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On decomposition numbers and bad primes

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If you have a question about this talk, please contact David Craven.

Let G be a finite group of Lie type defined over the field with q elements, with q = pf, and let U be a Sylow p-subgroup of G. The problem of determining the (shape of the) l-modular decomposition matrices of G when lp is more difficult in the case where p is a bad prime for G. For instance, if p is good then one is allowed to use the theory of GGG Rs, which is often crucial to obtain the unitriangularity of such matrices. The degrees of the irreducible characters of U are all powers of q when p is a good prime for G and rk(G) ≤ 6. On the other hand, if p is a bad prime for G then one always finds irreducible characters of U of degree of the form qn/p. The goal of this talk is to explain why such characters seem to play a major role towards the determination of the l-decomposition numbers of G when p is a bad prime, and how they have already been used to obtain new results on the l-modular decomposition matrices of SO+(8,p2f).

This talk is part of the Algebra seminar series.

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