Macdonald polynomials and decomposition numbers for finite unitary groups

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• Olivier Dudas (IMJ-PRG)
• Thursday 01 October 2020, 15:00-16:00
• https://bham-ac-uk.zoom.us/j/92610684996?pwd=K2FKV2dpczhXSUdvUllqNnlYMVlEUT09.

If you have a question about this talk, please contact Lewis Topley.

(work in progress with R. Rouquier) I will present a computational (yet conjectural) method to determine some decomposition matrices for finite groups of Lie type. These matrices encode how ordinary representations decompose when they are reduced to a field with positive characteristic \ell. There is an algorithm to compute them for GL(n,q) when \ell is large enough, but finding these matrices for other groups of Lie type is a very challenging problem.

In this talk I will focus on the finite general unitary group GU(n,q). I will first explain how one can produce a “natural” self-equivalence in the case of GL(n,q) coming from the topology of the Hilbert scheme of the complex plane . The combinatorial part of this equivalence is related to Macdonald’s theory of symmetric functions and gives (q,t)-decomposition numbers. The evidence suggests that the case of finite unitary groups is obtained by taking a suitable square root of that equivalence, which encodes the relation between GU(n,q) and GL(n,-q).

This talk is part of the Algebra seminar series.

This talk is included in these lists:

• Algebra seminar
• School of Mathematics events
• https://bham-ac-uk.zoom.us/j/92610684996?pwd=K2FKV2dpczhXSUdvUllqNnlYMVlEUT09

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