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 Sn. In 2016, Giannelli, Tent and the speaker established a canonical bijection realising A-M for Sn; 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 Sn, there is a 2-block of a maximal Young subgroup of Sn 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.
This video was produced by the Okinawa Institute of Science and Technology, as part of their OIST Representation Theory Seminar series.
