Transition matrices for symmetric and quasisymmetric Hall-Littlewood polynomials

TL;DR

This paper provides explicit combinatorial interpretations and expansions for transition matrices involving skew Hall-Littlewood polynomials and quasisymmetric Hall-Littlewood polynomials, enriching the understanding of their structure.

Contribution

It introduces new combinatorial interpretations and explicit expansions for various Hall-Littlewood and quasisymmetric polynomials, including starred tableaux for skew cases.

Findings

G-expansions of P_lambda, M_alpha, S_alpha, and K_alpha

Expansion of P_lambda/mu in terms of F_alpha

Introduction of starred tableaux for skew polynomial expansion

Abstract

We introduce explicit combinatorial interpretations for the coefficients in some of the transition matrices relating to skew Hall-Littlewood polynomials P_lambda/mu(x;t) and Hivert's quasisymmetric Hall-Littlewood polynomials G_gamma(x;t). More specifically, we provide: 1) the G-expansions of the Hall-Littlewood polynomials P_lambda, the monomial quasisymmetric polynomials M_alpha, the quasisymmetric Schur polynomials S_alpha, and the peak quasisymmetric functions K_alpha; 2) an expansion of P_lambda/mu in terms of the F_alpha's. The F-expansion of P_lambda/mu is facilitated by introducing starred tableaux.