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Manchester Algebra Seminar - John Murray

Dates:24 May 2022
Times:13:00 - 14:00
What is it:Seminar
Organiser:Department of Mathematics
Who is it for:University staff, Current University students
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  • Department of Mathematics

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  • In category "Seminar"
  • In group "(Maths) Algebra"
  • In group "(Maths) Maths seminar series"
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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

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Kieran Calvert

kieran.calvert@manchester.ac.uk

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