An iterative formula for the Kostka-Foulkes polynomials

TL;DR

This paper introduces an iterative formula for Kostka-Foulkes polynomials based on vertex operator realization, enabling efficient computation for large polynomials and revealing a stability property, with applications to hook-shaped partitions.

Contribution

It provides a novel iterative formula for Kostka-Foulkes polynomials using vertex operators, improving computational efficiency and understanding of their stability.

Findings

Operational formula handles large polynomials efficiently.

Proves a stability property for Kostka-Foulkes polynomials.

Derives a specific formula for hook-shaped partitions.

Abstract

An iterative formula for the Kostka-Foulkes polynomials is given using the vertex operator realization of the Hall-Littlewood polynomials. The operational formula can handle large Kostka-Foulkes polynomials, and a stability property for the Kostka-Foulkes polynomials is shown. We also use our algorithm to give a formula of $K_{\lambda\mu}(t)$ for $\mu$ being hook-shaped.