Tag - McKay conjecture

The relevance of the McKay Conjecture in the representation theory of finite groups has led to investigate how irreducible characters decompose when restricted to Sylow p-subgroups. In this talk we will focus on the symmetric groups. Since the linear constituents of the restriction to a Sylow p-subgroup has been studied a lot by E. Giannelli and S. Law, we will concentrate on constituents of higher degree. In particular, we will describe the set of the irreducible characters which allow a constituent of a fixed degree, separating the cases of p being odd and p=2.

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.