Affine charge and the $k$-bounded Pieri rule

Jennifer Morse,Anne Schilling

doi:10.46298/dmtcs.2470

Jennifer Morse, Anne Schilling

Open Access

https://doi.org/10.46298/dmtcs.2470

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Abstract

We provide a new description of the Pieri rule of the homology of the affine Grassmannian and an affineanalogue of the charge statistics in terms of bounded partitions. This makes it possible to extend the formulation ofthe Kostka–Foulkes polynomials in terms of solvable lattice models by Nakayashiki and Yamada to the affine setting. Nous proposons une nouvelle description de la règle de Pieri de l’homologie de la variété Grassmannienneaffine et un analogue affine de la statistique de charge en termes de partitions bornées . Il est ainsi possible d’étendreau cas affine la formulation due à Nakayashiki et Yamada des polynômes de Kostka–Foulkes en termes de modèlesde réseaux résolubles.

Highlights

Our study concerns an affine generalization of the Kostka–Foulkes polynomials
Kostka-Foulkes polynomials comprise a fascinating family of polynomials in a single parameter t that arises in diverse contexts such as cohomology of Springer fibers [25, 6], representation theory of GLn(Fq) [21, Ch
Lascoux and Schutzenberger [17, 1] proved that the Kostka–Foulkes polynomials can be expressed as the generating function of the set SSYT(λ, μ) of semi-standard Young tableaux of shape λ and content μ graded by the charge statistic

Summary

Introduction

Our study concerns an affine generalization of the Kostka–Foulkes polynomials. Kostka-Foulkes polynomials comprise a fascinating family of polynomials in a single parameter t that arises in diverse contexts such as cohomology of Springer fibers [25, 6], representation theory of GLn(Fq) [21, Ch. Lascoux and Schutzenberger [17, 1] proved that the Kostka–Foulkes polynomials can be expressed as the generating function of the set SSYT(λ, μ) of semi-standard Young tableaux of shape λ and content μ graded by the (non-negative integer) charge statistic. These form a basis for the subspace Λt(n) = span{Pλ(x; t)}λ1

Affine Pieri rule

Affine charge