Основы теории представлений

!" #"$ !"%! &...

Author: Шейнман О.К..

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@= 0 ;,=..2 0 *>*5,* 5.* -*/* .,< ,*2 SIH C 0 ;,=.. :,-5+? .,, " *+6 ;55,E-5 1 → SIH C" .,,2 < >97-/ A5*= ∈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| | − H C H!C

∈

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→ SIH C

> H C T H C 6 9:; ∈ " - 666 -5,E-< 55 -*/*+? .,,*2 2 . .,*-92 * 666 - -< 5,E-55 .,*-/" *;->*2 ,5-*+ ) 2 .,*66 - ;,-- -*/*+? .,,* ;< ,-9 .,*-/" - -5,E-5 =7=2 .,*-6 *< +96 ) H:*>*-1 ∼ T C" *B H< A-**-C *.,--5+? .,*-/ ;,=..+ :*< >-5 " - %

* + & + ! #$ " K, - :, 5,E-5 -*,-**+" < *2 B - *-? -: 666 *=+5 ..,,*52 -: . =97-5 .,*-5" . / )0123, ) #$ " @= H C T H C 2 - 0 ?,,--> *>*- " ; H − C H C T H CH − C2 " " − 05,E-5" UVWH − C T 2 .A5= − T H,5 !"(C" O55 '=, 0 **/ ,-,-/ *.,--5- ,-- .,< *-/" .,-2 **/ 5 ,B*-6 .,*-/ 0 .< ,*- -;**; H* * 6,*;C .97; .,," 4$ !$ * ) , '+ & 1 → → * & ∈ #$ " @= 1 → 0 -5,E-52 ; − 0 5,E-5 2 "" − T " 56 + !

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@= 0 .,*- 2 0 .,*- - T ⊕ " @,*- 2 .,** E,5=/ H CH ⊕ CT T H C ⊕ H C 2 *+6 ! .,*-/ - - :*>6 ⊕ " @,65=9 =55= A5.6, .,*-6 :*>-5 " @= 0 .,*- 2 0 .,*- - T ⊗ " @,*- ;,=..+ × 2 .,** E,5=/ H × CH ⊗ C T H C ⊗ H C H ∈ 2 ∈ 2 ∈ 2 ∈ C2 *+6 .,*-/ - - :*>6 ⊗ " ,5-* .,-5*6< 6 7 */ *;->*/ .,8--2 ,/ - 2 - 0 .,<

*-6 */ - / B ;,=..+ " @,- A52 . .,*-92 H ⊗ CH CH ⊗ C T H C ⊗ H C "

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C * T ⊕ !! % % , & + & ⊂ ! & T ⊕ ½ ⊕ ⊕ #$ " 5, +:-,5 - =-62 > ½ 0 5-5*6 *--56 .-5 H ,/*5 => *:, B6 .=+5C" ; 6 9:; T !

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∼ T & T ! #$ " >*9 -5,E-5 5B* >-2 > .,*-6 - /=9 *5 - 5 B .,,* 2 -2 **2 5+ -55 ,B*-6 .,65=9 =55= *< .,--5+? ..,,*1 T ⊕ ⊕ - T ⊕ ⊕ 2 > ∼ T - ∼ T " @,5 -*=8-/ . " @,-5*-5 55= !"! ..,,*= T " M T ⊕ ½ ⊕ ⊕ 2 ; ∼ ½ 0 *,6 > .,*-/ " ; T ∼ " **; . =-9 ,B*-6 T ⊕ ⊕ T ½ ⊕⊕ }−{ -55 ½ ⊕⊕ ∼ T ½ ⊕ ⊕ 2 ; { } T {!

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Q,62 > * ;,=..+ * 5*B 2 < - B5= ∈ .* :,B*- H C1 → 2 .,->5 H C T H CH H C C H C -- H C T H H CC H C H C" H H C 2 - H C :*>9 :, A5*< .,- :,B*-- H C"C :-? =>6? .,.;62 > H C T !" :,B*-6 H C > *+96 H*< *2 +5- -- .,+5-C" .,-2 5+ :=5 .- 5 H C - 5 H C" - * * ,==, .;->; .,,* H;< ; 5*;:,-62 *-->; 5*;:,-62 .,,* 5< ,/C2 -;- .,.;96 *.,,+*+5- H**2 ;< -5-2 *-->-5-2 -5,-5+5-C" + A5* ∈ *+6 5*B { | ∈ }" B5= /-9 * 5*B **+5 :,5 < = .,*- .,,* E=*8-/ * A5 5*B1 H CH C T H − C 6 ; /-6 H CH C T H C 6 .,; /-6

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@,-5,5 666 ! ;,=..+ 1 ^V[H C H C T H − C = -5 5 5J*- ,5-*2 * 55 ^V[ 66< 6 .,*-5 × 2 * " 5 * 5* 5+ :=5 .,-,B-< 6 ,-8-**/ ,5-*;--" ;,*->*- ^V[ * .;,=..+ × { } - { } × 9 < ** .,*-6 H C H C T H − C - H C H C T H C @, - *-? *+6 !2 , 0 !" *- /-* 6696 .,*-65- ;,=..+ 2 .,->5 = 5=2 0.,5= /-9 ;,=..+ * 5/ :" 56 ! U\]^V[ T | |

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55 .,*- 1 → SIH C" - +:, :- 2 5+ .=>-5 -5,E-5 SIH C → SIH C2 ; T U\] " */ ;< 55,E-5 → SIH C 5+ :=5 *+ 2 -56 -= . H C =B * 5 .,, .,< *-62 ; 5,-8=" M5*+ H C A/ 5,-8+2 ,5,-5+ E=*8-- * 2 *+96 ) .,< *-6 " :,*2 .= 5+ -55 ;55,E-5 → SIH C" +:-,6 :- - -.=6 *-97-/ .,- A5 -5,E-5 SIH C → SIH C2 .=>-5 .,*- " @,- +:, ,=;; :- .=>-6 A-** .,*-" -5 :,52 >- ,*-6 ,-.,*-/ * ,->-6 5B= .,,*/ - 5,->*/ E,5/ .,*-6" .,-2 5+ :=5 .,?- */ ,=;/ : :+? ;," B : ;, 5+ :=5 .,?- .,,* _ZUH C A*5,E-5 H.,,C .,,* =975= .,,*= 5,-8 `YWaH × C2 - A .,-- A-**+5 .,*-65"

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