Abstract
We give a new description of the Pieri rule for $k$-Schur functions using the Bruhat order on the affine type-$A$ Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of affine Grassmannians. We show how new combinatorics involved in our formulas gives the Kostka-Foulkes polynomials and discuss how this can be applied to study the transition matrices between Hall-Littlewood and $k$-Schur functions. Nous présentons une nouvelle description, issue de l'ordre de Bruhat du groupe de Weyl affine de type $A$, de la règle de Pieri pour les fonctions $k$-Schur. Ce faisant, nous obtenons une nouvelle formule combinatoire pour les représentants des classes de Schubert de la cohomologie des Grassmannienne affines. Nous décrivons aussi comment notre approche permet d'obtenir les polynômes de Kostka-Foulkes et comment elle peut être appliquée à l’étude des matrices de transition entre les polynômes de Hall-Littlewood et les fonctions $k$-Schur.
Highlights
The dual k-Schur functions arose in [LM08] where it was shown that their coproduct encodes structure constants of the Verlinde fusion algebra for the sln Wess–Zumino–Witten models [TUY89, Ver88] or equivalently, the 3-point Gromov-Witten invariants of genus zero (e.g. [Wit95])
Dual k-Schur functions are defined as the weight generating functions of k-tableaux; a combinatorial object encoding successions of saturated chains in weak order on the affine Weyl group Ak+1
We find that there is a natural realization of weak saturated chains of length as length k − saturated chains in the strong (Bruhat) order on Ak+1. This enables us to prove that the dual k-Schur functions S(λk) can be described in terms of a new combinatorial object called affine Bruhat counter-tableau, or ABC, that encodes successions of strong chains
Summary
The dual k-Schur functions arose in [LM08] where it was shown that their coproduct encodes structure constants of the Verlinde fusion algebra for the sln Wess–Zumino–Witten models [TUY89, Ver88] or equivalently, the 3-point Gromov-Witten invariants of genus zero (e.g. [Wit95]). Dual k-Schur functions are defined as the weight generating functions of k-tableaux; a combinatorial object encoding successions of saturated chains in weak order on the affine Weyl group Ak+1 (see § 2). We find that there is a natural realization of weak saturated chains of length as length k − saturated chains in the strong (Bruhat) order on Ak+1 This enables us to prove that the dual k-Schur functions S(λk) can be described in terms of a new combinatorial object called affine Bruhat counter-tableau, or ABC, that encodes successions of strong chains (see Definition 16). It is conjectured that A(μk)[X; 1] are the k-Schur functions s(μk) prompting our study of ABC combinatorics in the context of Hall-Littlewood polynomials The term affine Schur function is used for dual k-Schur function
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