A Refinement of the Murnaghan-Nakayama Rule by Descents for Border Strip Tableaux

TL;DR

This paper refines the Murnaghan-Nakayama rule by incorporating a new descent statistic for border strip tableaux, linking it to quasisymmetric functions and providing deeper combinatorial insights.

Contribution

It introduces a new descent statistic for border strip tableaux and connects it to quasisymmetric functions, extending classical results on the Murnaghan-Nakayama rule.

Findings

New descent statistic for border strip tableaux

Connection between border strip descents and quasisymmetric functions

Refined evaluation of fake degrees at roots of unity

Abstract

Lusztig's fake degree is the generating polynomial for the major index of standard Young tableaux of a given shape. Results of Springer and James & Kerber imply that, mysteriously, its evaluation at a $k$-th primitive root of unity yields the number of border strip tableaux with all strips of size $k$, up to sign. This is essentially the special case of the Murnaghan-Nakayama rule for evaluating an irreducible character of the symmetric group at a rectangular partition. We refine this result to standard Young tableaux and border strip tableaux with a given number of descents. To do so, we introduce a new statistic for border strip tableaux, extending the classical definition of descents in standard Young tableaux. Curiously, it turns out that our new statistic is very closely related to a descent set for tuples of standard Young tableaux appearing in the quasisymmetric expansion of LLT polynomials given by Haglund, Haiman and Loehr.