Kostka functions associated to complex reflection groups

TL;DR

This paper explores the combinatorial properties of Kostka functions linked to complex reflection groups, revealing a new combinatorial description for certain cases using geometric insights from the enhanced variety.

Contribution

It introduces a new combinatorial description of Kostka functions for specific cases, connecting algebraic and geometric perspectives.

Findings

K^-_{,}(t) has a Lascoux-Schfctzenberger type combinatorial description.

Establishes a connection between Kostka functions and intersection cohomology of the enhanced variety.

Provides new insights into the structure of Kostka functions associated with complex reflection groups.

Abstract

Kostka functions $K^{\pm}_{\lambda, \mu}(t)$ associated to complex reflection groups are a generalization of Kostka polynomials, which are indexed by a pair $\lambda, \mu$ of $r$-partitions and a sign $+, -$. It is expected that there exists a close connection between those Kostka functions and the intersection cohomology associated to the enhanced variety $X$ of level $r$. In this paper, we study combinatorial properties of Kostka functions by making use of the geometry of $X$. In particular, we show that if $\mu$ is of the form $\mu = (-,\dots, -, \xi)$ and $\lambda$ is arbitrary, $K^-_{\lambda, \mu}(t)$ has a Lascoux-Sch\"utzenberger type combinatorial description.