A multiparametric Murnaghan-Nakayama rule for Macdonald polynomials

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.Comment: 32 pp, 2 figure

Combinatorial Representation Theory

We attempt to survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. We give a personal viewpoint on the field, while remaining aware that there is much important and beautiful work that we have not been able to mention