Séminaire Lotharingien de Combinatoire, 85B.37 (2021), 12 pp.
Stephan Pfannerer
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 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.
Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.
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