Skew LittlewoodRichardson Rules from Hopf Algebras Aaron Lauve - PowerPoint PPT Presentation

Skew Littlewood–Richardson Rules from Hopf Algebras Aaron Lauve Texas A&M University Loyola University Chicago joint work with: Thomas Lam University of Michigan Frank Sottile Texas A&M University FPSAC 2010, San Francisco, CA

Hopf Algebras, *@#?% !

Hopf Structure of Λ As an algebra, . . . Λ = Z [ h 1 , h 2 , . . . ] complete homogeneous symmetric functions � h n := x i 1 x i 2 · · · x i n i 1 ≤ i 2 ≤···≤ i n = Z [ e 1 , e 2 , . . . ] elementary symmetric functions � x i 1 x i 2 · · · x i n e n := i 1 < i 2 < ··· < i n Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 3 / 20

Hopf Structure of Λ As an algebra, . . . Λ = Z [ h 1 , h 2 , . . . ] complete homogeneous symmetric functions � h n := x i 1 x i 2 · · · x i n i 1 ≤ i 2 ≤···≤ i n = Z [ e 1 , e 2 , . . . ] elementary symmetric functions � x i 1 x i 2 · · · x i n e n := i 1 < i 2 < ··· < i n � � = span Z Schur functions s λ (a nice basis) Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 3 / 20

Hopf Structure of Λ As a Hopf algebra, . . . we need more maps, Λ = ( Λ , · , ∆ , ε, S ) coproduct ∆ : Λ → Λ ⊗ Λ � ∆( h n ) = h j ⊗ h k j + k = n Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 4 / 20

Hopf Structure of Λ As a Hopf algebra, . . . we need more maps, Λ = ( Λ , · , ∆ , ε, S ) coproduct counit ∆ : Λ → Λ ⊗ Λ ε : Λ → Z � ∆( h n ) = h j ⊗ h k ε ( h n ) = δ n 0 j + k = n (put h 0 = e 0 = 1) Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 4 / 20

Hopf Structure of Λ As a Hopf algebra, . . . we need more maps, Λ = ( Λ , · , ∆ , ε, S ) coproduct counit antipode ∆ : Λ → Λ ⊗ Λ ε : Λ → Z S : Λ → Λ � S ( h k ) = ( − 1 ) k e k ∆( h n ) = h j ⊗ h k ε ( h n ) = δ n 0 j + k = n (put h 0 = e 0 = 1) Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 4 / 20

Hopf Structure of Λ As a Hopf algebra, . . . we need more maps, Λ = ( Λ , · , ∆ , ε, S ) coproduct counit antipode ∆ : Λ → Λ ⊗ Λ ε : Λ → Z S : Λ → Λ � S ( h k ) = ( − 1 ) k e k ∆( h n ) = h j ⊗ h k ε ( h n ) = δ n 0 j + k = n (put h 0 = e 0 = 1) together with some compatibility conditions (omitted) Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 4 / 20

Schur Functions � � s λ

Schur Functions A nice basis Definition. Given a partition λ , s λ is the generating function for the corresponding semistandard Young tableaux SSYT ( λ ) . a ≤ b Ferrers fillings satisfying . < c Example: 1 1 1 1 1 2 1 2 1 3 s = + + + + + · · · 2 3 2 3 2 x 12 x 2 + x 12 x 3 + x 1 x 22 + = + · · · 2 x 1 x 2 x 3 Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 6 / 20

Schur Functions A nice basis Definition. Given a partition λ , s λ is the generating function for the corresponding semistandard Young tableaux SSYT ( λ ) . a ≤ b Ferrers fillings satisfying . < c Example: 1 1 1 1 1 2 1 2 1 3 s = + + + + + · · · 2 3 2 3 2 x 12 x 2 + x 12 x 3 + x 1 x 22 + = + · · · 2 x 1 x 2 x 3 Worth noting: s = h n and s = e n . . . . . . . Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 6 / 20

Schur Functions Classical problem Problem. Understand the coefficients c ν λ, μ in � c ν s λ · s μ = λ, μ s ν . ν Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 7 / 20

Schur Functions Classical problem Problem. Understand the coefficients c ν λ, μ in � c ν s λ · s μ = λ, μ s ν . ν sum over all ways � ( λ + ) to add j Special case (Pieri rule). s λ · h j = s λ + boxes in a λ + j −→ h λ + horizontal strip to the diagram λ . Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 7 / 20

Schur Functions Classical problem Problem. Understand the coefficients c ν λ, μ in � c ν s λ · s μ = λ, μ s ν . ν sum over all ways � ( λ + ) to add j Special case (Pieri rule). s λ · h j = s λ + boxes in a λ + j −→ h λ + horizontal strip to the diagram λ . Example ( j = 3 ) : Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 7 / 20

Schur Functions Nice answer! � ν c ν Problem. Understand the coefficients c ν λ, μ in s λ · s μ = λ, μ s ν . Theorem (Littlewood–Richardson rule) Fix T ∈ SSYT ( ν ) . Then play jeu-de-taquin) c ν � � ( R , S ): R ∈ SSYT ( λ ) , S ∈ SSYT ( μ ) , R ∗ S = T λ, μ = # . Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 8 / 20

Schur Functions Nice answer! � ν c ν Problem. Understand the coefficients c ν λ, μ in s λ · s μ = λ, μ s ν . Theorem (Littlewood–Richardson rule) Fix T ∈ SSYT ( ν ) . Then play jeu-de-taquin) c ν � � ( R , S ): R ∈ SSYT ( λ ) , S ∈ SSYT ( μ ) , R ∗ S = T λ, μ = # . 1 1 Example: Pick T = . Guess R = 1 and S = 1 . 2 2 1 2 ∗ 1 Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 8 / 20

Schur Functions Nice answer! � ν c ν Problem. Understand the coefficients c ν λ, μ in s λ · s μ = λ, μ s ν . Theorem (Littlewood–Richardson rule) Fix T ∈ SSYT ( ν ) . Then play jeu-de-taquin) c ν � � ( R , S ): R ∈ SSYT ( λ ) , S ∈ SSYT ( μ ) , R ∗ S = T λ, μ = # . 1 1 Example: Pick T = . Guess R = 1 and S = 1 . 2 2 1 1 �−→ 1 2 1 2 Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 8 / 20

Schur Functions Nice answer! � ν c ν Problem. Understand the coefficients c ν λ, μ in s λ · s μ = λ, μ s ν . Theorem (Littlewood–Richardson rule) Fix T ∈ SSYT ( ν ) . Then play jeu-de-taquin) c ν � � ( R , S ): R ∈ SSYT ( λ ) , S ∈ SSYT ( μ ) , R ∗ S = T λ, μ = # . 1 1 Example: Pick T = . Guess R = 1 and S = 1 . 2 2 1 1 1 1 �−→ �−→ 1 2 1 2 2 Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 8 / 20

Schur Functions Nice answer! � ν c ν Problem. Understand the coefficients c ν λ, μ in s λ · s μ = λ, μ s ν . Theorem (Littlewood–Richardson rule) Fix T ∈ SSYT ( ν ) . Then play jeu-de-taquin) c ν � � ( R , S ): R ∈ SSYT ( λ ) , S ∈ SSYT ( μ ) , R ∗ S = T λ, μ = # . 1 1 Example: Pick T = . Guess R = 1 and S = 1 . 2 2 1 1 1 1 1 1 �−→ �−→ �−→ 1 2 1 2 2 2 Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 8 / 20

Schur Functions Nice answer! � ν c ν Problem. Understand the coefficients c ν λ, μ in s λ · s μ = λ, μ s ν . Theorem (Littlewood–Richardson rule) Fix T ∈ SSYT ( ν ) . Then play jeu-de-taquin) c ν � � ( R , S ): R ∈ SSYT ( λ ) , S ∈ SSYT ( μ ) , R ∗ S = T λ, μ = # . 1 1 Example: Pick T = . Guess R = 1 and S = 1 . 2 2 1 1 1 1 1 1 �−→ �−→ �−→ 1 2 1 2 2 2 c 21 2 , 1 ≥ 1 Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 8 / 20

Schur Functions More nice facts More Facts. � Same coefficients as for product! c ν ∆( s ν ) = λ, μ s λ ⊗ s μ ( Λ is a self-dual Hopf algebra.) λ, μ Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 9 / 20

Schur Functions More nice facts More Facts. � Same coefficients as for product! c ν ∆( s ν ) = λ, μ s λ ⊗ s μ ( Λ is a self-dual Hopf algebra.) λ, μ � “Skew ν by μ ” ≡ “collect terms = s ν/μ ⊗ s μ ( − ) ⊗ s μ in the coproduct.” μ Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 9 / 20

Schur Functions More nice facts More Facts. � Same coefficients as for product! c ν ∆( s ν ) = λ, μ s λ ⊗ s μ ( Λ is a self-dual Hopf algebra.) λ, μ � “Skew ν by μ ” ≡ “collect terms = s ν/μ ⊗ s μ ( − ) ⊗ s μ in the coproduct.” μ | λ | s λ ′ � � S ( s λ ) = ( − 1 ) = − s E.g., S s Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 9 / 20

Skew Schur Functions � � s λ/μ | μ ⊆ λ

Skew Schur Functions Assaf-McNamara problem Problem. Understand the coefficients in � s λ/μ · s σ/τ = d . Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 11 / 20

Skew Schur Functions Assaf-McNamara problem Problem. Understand the coefficients in � s λ/μ · s σ/τ = d . Natural to take to be Schur functions. Assaf-McNamara take to be skew Schur functions. Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 11 / 20

Skew Schur Functions Assaf-McNamara problem Problem. Understand the coefficients in � s λ/μ · s σ/τ = d . Natural to take to be Schur functions. Assaf-McNamara take to be skew Schur functions. in the spirit of Pieri rule. . . μ λ Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 11 / 20