# Manchester Algebra Seminar - John Murray

Dates: | 24 May 2022 |
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Times: | 13:00 - 14:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, Current University students |

Speaker: John Murray

Title: From the Alperin-McKay conjecture to Berstein-Zelevinsky Triangles

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

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 powers of 2, and this parametrization is compatible with restriction to the defect group.

Given a 2-block of $S_n$, there is a corresponding 2-block of a certain maximal Young subgroup of $S_n$. It is an obvious question to ask whether our canonical bijection is compatible with restriction between these blocks.

Attempting to prove this compatibility lead me to a conjecture which asserts that certain differences of skew-Schur functions are Schur positive. The corresponding skew-shapes have triangular inner-shape, but otherwise are independent of 2-block theory.

I will describe my conjecture, give positive evidence in its favour and outline a possible proof involving Berstein-Zelevinsky triangles.

- subject to equipment and connection

Tea and biscuits 12:45 in the foyer