Schur Weyl Duality

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Niveau: Supérieur, Doctorat, Bac+8
Schur-Weyl Duality Bruno Duchesne January 30 2006 Introduction For any given complex vector space V of finite dimension k, we will consider the two natural actions of GL(V ) and Sd on V ?d: { GL(V ) V ?d : ?g ? GL(V ), g · (v1 ? · · · ? vd) = gv1 ? · · · ? gvd Sd V ?d : ?? ? Sd, ? · (v1 ? · · · ? vd) = v?(1) ? · · · ? v?(d) The first thing to notice is that these two actions commute: ??, g; g · (? · (v1 ? · · · ? vd)) = g · ( v?(1) ? · · · ? v?(d) ) = gv?(1) ? · · · ? gv?(d) = ? · (gv1 ? · · · ? gvd) = ? · (g · (v1 ? · · · ? vd)) We want to study these two representations and more precisely to see the links between the irreducible representations of GL(V ) and the ones of Sd. We begin by giving a way to obtain every irreducible representation of Sd thanks to Young diagrams. Then we introduce Weyl's construction and Schur functors that make the link between GL(V ) and Sd.

  • groups say

  • conjugacy classes

  • s?v

  • sym?kv ?

  • young diagrams

  • between conjugacy

  • partition ?

  • schur polynomial


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V k
⊗dGL(V) S Vd

⊗dGL(V)V : ∀g ∈GL(V), g·(v ⊗···⊗v ) =gv ⊗···⊗gv1 d 1 d
⊗dS V : ∀σ ∈S , σ·(v ⊗···⊗v ) =v ⊗···⊗vd d 1 d σ(1) σ(d)

∀σ,g; g·(σ·(v ⊗···⊗v )) = g· v ⊗···⊗v =gv ⊗···⊗gv1 d σ(1) σ(d) σ(1) σ(d)
=σ·(gv ⊗···⊗gv ) =σ·(g·(v ⊗···⊗v ))1 d 1 d
GL(V) Sd
Sd
GL(V) S GL(V)d
C[S ]d
Sd
S p(d)d
d
p(d) = #{(λ ,··· ,λ ), d =λ +···+λ λ ≥···≥λ ≥ 0}1 k 1 k 1 k
λ = (λ ,··· ,λ ) λ i1 k i
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(5,3,2)
Sd
P Q Sd
P =P ={σ ∈S , σ }λ d
Q =Q ={σ ∈S , σ }λ d
C[S ]d
P Q
X X
a = e b = sgn(σ)eλ σ λ σ
σ∈P σ∈Q
⊗dV
O O O
λ λ λ ⊗d1 2 k(a ) =Sym V Sym V ··· Sym V ⊂Vλ
μ λ
O O O
μ μ μ ⊗d01 2 k(b ) =∧ V ∧ V ··· ∧ V ⊂Vλ
c λλ
c =a ·b ∈C[S ]λ λ λ d
umeratedhpreservrows:wwartitioneptoonjugatesymmetriccwthepobtain(aouxesyiscolumns,:andandwswrotherehangenamepreservtionesImeacthh,columnerterctheIntothetoeacgroupbalgebraofinsubgroupsoudeneyyitaIfoewllthedemonstraterillustratewithexamples:vooungducetablethewtfollwcutivobelemenumtussoungcorrespassoonding.to.therepresensubgroupscanthenalsoandgroupandthe:etationarepresenoregularttheoducibltirreaIncanonicalThisisleadxes),thebwIfeeopnjectionconjugateouraexampleticofprowehadescribe:anddiagramconstructytoaugivthisWwieNocanoobservaseelytheconseactionsoofthethesebtnwLetodeneelemenYtssymmetrizeronyhangeciatedtoa.wandTheso:.esolofImtationsYeoungdiagramsthedenitionfollototherwingforowillemthatanderatorswi,notwbutewithin2trc Vλ λ
S Sd d
S Sd dP
λ = (d) c =a = eλ λ σσ∈Sd
X X
V =C[S ]· e =C· e ,λ d σ σ
σ∈S σ∈Sd d
P
λ = (1,··· ,1) c =b = sgn(σ)eλ λ σσ∈Sd
X X
V =C[S ]· sgn(σ)e =C· sgn(σ)e ,λ d σ σ
σ∈S σ∈Sd d
c λ dλ
⊗dGL(V) S Vd
⊗dc V S Vλ λ
S V = (c | ⊗d)λ λ V
GL(V) S V 7−→S Vλ λ
λ
dλ = (d) V 7−→ Sym V
dλ = (1,··· ,1) V 7−→ ∧ V g ∈ GL(V)
⊗d ⊗dS V ⊂ V V GL(V)λ
χ (g) x ,··· ,xS 1 kλ
g V k =dim(V) λ = (d)
dS V =Sym V χ (g) =H (x ,··· ,x ),λ S V d 1 kλ
H d λ = (1,··· ,1)d
dS V =∧ V χ (g) =E (x ,··· ,x ),λ S V d 1 kλ
Ed
χ (g) =S (x ,··· ,x )S V λ 1 kλ
S (x ,··· ,x )λ 1 k

λ +k−ii(x ) 1≤i,j≤kj
S (x ,··· ,x ) =