Branching algebras for classical groups Soo Teck Lee National - PowerPoint PPT Presentation

Branching algebras for classical groups Soo Teck Lee National University of Singapore Survey on some of the works done by Roger Howe and his collab- orators (Jackson, Kim, Lee, Tan, Wang, Willenbring) on branch- ing algebras. 1

Setting: G : complex classical group H : certain subgroup of G (mostly symmetric subgroup) Examples of ( G , H ): (GL n , O n ), (Sp 2 n , GL n ), (GL n × GL n , GL n ) 2

Setting: G : complex classical group H : certain subgroup of G (mostly symmetric subgroup) Examples of ( G , H ): (GL n , O n ), (Sp 2 n , GL n ), (GL n × GL n , GL n ) Branching problem for ( G , H ) If V be an irreducible rational G module, what is V | H ? (1) We have � V | H = m U , V U U where the U s are irreducible H modules. Determine the branching multiplicities m ( U , V ). (2) Describe the H submodules of V . 3

Use highest weight theory: Let B H = A H U H be a Borel subgroup of H , and consider V U H = { v : g . v = v ∀ g ∈ U H } . This is a module for A H , and � V U H = ( V U H ) λ λ where ( V U H ) λ = { v ∈ V U H : a . v = λ ( a ) v ∀ a ∈ A H } ( H highest weight vectors of weight λ ) Then � (dim( V U H ) λ ) U λ V | H ≃ λ where U λ = irreducible H module with highest weight λ . 4

� (dim( V U H ) λ ) U λ Branching rule G ↓ H : V | H ≃ λ Questions: 1. How to calculate dim( V U H ) λ ? 2. Can we describe a basis for ( V U H ) λ ? 5

Howe’s approach: (i) Consider a “concrete” algebra R G with an G action such that R G is decomposed as a multiplicity free sum of irreducible G submodules as � R G = V i . i 6

Howe’s approach: (i) Consider a “concrete” algebra R G with an G action such that R G is decomposed as a multiplicity free sum of irreducible G submodules as � R G = V i . i (ii) Consider the subalgebra of U H invariants: � A ( G , H ) : = R U H V U H = . G i i It is a A H module. 7

Howe’s approach: (i) Consider a “concrete” algebra R G with an G action such that R G is decomposed as a multiplicity free sum of irreducible G submodules as � R G = V i . i (ii) Consider the subalgebra of U H invariants: � A ( G , H ) : = R U H V U H = . G i i It is a A H module. (iii) The structure of A ( G , H ) encodes part of the branching rule from G to H , so call it a branching algebra for ( G , H ). 8

Howe’s approach: (i) Consider a “concrete” algebra R G with an G action such that R G is decomposed as a multiplicity free sum of irreducible G submodules as � R G = V i . i (ii) Consider the subalgebra of U H invariants: � A ( G , H ) : = R U H V U H = . G i i It is a A H module. (iii) The structure of A ( G , H ) encodes part of the branching rule from G to H , so call it a branching algebra for ( G , H ). (iv) Study the branching algebra A ( G , H ) . 9

Basic example: G = GL n × GL n , H = ∆ (GL n ) = { ( g , g ) : g ∈ GL n } . 10

Basic example: G = GL n × GL n , H = ∆ (GL n ) = { ( g , g ) : g ∈ GL n } . Polynomial representations of GL n are parametrized by Young di- agrams with at most n rows (i.e. with depth ≤ n ). D (Young diagram) −→ ρ D n (representation of GL n ) . 11

Basic example: G = GL n × GL n , H = ∆ (GL n ) = { ( g , g ) : g ∈ GL n } . Polynomial representations of GL n are parametrized by Young di- agrams with at most n rows (i.e. with depth ≤ n ). D (Young diagram) −→ ρ D n (representation of GL n ) . Example of a Young diagram: D = = (6,4,4,2) or (6,4,4,2,0) etc 12

Branching problem for ( G , H ) = (GL n × GL n , GL n ) : For Young diagrams D and E , ρ D n ⊗ ρ E n is an irreducible module for GL n × GL n . Restrict the action to GL n = ∆ (GL n ), and describe the GL n mod- ule structure of ρ D n ⊗ ρ E n . 13

Branching problem for ( G , H ) = (GL n × GL n , GL n ) : For Young diagrams D and E , ρ D n ⊗ ρ E n is an irreducible module for GL n × GL n . Restrict the action to GL n = ∆ (GL n ), and describe the GL n mod- ule structure of ρ D n ⊗ ρ E n . In other wrods, we want to decompose the GL n tensor product ρ D n ⊗ ρ E n . 14

Branching problem for ( G , H ) = (GL n × GL n , GL n ) : For Young diagrams D and E , ρ D n ⊗ ρ E n is an irreducible module for GL n × GL n . Restrict the action to GL n = ∆ (GL n ), and describe the GL n mod- ule structure of ρ D n ⊗ ρ E n . In other wrods, we want to decompose the GL n tensor product ρ D n ⊗ ρ E n . So the branching rule in this case is the Littlewood-Richardson (LR) Rule: � ρ D n ⊗ ρ E c F D , E ρ F n = n , F where c F D , E is the number of LR tableaux of shape F / D and con- tent E . 15

