xn )2k+1 = Vk (t, u), where t = 2 a (a2 + 4 b) xn yn yn+1 and u = (2 yn+1 )2 − a (a2 + 4 b) xn yn ; d) for x = a xn and y = yn : (2 xn+1 )2k+1 − (a xn )2k+1 − yn2k+1 = Vk (t, u), where t = 2 a xn yn xn+1 and u = (2 xn+1 )2 − a xn yn ; e) for x = zn+3 , y = −a b zn and z ∈ {x, y}: 2k+1 ((a2 + b) zn+1 )2k+1 − zn+3 + (a b zn )2k+1 = Vk (t, u),
where t = −a b (a2 + b) zn zn+1 zn+3 and u = ((a2 + b) zn+1 )2 + a b zn zn+3 ; f) for x = a zn+3 , y = b2 zn and z ∈ {x, y}: ((a2 + b) zn+2 )2k+1 − (a zn+3 )2k+1 − (b2 zn )2k+1 = Vk (t, u), where t = a b2 (a2 + b) zn zn+2 zn+3 and u = ((a2 + b) zn+2 )2 − a b2 zn zn+3 ; g) for x = yn+1 and y = −a xn+1 : 2k+1 (2 b xn )2k+1 − yn+1 + (a xn+1 )2k+1 = Vk (t, u),
where t = −2 a b xn xn+1 yn+1 and u = (2 b xn )2 + a xn+1 yn+1 ; h) for x = (a2 + 4 b) xn+1 and y = −a yn+1 : (2 b yn )2k+1 − ((a2 + 4 b) xn+1 )2k+1 + (a yn+1 )2k+1 = Vk (t, u), where t = −2 a b (a2 + 4 b) yn xn+1 yn+1 and u = (2 b yn )2 + a (a2 + 4 b) xn+1 yn+1 .
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Some relationship for conjugate recurrence sequences of the third order
The following lemma is a generalization of Lemma 4.1 for recurrence sequences of the third order: Lemma 5.1. Let a, b, A, B, w0 , w1 , w2 ∈ C for every w ∈ {x, y, z}. Let us set wn+2 = awn+1 + bwn + wn−1
(13)
for every w ∈ {x, y, z} and n ∈ N and suppose that the following conjugate conditions are satisfied xn+2 + A xn = yn+1 (14) yn+2 + A yn = zn+1
(15)
zn+2 + A zn = B xn+1
(16)
for every n ∈ N. Then the natural condition: αxn+1 + βxn + γxn−1 = 0 for sufficiently large n ∈ N ⇒ α = β = γ = 0 implies the following relations: a) B = (1 + a b)(1 + A3 ), b) if A3 = 1 then b2 = −3 A2 and a2 = −3 A, c) if A3 = 1 then b = (a2 + 3 A)/(A3 − 1) and (b2 − a) A3 + 3 A2 + a = 0 or, equivalently: −a A9 + 3 A8 + 3 a A6 + 3 A5 + 6 a2 A4 + (a4 − 3 a) A3 + 3 A2 + a = 0. Proof. Apply to equality (16) identities (15), (14) and (13) for w = x we obtain the identity (−b A3 + 3 A + a2 + b) xn+2 + ((1 + a b) (1 + A3 ) − B) xn+1 + ((b2 − a) A3 + 3 A2 + a) xn = 0 which implies
⎧ ⎪ ⎪ −b A3 + 3 A + a2 + b = 0 ⎪ ⎪ ⎨ B = (1 + a b) (1 + A3 ) ⎪ ⎪ ⎪ ⎪ ⎩ (b2 − a) A3 + 3 A2 + a = 0.
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Corollary 5.2. If elements of sequences {xn }, {yn } and {zn } satisfy conditions (13)– (16) then the following identities hold true (only three identities – generalizations of the identities from Section 3 are presented below): a) for x = xn+2 and y = A xn : 2k+1 2k+1 − xn+2 − (A xn )2k+1 = Vk (t, u), yn+1 2 − A xn xn+2 ; where t = A xn+2 yn+1 xn and u = yn+1 b) for x = yn+2 and y = A yn : 2k+1 2k+1 − yn+2 − (A yn )2k+1 = Vk (t, u), zn+1 2 where t = A yn xn+1 yn+2 and u = zn+1 − A yn yn+2 ; c) for x = zn+2 and y = A zn : 2k+1 − (A zn )2k+1 = Vk (t, u), (B xn+1 )2k+1 − zn+2
where t = A B zn xn+1 zn+2 and u = (B xn+1 )2 − A zn zn+2 .
