Graded characters, demazure multiplicities, and chebyshev polynomials

TL;DR

This paper investigates the multiplicities of Demazure modules within certain $ ext{sl}_2[t]$-modules, expressing generating functions via Chebyshev polynomials and providing new proofs for graded multiplicities using elementary recursive methods.

Contribution

It introduces a novel elementary approach to compute Demazure module multiplicities and graded characters, generalizing previous results and connecting them to Chebyshev and Hall--Littlewood polynomials.

Findings

Generating functions for Demazure multiplicities expressed via Chebyshev polynomial quotients.

New proof of graded multiplicities using recursive structures and cocharge Kostka--Foulkes polynomials.

Explicit formulas for graded characters in terms of Hall--Littlewood polynomials.

Abstract

In this paper, we study numerical multiplicities of Demazure modules in the excellent filtration of $\mathfrak{sl}_2[t]$-modules $V(\xi)$, where $V(\xi)$ denotes the fusion product associated to a partition $\xi$. We express generating functions for the numerical multiplicities of level $m$ Demazure modules in excellent filtrations of $V(\xi)$ in terms of quotients of Chebyshev polynomials, thereby generalizing earlier results for fat hook partitions. We also revisit the graded multiplicities of irreducible $\mathfrak{sl}_2$-modules in $V(\xi)$ and provide a new and self-contained proof of their description in terms of cocharge Kostka--Foulkes polynomials. While this connection has been established in earlier works, our approach is elementary and relies only on recursive structures arising from short exact sequences of fusion products. As a consequence, we obtain a direct and self-contained derivation of the graded characters of $V(\xi)$ in terms of Hall--Littlewood polynomials with coefficients given by cocharge Kostka--Foulkes polynomials.