# 2022-23

### Seminars are held on Mondays at 17:00 in B3.02

#### Organisers: Adam Thomas and Gareth Tracey

(To see the abstract and title of a talk in the past, click on the speakers name to expand it)

#### Term 1:

10th October: Gareth Tracey (Warwick)

17th October: Kamilla Rekvényi (Imperial College London)

24th October: Alexandre Zalesski (UEA)

31st October: Sean Eberhard (Queen's University Belfast)

14th November: Jay Taylor (Manchester)

21st November: Diego Martin Duro (Warwick)

28th November: Rachel Pengelly (Birmingham)

5th December: Ana Retegan (Birmingham)

Title: Minimum eigenspace codimension in irreducible representations of simple classical linear algebraic groups

Let $k$ be an algebraically closed field of characteristic $p \geq 0$, let $G$ be a simple simply connected classical linear algebraic group of rank $\ell$ and let $T$ be a maximal torus in $G$ with rational character group $X(T)$. For a nonzero $p$-restricted dominant weight $\lambda\in X(T)$, let $V$ be the associated irreducible $kG$-module. Let $V_{g}(\mu)$ denote the eigenspace corresponding to the eigenvalue $\mu\in k^{*}$ of $g\in G$ on $V$ and define $\nu_{G}(V)=\min\{\text{dim}(V)-\text{dim}(V_{g}(\mu))\mid g\in G\setminus Z(G), \ \mu \in k^{*}\}$ to be the minimum eigenspace codimension on $V$. In this talk, we determine $\nu_{G}(V)$ for $G$ of type $A_{\ell}$, $\ell\geq 16$ and $\dim(V)\leq \frac{\ell^{3}}{2}$; for $G$ of type $B_{\ell}$, $C_{\ell}$, $\ell\geq 14$ and $\dim(V)\leq 4\ell^{3}$; and for $G$ of type $D_{\ell}$, $\ell\geq 16$ and $\dim(V)\leq 4\ell^{3}$. Moreover, for the groups of smaller rank and their corresponding irreducible modules with dimension satisfying the above bounds, we determine lower-bounds for $\nu_{G}(V)$.

#### Term 2:

16th January: Scott Harper (St Andrews)

23rd January: Michael Bate (York)

30th January: Miriam Norris (Manchester)

6th February: Alastair Litterick (Essex)

13th February: