### 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)*

Title: Primitive amalgams and the Goldschmidt-Sims conjecture

Abstract: The Classification of Finite Simple Groups has led to substantial progress on deriving sharp order bounds in various natural families of finite groups. One of the most well-known instances of this is Sims' conjecture, which states that a point stabiliser in a primitive permutation group has order bounded in terms of its smallest non-trivial orbit length (this was proved by Cameron, Praeger, Saxl and Seitz using the CFSG in 1983). In the meantime, Goldschmidt observed that a generalised version of Sims' conjecture, which we now call the Goldschmidt--Sims conjecture, would lead to important applications in graph theory. In this talk, we will describe the conjecture, and discuss some recent progress. Joint work with L. Pyber.

**17th October: ***Kamilla Rekvényi (Imperial College London)*

Title: The Orbital Diameter of Primitive Permutation Groups

Abstract: Let G be a group acting transitively on a finite set Ω. Then G acts on ΩxΩ component wise. Define the orbitals to be the orbits of G on ΩxΩ. The diagonal orbital is the orbital of the form ∆ = {(α, α)|α ∈ Ω}. The others are called non-diagonal orbitals. Let Γ be a non-diagonal orbital. Define an orbital graph to be the non-directed graph with vertex set Ω and edge set (α,β)∈ Γ with α,β∈ Ω. If the action of G on Ω is primitive, then all non-diagonal orbital graphs are connected. The orbital diameter of a primitive permutation group is the supremum of the diameters of its non-diagonal orbital graphs.

There has been a lot of interest in finding bounds on the orbital diameter of primitive permutation groups. In my talk I will outline some important background information and the progress made towards finding explicit bounds on the orbital diameter. In particular, I will discuss some results on the orbital diameter of the groups of simple diagonal type and their connection to the covering number of finite simple groups. I will also discuss some results for affine groups, which provides a nice connection to the representation theory of quasisimple groups.

**24th October: ***Alexandre Zalesski (UEA)*

Title: Some problems on representations of simple algebraic groups

Abstract: Some open questions on the weight structure of tensor-decomposable representations of simple algebraic groups will be discussed.

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

Title: The Boston--Shalev conjecture for conjugacy classes

Abstract: The Boston--Shalev conjecture (proved by Fulman and Guralnick in 2015) asserts that in any nonabelian simple group G in any nontrivial permutation action the proportion of derangements is at least some absolute constant c > 0. Since the set of derangements is closed under conjugacy it is also natural to ask about the proportion of *conjugacy classes* containing derangements. It is easy to see that this version of the question has a negative answer for alternating groups, but Guralnick and Zalesski asked whether it holds for groups of Lie type. I will outline a proof. We can also (1) extend to the case of almost simple groups, which is not true for the original conjecture, and (2) deduce the original conjecture, which amounts to a simplification of the Fulman--Guralnick proof. The key turns out to be a kind of analytic number theory for palindromic polynomials. This is ongoing work with Daniele Garzoni.

**14th November: ***Jay Taylor (Manchester)*

Title: Bounding Character Values at Regular Semisimple Elements

Abstract: If $\chi : G \to \mathbb{C}$ is an irreducible character of a finite group $G$ then one can ask how large the absolute value $|\chi(g)| \in \mathbb{R}$ at a given element $g \in G$ is. Typically one wants to obtain some $C \in \mathbb{R}$ such that the bound $|\chi(g)| < C$ holds as $g$ varies over a suitable subset of elements. In this talk we’ll try to explain why this question is important and discuss recent joint work with P. H. Tiep and M. Larsen obtaining new bounds in the case where $G$ is a finite reductive group and $g$ varies over the regular semisimple elements.

**21st November: ***Diego Martin Duro (Warwick)*

Title: Knutson's Conjecture on the representation ring

Abstract: Donald Knutson proposed the conjecture, later disproven and refined by Savitskii, that for every irreducible character of a finite group, there existed a virtual character such their tensor product was the regular character. In this talk, we disprove both this conjecture and its refinement. We then introduce the Knutson Index as a measure of the failure of Knutson's Conjecture and discuss its algebraic properties.

**28th November: ***Rachel Pengelly (Birmingham)*

Title: Comparing $\mathfrak{sl}_2$-subalgebras and $\text{SL}_2$-subgroups of classical algebraic groups

Abstract: Let $G$ be a classical algebraic group, and take $\mathfrak{g}$ to be its Lie algebra. There is a Springer isomorphism from the variety of unipotent elements in $G$ to the unipotent elements in $\mathfrak{g}$. Hence it is natural to consider to what extent results which hold for nilpotent elements in $\mathfrak{g}$ hold for unipotent elements in $G$. In this talk I will discuss when there is a unique $\mathfrak{sl}_2$-subalgebra for a given nilpotent orbit in $\mathfrak{g}$, and compare this to when there is a unique $\text{SL}_2$-subgroup for a given unipotent orbit in $G$.

**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: **

**20th February: **Nadia Mazza (Lancaster)

**27th February:**

**6th March:**

**13th March:**