Skew Littlewood Richardson Rules from Hopf Algebras

Transcription

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

2 Hopf Algebras,

3 Hopf Structure of Λ As an algebra,... Λ = Z[h 1, h 2,...] h n := x i1 x i2 x in i 1 i 2 i n complete homogeneous symmetric functions = Z[e 1, e 2,...] elementary symmetric functions e n := x i1 x i2 x in i 1 <i 2 < <i n = span Z Schur functions sλ (a nice basis) Aaron Lauve (TAMU, LUC) Skew Littlewood Richardson Rules 4 August / 20

4 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 (h n ) = h j h k ε(h n ) =δ n0 S(h k ) = ( 1) k e k 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 / 20

5 Schur Functions sλ

6 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: < s = = x 1 2 x 2 + x 1 2 x 3 + x 1 x 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 / 20

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

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

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

10 Skew Schur Functions sλ/μ μ λ

11 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 μ μ λ to be skew Schur functions. in the spirit of Pieri rule... λ + sum over ways to grow (λ λ + ) and shrink (μ μ ) the skew-shape λ/ μ Aaron Lauve (TAMU, LUC) Skew Littlewood Richardson Rules 4 August / 20

12 Skew Schur Functions Assaf-McNamara problem Problem. Understand the coefficients in s λ/μ s σ/τ = d s. Special case (Assaf McNamara rule). Given μ λ and n 0, s λ/μ h n = j+k=n λ +j h λ + ( 1) k s λ + /μ. add (+j) a horizontal strip toλ μ k v μ remove ( k) a vertical strip fromμ Proof: a sign-reversing involution. Aaron Lauve (TAMU, LUC) Skew Littlewood Richardson Rules 4 August / 20

13 Skew Schur Functions Assaf-McNamara problem Problem. Understand the coefficients in s λ/μ s σ/τ = d s. Theorem (Lam L Sottile) Fix T SSYT(σ). Then s λ/μ s σ/τ = ( 1) R s λ + /μ, the sum taken over R SSYT((μ/μ ) ), R + SSYT(λ + /λ), and S SSYT(τ) such that R R + S = T. Proof: a Hopf formula. Aaron Lauve (TAMU, LUC) Skew Littlewood Richardson Rules 4 August / 20

14 A Hopf Formula

15 Dual Pairs of Hopf Algebras The harpoon action Fix H and its dual H under a bilinear pairing, : H H. For h H and a H, put (h) = (h) (1) h (2) and (a) = (a) (1) a (2). Definition. Construct left actions (): coproduct + evaluation. ah := (h) h(2), a h (1) and ha := Examples: Take H = Λ ( Λ under Hall inner product ) s μ s λ = id, s μ s λ/κ s κ = s λ/μ e k s μ = s μ/. = κ s μ μ v k μ (a) a dual Pieri rule h, a(2) a(1). Aaron Lauve (TAMU, LUC) Skew Littlewood Richardson Rules 4 August / 20

16 Dual Pairs of Hopf Algebras A Hopf formula Lemma For all g, h H, for all a H, we have (ag) h = S(h(2) )a (g h (1) ). (h) Example: Take a = s μ, g = s λ, and h = h n. LHS: (s μ s the left-hand side in the λ) h n = s λ/μ h n Assaf McNamara Pieri rule RHS: S(hk )s μ (sλ h j ) (unravelling coproduct) j+k=n Aaron Lauve (TAMU, LUC) Skew Littlewood Richardson Rules 4 August / 20

17 Dual Pairs of Hopf Algebras A Hopf formula RHS = j+k=n ( 1) k e k s μ (sλ h j ) (antipode) RHS = RHS = j+k=n j+k=n ( 1) k e k s μ ( 1) k μ k v μ s μ λ +j h λ + s λ+ λ +j h λ + s λ+ (Pieri rule) (dual Pieri rule) RHS = j+k=n λ +j h λ + μ k v μ ( 1) k s λ + /μ the right-hand side in the Assaf McNamara Pieri rule Aaron Lauve (TAMU, LUC) Skew Littlewood Richardson Rules 4 August / 20

18 Dual Pairs of Hopf Algebras More generally... Fix basis indexing set P Fix bases H = span L λ λ P and H = span R λ λ P Define skewing as harpooning: L λ/μ := R μ L λ Find structure coefficients: L λ L μ = ν cν λ,μ L ν and R λ R μ = Theorem IF: need not be the same! ν dν λ,μ R ν S(L λ ) = e(λ) L T(λ) for functions e :P and T :P P, THEN: L λ/μ L σ/τ = μ,λ +,π,κ,ρ e(ρ) d κ π,ρ dσ κ,τ c μ μ,t(ρ) c λ+ λ,π L λ + /μ Aaron Lauve (TAMU, LUC) Skew Littlewood Richardson Rules 4 August / 20

19 Dual Pairs of Hopf Algebras More generally... We have found combinatorial interpretations for skew product coefficients in several settings arising in algebraic combinatorics: Schur functions Schur P-functions and Q-functions Noncommutative ribbon Schur functions ( Gessel quasi-symmetric functions) k-schur functions (Pieri rule only) Schubert classes for affine Gr Sp(2n,C) (Pieri rule only) Aaron Lauve (TAMU, LUC) Skew Littlewood Richardson Rules 4 August / 20

20 T H E E N D Thank You! References Assaf, McNamara, A Pieri rule for skew shapes, FPSAC Lam, Lauve, Sottile, Skew Littlewood Richardson rules from Hopf algebras, FPSAC Macdonald, Symmetric functions and Hall polynomials, 2 ed., Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series, v.82., Aaron Lauve (TAMU, LUC) Skew Littlewood Richardson Rules 4 August / 20