Spin Kostka polynomials and vertex operators

Naihuan Jing and Ning Liu

Vol. 325 (2023), No. 1, 127–146

DOI: 10.2140/pjm.2023.325.127

Abstract

An algebraic iterative formula for the spin Kostka–Foulkes polynomial $K_{\xi \mu }^{-}(t)$ is given using vertex operator realizations of Hall–Littlewood symmetric functions and Schur $Q$-functions. Based on the operational formula, more favorable properties are obtained parallel to the Kostka polynomial. In particular, we obtain some formulae for the number of (unshifted) marked tableaux. As an application, we confirmed a conjecture of Aokage on the expansion of the Schur $P$-function in terms of Schur functions. Tables of $K_{\xi \mu }^{-}(t)$ for $\left|\xi \right|\le 6$ are listed.

Keywords

spin Kostka polynomials, vertex operators

Mathematical Subject Classification

Primary: 05E05, 17B69

Secondary: 20C25, 20C30

Milestones

Received: 25 January 2022

Revised: 2 October 2022

Accepted: 18 March 2023

Published: 3 September 2023