We want to construct a branching algebra A ( G , H ) which encodes the LR rule. 16

We want to construct a branching algebra A ( G , H ) which encodes the LR rule. � ρ D n ⊗ ρ E First we need an algebra R G = n . D , E 17

We want to construct a branching algebra A ( G , H ) which encodes the LR rule. � ρ D n ⊗ ρ E First we need an algebra R G = n . D , E Then     1                 ∗  1    A ( G , H ) : = R U H         where U H = U n = ∈ GL n .    ...  G                     0     1   18

The construction of R G : GL n × GL k acts on the algebra P (M nk ) of polynomial functions on M nk ( C ): � ρ D n ⊗ ρ D P (M nk ) � (GL n , GL k ) duality) k D 19

The construction of R G : GL n × GL k acts on the algebra P (M nk ) of polynomial functions on M nk ( C ): � ρ D n ⊗ ρ D P (M nk ) � (GL n , GL k ) duality) k D Extracting U k invariants: � � � U k ≃ � P (M nk ) U k ≃ ρ D ρ D ρ D n ⊗ n . k D D 20

The construction of R G : GL n × GL k acts on the algebra P (M nk ) of polynomial functions on M nk ( C ): � ρ D n ⊗ ρ D P (M nk ) � (GL n , GL k ) duality) k D Extracting U k invariants: � � � U k ≃ � P (M nk ) U k ≃ ρ D ρ D ρ D n ⊗ n . k D D Take another copy: � � � U ℓ ≃ � P (M n ℓ ) U ℓ ≃ ρ E ρ E ρ E n ⊗ n . ℓ E E 21

Form the tensor product:     � � �     R G : = P (M nk ) U k ⊗ P (M n ℓ ) U ℓ ≃     ρ D ρ E ρ D n ⊗ ρ E  ⊗  ≃          n   n  n   D E D , E 22

Form the tensor product:     � � �     R G : = P (M nk ) U k ⊗ P (M n ℓ ) U ℓ ≃     ρ D ρ E ρ D n ⊗ ρ E  ⊗  ≃          n   n  n   D E D , E Extract the U n = ∆ ( U n ) invariants: P (M nk ) U k ⊗ P (M n ℓ ) U ℓ � U n ≃ � � U n . � � A ( G , H ) : = R U H ρ D n ⊗ ρ E = n G D , E 23

Form the tensor product:     � � �     R G : = P (M nk ) U k ⊗ P (M n ℓ ) U ℓ ≃  ρ D   ρ E  ρ D n ⊗ ρ E   ⊗     ≃      n   n  n   D E D , E Extract the U n = ∆ ( U n ) invariants: P (M nk ) U k ⊗ P (M n ℓ ) U ℓ � U n ≃ � � U n . � � A ( G , H ) : = R U H ρ D n ⊗ ρ E = n G D , E It can be further decomposed as   �  �  � � � U n   A ( D , E , F )  ρ D n ⊗ ρ E  A ( G , H ) ≃  = n   ( G , H ) F    D , E F D , E , F where � � U n A ( D , E , F ) ρ D n ⊗ ρ E F = highest weight vectors of weigth F in ρ D n ⊗ ρ E = n n ( G , H ) dim A ( D , E , F ) = multiplicity of ρ F n in ρ D n ⊗ ρ E n ( G , H ) Howe et al. call A ( G , H ) a GL n tensor product algebra. 24

It turns out that A ( G , H ) also encodes another branching rule: P (M nk ) U k ⊗ P (M n ℓ ) U ℓ � U n ≃ P (M nk ⊕ M n ℓ ) U n × U k × U ℓ � A ( G , H ) = R U H = G U n × U k × U ℓ    �    ≃ P (M n ( k + ℓ ) ) U n × U k × U ℓ ≃ ρ F n ⊗ ρ F       k + ℓ     F � � � U k × U ℓ ≃ � U k × U ℓ . � U n ⊗ � � � ρ F ρ F ρ F ≃ n k + ℓ k + ℓ F F 25

It turns out that A ( G , H ) also encodes another branching rule: P (M nk ) U k ⊗ P (M n ℓ ) U ℓ � U n ≃ P (M nk ⊕ M n ℓ ) U n × U k × U ℓ � A ( G , H ) = R U H = G U n × U k × U ℓ    �    ≃ P (M n ( k + ℓ ) ) U n × U k × U ℓ ≃ ρ F n ⊗ ρ F       k + ℓ     F � � � U k × U ℓ ≃ � U k × U ℓ . � U n ⊗ � � � ρ F ρ F ρ F ≃ n k + ℓ k + ℓ F F A ( G , H ) encodes the branching rule for GL k + ℓ ↓ GL k × GL ℓ . So the algebra A ( G , H ) encodes two branching rules. 26