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arXiv · 2105.13750

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

Stephan Pfannerer
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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.

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Stephan Pfannerer. 2021-05-28. A Refinement of the Murnaghan-Nakayama Rule by Descents for Border Strip Tableaux. https://arxiv.org/abs/2105.13750

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