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.

Source : oai:HAL:hal-01283121v1

Volume: DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)

Section: Proceedings

Published on: January 1, 2012

Submitted on: January 31, 2017

Keywords: Bruhat order, Macdonald polynomials, Hall-Littlewood polynomials, $k$-tableaux,$k$-Schur functions, Pieri rule,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

This page has been seen 75 times.

This article's PDF has been downloaded 84 times.