Q-Kostka polynomials and spin Green polynomials

Abstract

We study the $Q$-Kostka polynomials $L_{\lambda \mu }(t)$ by the vertex operator realization of the $Q$-Hall-Littlewood functions $G_{\lambda }(x;t)$ and derive new formulae for $L_{\lambda \mu }(t)$. In particular, we have established stability property for the Q-Kostka polynomials. We also introduce spin Green polynomials $Y_{\mu }^{\lambda }(t)$ as both an analogue of the Green polynomials and deformation of the spin irreducible characters of ${\mathfrak{S}}_n$. Iterative formulas of the spin Green polynomials are given and some favorable properties parallel to the Green polynomials are obtained. Tables of $Y_{\mu }^{\lambda }(t)$ are included for $n\le 7.$

5 tables