Generating functions for fixed points of the Mullineux map

Document Type

Article

Publication Date

6-2025

Department

Department of Mathematical Sciences

Abstract

Mullineux defined an involution on the set of e-regular partitions of n. When e=p is prime, these partitions label irreducible symmetric group modules in characteristic p. Mullineux's conjecture, since proven, was that this “Mullineux map” described the effect on the labels of taking the tensor product with the one-dimensional signature representation. Counting irreducible modules fixed by this tensor product is related to counting irreducible modules for the alternating group An in prime characteristic. In 1991, Andrews and Olsson worked out the generating function counting fixed points of Mullineux's map when e=p is an odd prime (providing evidence in support of Mullineux's conjecture). In 1998, Bessenrodt and Olsson counted the fixed points in a p-block of weight w. We extend both results to arbitrary e, and determine the corresponding generating functions. When e is odd but not prime the extension is immediate, while e even requires additional work and the results, which are different, have not appeared in the literature.

Publication Title

European Journal of Combinatorics

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