A multiparametric Murnaghan-Nakayama rule for Macdonald polynomials

Abstract

We introduce a new family of operators as multi-parameter deformation of the one-row Macdonald polynomials. The matrix coefficients of these operators acting on the space of symmetric functions with rational coefficients in two parameters $q,t$ (denoted by $\Lambda[q,t]$) are computed by assigning some values to skew Macdonald polynomials in $\lambda$-ring notation. The new rule is utilized to provide new iterative formulas and also recover various existing formulas in a unified manner. Specifically the following applications are discussed: (i) A $(q,t)$-Murnaghan-Nakayama rule for Macdonald functions is given as a generalization of the $q$-Murnaghan-Nakayama rule; (ii) An iterative formula for the $(q,t)$-Green polynomial is deduced; (iii) A simple proof of the Murnaghan-Nakayama rule for the Hecke algebra and the Hecke-Clifford algebra is offered; (iv) A combinatorial inversion of the Pieri rule for Hall-Littlewood functions is derived with the help of the vertex operator realization of the Hall-Littlewood functions; (v) Two iterative formulae for the $(q,t)$-Kostka polynomials $K_{\lambda\mu}(q,t)$ are obtained from the dual version of our multiparametric Murnaghan-Nakayama rule, one of which yields an explicit formula for arbitrary $\lambda$ and $\mu$ in terms of the generalized $(q, t)$-binomial coefficient introduced independently by Lassalle and Okounkov.