6
Third basic identity
The following identity relates to decompositions (6) and (7) and shall be used to generate a set of the identities for the powers of Fibonacci and Lucas numbers. Lemma 6.1. The following identity hold x2n + y 2n =
n r=0
r 2n − l n − l 2n (xy)r (x2 + xy + y 2 )n−r . (−1)l 2n − l l r − l l=0
(17)
ωr,n
Remark 6.2. For small values r (= 1, 2, . . .) coefficients ωr,n of decomposition of polynomial x2n + y 2n from Lemma 6.1 have the form: ω0,n = 1,
ω4,n =
ω1,n = −n,
1 ω2,n = n(n − 3), 2
1 n(n − 3)(n2 − 15n + 38), 24
1 ω3,n = − n(n − 2)(n − 7), 6
1 ω5,n = − n(n − 3)(n − 4)(n − 6)(n − 17). 5!
Coefficients ωr,n for small values of |n − r| it will be generated in Remark 6.5.
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Proof (of Lemma 6.1). We have
x
2n
+y
2n
n 2n x+y 2n − k n k = Ω2n √ (−1) (xy) = (xy)k (x + y)2n−2k xy 2n − k k k=0 n n−k 2n − k 2n k = (−1) (xy)k (x2 + xy + y 2 ) + xy 2n − k k k=0 n n−k 2n − k n−k 2n k = (−1) (xy)k+i (x2 + xy + y 2 )n−k 2n − k k i k=0 i=0 n r 2n − l n − l 2n (xy)r (x2 + xy + y 2 )n−r . = (−1)l 2n − l l r − l r=0 l=0
where Ωn (x) := 2Tn (x/2) is the so called: modified Chebyshev polynomial of the first kind (see [16] where many interesting properties of these polynomials are discussed). Corollary 6.3. We present an explicit form of identities (17) for n = 4, 5, 7. The identities were subsequently transformed to derive at the equivalent yet more attractive forms for further applications: x8 + x4 y 4 + y 8 = (x2 + xy + y 2 )4 − 4xy(x2 + xy + y 2 )3 + + 2x2 y 2 (x2 + xy + y 2 )2 + 4x3 y 3 (x2 + xy + y 2 ) 3 = 3(x2 + xy + y 2 )4 − 2 (x2 + kxy + y 2 ),
(18)
k=0 10
5 5
x +x y +y
10
2
2 5
= (x + xy + y ) − 5xy(x2 + xy + y 2 )4 + + 5x2 y 2 (x2 + xy + y 2 )3 + 5x3 y 3 (x2 + xy + y 2 )2 − − 5x4 y 4 (x2 + xy + y 2 ) = (x2 + xy + y 2 )5 − 5xy(x2 + xy + y 2 )(x + y)2 (x2 + y 2 )2 ,
(19)
x14 + (xy)7 + y 14 = (x2 + xy + y 2 )7 − 7xy(x2 + xy + y 2 )6 + + 14x2 y 2 (x2 + xy + y 2 )5 − 21x4 y 4 (x2 + xy + y 2 )3 + + 7x5 y 5 (x2 + xy + y 2 )2 + 7x6 y 6 (x2 + xy + y 2 ) = (x2 + xy + y 2 )7 − 7xy(x2 + xy + y 2 )(x + y)2 (x2 + y 2 )× × (x2 + y 2 )2 (x2 + xy + y 2 ) + x3 y 3 ,
(20)
Remark 6.4. The identities generated in Corollary 6.3 bring to mind easily verifiable divisibility relations. Accordingly, for each n ∈ N, 3 | n, there is a polynomial pn ∈ Z[x, y], such as: pn (x, y)
2 k=0
4n x2 + k x y + y 2 = 3 x 2 + x y + y 2 − x8n − (x y)4n − y 8n .
