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

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.