University of Birmingham > Talks@bham > Algebra seminar > Qualitative results on the dimensions of irreducible representations of linear groups over local rings.

## Qualitative results on the dimensions of irreducible representations of linear groups over local rings.Add to your list(s) Download to your calendar using vCal - 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 _{n}(O/P), where ^{r}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, as well as the number of irreducible representations of a fixed dimension, are given by evaluating finitely many polynomials (only depending on _{r}n and r) at the residue field cardinality q. In particular, it is conjectured that the two groups GL_{n}(𝔽_{p}[t]/t) and GL^{r}_{n}(ℤ/p) have the same number of irreducible representations of dimension ^{r}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(ℤ/p) have the same number of irreducible representations of dimension ^{r}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. ## This talk is included in these lists:Note that ex-directory lists are not shown. |
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