Abstract
We use Hopf algebras to prove a version of the Littlewood―Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara. We also establish skew Littlewood―Richardson rules for Schur $P-$ and $Q-$functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for k-Schur functions, dual k-Schur functions, and for the homology of the affine Grassmannian of the symplectic group. Nous utilisons des algèbres de Hopf pour prouver une version de la règle de Littlewood―Richardson pour les fonctions de Schur gauches, qui implique une conjecture d'Assaf et McNamara. Nous établissons également des règles de Littlewood―Richardson gauches pour les $P-$ et $Q-$fonctions de Schur et les fonctions de Schur rubbans non commutatives, ainsi que des règles de Pieri gauches pour les $k-$fonctions de Schur, les $k-$fonctions de Schur duales, et pour l'homologie de la Grassmannienne affine du groupe symplectique.
Highlights
We use Hopf algebras to prove a version of the Littlewood–Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara
Their paper included a conjectural skew version of the Littlewood–Richardson rule, and an appendix by one of us (Lam) with a simple algebraic proof of their formula. We show how these formulas and much more are special cases of a simple formula that holds for any pair of dual Hopf algebras
We prove first the formula (h b) · a = h1 (b · (S(h2) a))
Summary
We assume basic familiarity with Hopf algebras, as found in the opening chapters of the book [Mon93]. Let H, H∗ be a pair of dual Hopf algebras over a field k. This means that there is a nondegenerate pairing. H could be finite-dimensional and H∗ its linear dual, or H could be graded with each component finitedimensional and H∗ its graded dual These algebras naturally act on each other [Mon93, 1.6.5]: suppose that h ∈ H and a ∈ H∗ and set h a := h, a2 a1 and a h := h2, a h1. (We use Sweedler notation for the coproduct, ∆h = h1 ⊗ h2.) These left actions are the adjoints of right multiplication: for g, h ∈ H and a, b ∈ H∗, g, h a = g · h, a and a h, b = h, b · a This shows that H∗ is a left H-module under the action in (1). Remark 2 This proof is identical to the argument in the appendix to [AM], where h was a complete homogeneous symmetric function in the Hopf algebra H of symmetric functions
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