A $q$-multinomial expansion of LLT coefficients and plethysm multiplicities

TL;DR

This paper proves a $q$-multinomial expansion for LLT polynomial coefficients in a specific case and introduces a $q$-analog of plethysm multiplicities, advancing understanding of symmetric functions.

Contribution

It provides a new $q$-multinomial expansion for LLT coefficients and defines a $q$-analog of plethysm multiplicities, offering novel insights into symmetric polynomial theory.

Findings

Established a $q$-multinomial expansion for LLT coefficients

Defined a $q$-analog of plethysm multiplicities

Enhanced understanding of symmetric polynomial structures

Abstract

Lascoux, Leclerc and Thibon\cite{LLT} introduced a family of symmetric polynomials, called LLT polynomials. We prove a $q$-multinomial expansion of the coefficients of LLT polynomials in the case where $ \boldsymbol{\mu} = \underbrace{(\mu,...,\mu)}_{n}$ and define a $q$-analog of a sum of the plethysm multiplicities.