A refinement of the Murnaghan-Nakayama rule by descents for border strip tableaux

Abstract

Lusztig's fake degree is the generating polynomial for the major index of standard Young tableaux of a given shape. Results of Springer (1974) and James & Kerber (1984) 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 (2005).Mathematics Subject Classifications: 05A19, 05E10Keywords: Border strip tableaux, descents, Murnaghan-Nakayama rule, fake degree