(21)
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For example, we derive: p1 (x, y) = 2 (x2 + 3 x y + y 2 ), p2 (x, y) = 2 (x4 + 3 x2 y 2 + y 4 ) (x6 + 9 x5 y + 19 x4 y 2 + 24 x3 y 3 + 19 x2 y 4 + 9 x y 5 + y 6 ) = 2 (x4 + 3 x2 y 2 + y 4 ) (x2 + x y + y 2 )2 (x2 + 7 x y + y 2 ) + x2 y 2 (x2 − x y + y 2 ) . Conversely, for each n ∈ N there are polynomials qn± ∈ Z[x, y], such that: xy
qn± (x, y)
2
6n±1 x2 + k x y + y 2 = x 2 + x y + y 2 − x12n±1 − (x y)6n±1 − y 12n±1 . (22)
k=0
Additionally, if 6n + 1 or 6n − 1 is a prime number, the left side of equation (22) may be supplemented by multiplier 6n + 1 or 6n − 1, respectively. For example: q1+ (x, y) = 7 (x6 + x5 y + 3 x4 y 2 + 3 x3 y 3 + 3 x2 y 4 + x y 5 + y 6 ),
q1− (x, y) = 5 (x2 + y 2 ).
Proof. We have x2 + y 2 = 0
⇐⇒
y = ±i x
and then we get 4n 3 x2 + x y + y 2 − x8n − (x y)4n − y 8n = 4n = 3 ± i x2 − x8n − (±i x2 )4n − (±i x)8n = 0. Now, we have x2 + x y + y 2 = 0
⇐⇒
⎧ ⎪ ⎨ x3 − y 3 = 0, ⎪ ⎩ x − y = 0,
⇐⇒
y = x exp ± i 23 π .
Hence, we obtain: 4n 3 x2 + x y + y 2 − x8n − (x y)4n − y 8n =
8n 8 16 = −x 1 + exp ± i 3 n π + exp ± i 3 n π =
n π
8n n n 8n n nπ nπ = −x = 1+(−1) exp ∓i 3 +(−1) exp ±i 3 1+2 (−1) cos 3 = −x nπ (but cos 3 = (−1)n−1 /2 whenever 3 | n) = −x8n 1 + (−1)2n−1 = 0. Moreover, we get
and
2 2 4n 8n 4n 8n 3 x + xy + y − x − (x y) − y
=0 y=−x
∂ 2 2 4n 8n 4n 8n 3 x + xy + y − x − (x y) − y = 0. ∂x y=−x
2
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6.1 Some Applications Identities (18), (19) and (20) (respectively) can be used to generate some identities for Lucas and Fibonacci numbers. First, for x = Ln , y = Ln+1 : 4 4 3 2 L2n+2 + (−1)n − L8n − Ln Ln+1 − L8n+1 = = 10 F2n+1 L2n+2 2 L2n+2 + (−1)n 2 L2n+3 + 3(−1)n ,
2 L2n+2 + (−1)n
5
5 − L10 − L10 n − Ln Ln+1 n+1 =
2 2 L2n+2 + (−1)n ; = 125 Ln Ln+1 L2n+2 F2n+1
next, for x = Fn , y = Fn+1 : 4 4 8 = 3 2 L2n+2 − (−1)n − 54 Fn8 + Fn Fn+1 + Fn+1 2 2 L2n+2 − (−1)n 2 L2n+3 − 3(−1)n , = 50 F2n+1 Fn+2
2 L2n+2 − (−1)n
5
5 10 = − 55 Fn10 + Fn Fn+1 + Fn+1
2 2 = 55 Fn Fn+1 Fn+2 2 L2n+2 − (−1)n ; F2n+1
now, for x = Ln , y = Ln+3 :
4 L2n+2 − Ln Ln+3
5
5 − L10 − L10 n − Ln Ln+3 n+3 =
2 2 4 Ln+2 − Ln Ln+3 , = 2000 Ln Ln+3 L2n+2 F2n+3
7 2 − L14 − L14 n − Ln Ln+3 n+3 = 280 Ln Ln+3 Ln+2 × 2 2 3 2 × F2n+3 4 Ln+2 − Ln Ln+3 100 F2n+3 4 Ln+2 − Ln Ln+3 + Ln Ln+3 ;
4 L2n+2 − Ln Ln+3
7
and finally, for x = Fn , y = Fn+3 :
2 4 Fn+2 − Fn Fn+3
5
5 10 − Fn10 − Fn Fn+3 − Fn+3 =
2 2 2 = 80 Fn Fn+3 Fn+2 4 Fn+2 − Fn Fn+3 , F2n+3
7 14 − Fn14 − Fn Fn+3 − Fn+3 = 2 3 2 2 2 = 56 Fn Fn+3 Fn+2 F2n+3 4 Fn+2 − Fn Fn+3 4 F2n+3 4 Fn+2 − Fn Fn+3 + Fn Fn+3 . 2 4 Fn+2 − Fn Fn+3
7
Remark 6.5. The coefficients ωr,n from Lemma 6.1 for the values of r near to n are related to Chebyshev polynomials of the first kind Tn (x) = cos(n arccos x), x ∈ [−1, 1] (see [12, 15]), because the following lemma holds:
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Lemma 6.6. We have n/2 n (−1)k n − k (2x)n−2k , Tn (x) = 2 k=0 n − k k
x ∈ R.
