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 $\mathbf{\mu }=\underset{n}{\underbrace{(\mu ,...,\mu )}}$ and define a $q$-analog of a sum of the plethysm multiplicities.