OIST Representation Theory Seminar

John Murray, Maynooth University

Title: A Schur-positivity conjecture inspired by the Alperin-Mckay conjecture

Abstract: The McKay conjecture asserts that a finite group has the same number of odd degree irreducible characters as the normalizer of a Sylow 2-subgroup. The Alperin-McKay (A-M) conjecture generalizes this to the height-zero characters in 2-blocks.

In his original paper, McKay already showed that his conjecture holds for the finite symmetric groups S_n. In 2016, Giannelli, Tent and the speaker established a canonical bijection realising A-M for S_n; the height-zero irreducible characters in a 2-block are naturally parametrized by tuples of hooks whose lengths are certain powers of 2, and this parametrization is compatible with restriction to an appropriate 2-local subgroup.

Now corresponding to a 2-block of the symmetric group S_n, there is a 2-block of a maximal Young subgroup of S_n of the same weight. An obvious question is whether our canonical bijection is compatible with restriction of height-zero characters between these blocks.

Attempting to prove this compatibility lead me to formulate a conjecture asserting the Schur-positivity of certain differences of skew-Schur functions. The corresponding skew-shapes have triangular inner-shape, but otherwise do not refer to the 2-modular theory. I will describe my conjecture and give positive evidence in its favour.