(23)
Corollary 6.7. By differentiating the identity (23) for x ∈ (−1, 1) we obtain the following formulas: n sin(n arccos x) √ Tn (x) = =n 1 − x2
Tn (x) =
Tn (x) =
(n−1)/2
k=0
− 2k n − k (−1) (2x)n−2k−1 , n−k k kn
(24)
−n2 cos(n arccos x) n x sin(n arccos x) + = 1 − x2 (1 − x2 )3 n/2−1 n−k n − 2k 1 k (−1) (2x)n−2k−2 ; (25) = 4n n − k k 2 k=0 (n − n3 ) sin(n arccos x) 3 n2 x cos(n arccos x) 3 n x2 sin(n arccos x) − + = (1 − x2 )2 (1 − x2 )3 (1 − x2 )5 (n−3)/2 n−k n − 2k 1 k (−1) (2x)n−2k−3 ; (26) = 24 n n − k k 3 k=0
etc. Using identities (23)–(26), for x = 12 , the following ones can be deduced: (−1)k n− k 2 Tn 2 = 2 cos =n ; n− k k k=0 m 2 2m− l l 2m ωm,m = 2 cos 3 πm = ; (−1) 2m− l l l=0 1
n π3
n/2
(27) (28)
m−1 2 1 1 2√ 2m− k k m− k ωm−1,m = T2m 2 = (−1) 3m sin 3 πm = 2m ; (29) 2 3 2m− k k k=0 √ n/2−1 π π n2 3 n− k n− 2k 1 1 1 k = − cos n 3 + (−1) T n sin n 3 = n ; 4 n 2 3 9 n− k k 2 k=0 (30)
ωm−2,m
√ 2 m2 3 1 1 1 1 1
= T2m 2 − T2m 2 = − cos 3 πm − m sin 23 πm = 4 4 2 3 9 m−2 2m − l m − l l 2m = (−1) ; (31) 2m − l l 2 l=0
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4 4√ 1 1 3m(1 − 2m2 ) sin 23 πm − m2 cos 23 πm = T2m 2 = 24 27 9 m−2 2m − k 2m − 2k 2m k (−1) ; (32) = 2m − k k 3 k=0 ωm−3,m =
1 1 1 1
1 1 1 1 − T2m 12 + T2m 12 = T2m 2 − 4 ωm−2,m = T 8 24 8 24 2m 2 4 2 1√ 1 = 3m(2 − m2 ) sin 23 πm + m2 cos 23 πm = 27 9 m−3 2m − l m − l l 2m = (−1) . (33) 2m − l l 3 l=0
Moreover, from identities (24)-(26) it is possible to generate the following identities between the derivatives of Chebyshev polynomials of the first kind. Lemma 6.8. We have
n/2 Tn (x) k n−k Tn (x) k − 2n = −2 n (−1) (2x)−2k , (2x)n−1 (2x)n n − k k k=1
n/2 (−1)k k 2 n − k Tn (x) Tn (x) 2 Tn (x) (2x)−2k , − 2 (2 n − 1) + 4n = 8n (2x)n−2 (2x)n−1 (2x)n n − k k k=1
(34)
(35)
Tn (x) T (x) T (x) Tn (x) − 2 (3 n − 3) n n−2 + 4 (3 n2 − 3 n + 1) n n−1 − 8 n3 = n−3 (2x) (2x) (2x) (2x)n n/2 3 n−k k k = −32 n (−1) (2x)−2k , (36) n − k k k=1 and the following general formula hold: n/2 l (l−k) l Tn (x) n−k k 2l−1 k+l k (−2) pk,l (n) =2 n (−1) (2x)−2k , n−l+k (2x) n − k k k=0 k=1
(37)
where p0,l (n) = 1,
pl,l (n) = nl ,
pk+1,l+1 (n) = (n − l + k) pk,l (n) + pk+1,l (n),
k = 0, 1, . . . , l − 1.
