Qualitative results on the dimensions of irreducible representations of linear groups over local rings.

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• Alexander Stasinski (Durham)
• Thursday 16 November 2023, 15:00-16:00
• LG06 Old Gym.

If you have a question about this talk, please contact Chris Parker.

Let G_r = GL_n(O/P^r), where O is the ring of integers of a local field with finite residue field 𝔽_q of characteristic p, P is the maximal ideal and r is a positive integer. It has been conjectured by U. Onn that the dimensions of the irreducible representations of G_r, as well as the number of irreducible representations of a fixed dimension, are given by evaluating finitely many polynomials (only depending on n and r) at the residue field cardinality q. In particular, it is conjectured that the two groups GL_n(𝔽_p[t]/t^r) and GL_n(ℤ/p^r) have the same number of irreducible representations of dimension d, for each d. These conjectures can be generalised by allowing other (reductive) group schemes than GL_n.

I will report on some recent progress on the polynomiality of the representation dimensions in joint work with A. Jackson as well as some independent related work by I. Hadas. The latter proved that for any affine group scheme G of finite type over ℤ, all r and all large enough p (depending on G and r), the groups G(𝔽_p[t]/t^r) and G(ℤ/p^r) have the same number of irreducible representations of dimension d, for each d. A crucial intermediate result is that the stabilisers of representations of certain finite groups are the 𝔽_q-points of algebraic groups with boundedly many geometric connected components.

This talk is part of the Algebra seminar series.

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• Algebra seminar
• LG06 Old Gym
• School of Mathematics events

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