Hence, for x = 12 we get the following identities: √ n/2 π k 3 π n−k k cos n (−1) − sin n = , 3 3 3 n−k k k=1 √ n/2 2 π π n−k 3 n k k (−1) cos n − (3n − 2) sin n = , 3 3 9 3 n−k k k=1 n/2 √ 3 π π n−k 3 n 2 k k (−1) cos n − (2n − 3n + 2) sin n = . 3 3 9 3 n − k k k=1
(38)
(39)
(40)
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Remark 6.9. Because the derivatives of polynomials Tn (x) are recurrently related [12]: (1 − x2 )Tn(m+2) (x) = (2m + 1) x Tn(m+1) (x) + (m2 − n2 ) Tn(m) (x),
m = 0, 1, . . .
thus, for x = 12 , the following identity is derived: 3 4
Tn(m+2)
1 2
= 12 (2m + 1) Tn(m+1)
1 2
+ (m2 − n2 ) Tn(m)
1 2
hence, the following formulas can be obtained π 2√ 1 1 π Tn = cos n , Tn = , 3 n sin n 2 3 2 3 3 1 π π 4√ 4 2 3 n sin n Tn = − n cos n 2 9 3 3 3 and the general formula: π π (m) 1 = α(m) pm (n) sin n + β(m) qm (n) cos n , Tn 2 3 3 √ where α(m), β(m) ∈ Q, α(m) = 2a(m) 3b(m) 3, β(m) = 2c(m) 3d(m) , and pm (n), qm (n) ∈ Z[n] are polynomials, even and odd degree, respectively.
References [1] L. Carlitz and J.A.H. Hunter: “Sums of Powers of Fibonacci and Lucas Numbers”, Fibonacci Quart., Vol. 7, (1969), pp. 467–473. [2] A.F. Horadam: “Basic properties of a certain generalized sequence of numbers”, Fibonacci Quart., Vol. 3, (1965), pp. 161–176. [3] A.F. Horadam: “Generating functions for powers of a certain generalised sequence of numbers”, Duke Math. J., Vol. 32, (1965), pp. 437–446. [4] T. Koshy: Fibonacci and Lucas Numbers with Applications, Wiley, New York, 2001. [5] T. Mansour: “A formula for the generating functions of powers of Horadam’s sequence”, Australas. J. Combin., Vol. 30, (2004), pp. 207–212. [6] R.S. Melham: “Sums of Certain Products of Fibonacci and Lucas Numbers – Part I”, Fibonacci Quart., Vol. 37, (1999), pp. 248–251. [7] R.S. Melham: “Families of Identities Involving Sums of Powers of the Fibonacci and Lucas Numbers”, Fibonacci Quart., Vol. 37, (1999), pp. 315–319. [8] R.S. Melham: “Sums of Certain Products of Fibonacci and Lucas Numbers – Part II”, Fibonacci Quart., Vol. 38, (2000), pp. 3–7. [9] R.S. Melham: “Alternating Sums of Fourth Powers of Fibonacci and Lucas Numbers”, Fibonacci Quart., Vol. 38, (2000), pp. 254–259. [10] J. Morgado: “Note on some results of A.F. Horadam and A.G. Shannon concerning a Catalan’s identity on Fibonacci Numbers”, Portugal. Math., Vol. 44, (1987), pp. 243–252.
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[11] J. Morgado: “Note on the Chebyshev polynomials and applications to the Fibonacci numbers”, Portugal. Math., Vol. 52, (1995), pp. 363–378. [12] S. Paszkowski: Numerical Applications of Chebyshev Polynomials and Series, PWN, Warsaw, 1975 (in Polish). [13] P. Ribenboim: Fermat’s Last Theorem For Amateurs, Springer, New York 1999. [14] J. Riordan: “Generating functions for powers of Fibonacci numbers”, Duke Math. J., Vol. 29, (1962), pp. 5–12. [15] T. Rivlin: Chebyshev Polynomials from Approximation Theory to Algebra and Number Theory, 2nd ed., Wiley, New York, 1990. [16] R. Witula and D. Slota: “On Modified Chebyshev Polynomials”, J. Math. Anal. Appl., (2006), in print.
