# 2021-22

### In Term 2 seminars will be held on Mondays at 17:00 sometimes in B3.02 and via MSTeams (join the Warwick Algebra Seminar Team).

#### Organisers: Dmitriy Rumynin, Adam Thomas.

#### Term 2:

**17th January (B3.02 and online): **Dave Benson (Aberdeen)

Title: Blocks with normal defect groups.

Abstract: Let G be a finite group and k a field of characteristic p. Suppose that G has a block with normal defect group and abelian p’-inertial quotient. Then I shall describe the structure of the block as a quantum deformation of the group algebra of the defect group. This is joint work with Radha Kessar and Markus Linckelmann.

**24th January (online): **Vincent Knibbeler (Loughborough)

Title: Automorphic Lie algebras and modular forms

Abstract: We revisit twisted loop algebras and the classical Onsager algebra that appeared in statistical mechanics. Automorphic Lie algebras are generalisations of these two examples. In recent work we describe the automorphic Lie algebras consisting of holomorphic maps from the complex upper half plane to a complex finite dimensional Lie algebra, which are equivariant with respect to the modular group SL(2,Z). We obtain an analogue of a classical theorem of V.G.Kac on twisted loop algebras and a realisation of the Onsager algebra in terms of matrix-valued modular forms. The talk is based on joint work with Sara Lombardo and Alexander Veselov.

**7th February (B3.02 online): **Nick Gill (Open University)

Title: Subgroup lattices in PSL(2,q)

Abstract: I will report on some recent work with my PhD student, Scott Hudson. We consider the action of G=PSL(2,q) on the cosets of a subgroup H. We are interested in various properties of the associated "stabilizer lattice", i.e. the lattice of subgroups that crop up as intersections of conjugates of H. The results that I will present lead naturally to conjectures about such lattices in other simple groups of Lie type.

**14th February (online):** Emily Hall (Bristol)

Title: Almost Elusive Groups

Abstract: Let G be a transitive permutation group acting on a finite set X with |X|>1. A derangement in G is an element of G that has no fixed points on X, and as a consequence of the orbit-counting lemma we know that such elements always exist in G. But what happens if we seek derangements with special properties i.e specific order? In this talk I will discuss this question and introduce the notion of Almost elusive groups. I will provide motivation behind the concept of Almost elusive groups and discuss key concepts behind the classification of these groups in the primitive setting.

**28th February:** Gareth Tracey (Birmingham)

Title: How many subgroups are there in a finite group?

Abstract: Counting the number of subgroups in a finite group has numerous applications, ranging from enumerating certain classes of finite graphs (up to isomorphism), to counting how many isomorphism classes of finite groups there are of a given order. In this talk, I will discuss the history behind the question; why it is important; and what we currently know.

**14th March: ** Martin Liebeck (Imperial)

Title: Orbits of compact linear groups

Abstract: For a compact subgroup G of GL(n,R), define the vector closure of G to be the largest subgroup of GL(n,R) that has the same orbits on vectors as G. Subgroups G that are equal to their vector closures are particularly interesting, as these are isometry groups of norms. I will present some examples and results on such vector closures.

#### Term 1 (seminars were held on Tuesdays at 15:00 on Zoom jointly with Birmingham):

**16th November:** Lucia Morotti (Hannover University).

Title: Decomposition of spin representations of symmetric groups in characteristic 2

Abstract: Any representation of a double cover of a symmetric group can also be viewed as a representation of S_{n} when reduced to characteristic 2. However not much is known about the corresponding decomposition matrices. For example, while decomposition numbers of Specht modules indexed by 2-parts partitions are known, the decomposition numbers of spin irreducible modules indexed by 2-parts partitions are still mostly unknown, with in most cases only multiplicities of maximal composition factors (under a certain ordering of the modular irreducible representations) being known.

In this talk I will characterise irreducible representations that appear when reducing 2-parts spin representations to characteristic 2 and describe part of the corresponding rows of the decomposition matrices.

**30th November:** Dávid Szabó (Rényi Institute, Budapest).

Title: Class 2 nilpotent and twisted Heisenberg groups

property of various transformation groups.

**7th December:** Barbara Baumeister (Bielefeld University).

Title: The dual approach to Coxeter and Artin groups

Abstract: Independently Brady and Watt as well as Bessis, Digne and Michel started to study Coxeter systems (W,S) and Artin groups by replacing the simple system S by the set of all reflections. In particular this provides new presentations for the Artin groups of spherical type. I will introduce into this fascinating world. I also will present a slight modification of the new concept.

**14th December:** Ralf Koehl (Giessen University).

Title: Statistical topological data analysis: some musings about networks and applications

Abstract: I will start with an overview how linear algebra (in particular eigenvalue techniques) help with the understanding of networks. Then I mention random walks (which for me is a combination of linear algebra with limit arguments). Then I go to the core topic of the talk: persistent homology, starting with plenty of examples. Then I mention how a group-theorist can end up working with networks. And finally I explain how a pure mathematician can train themselves for applying methods from network theory by studying properties of Riemannian manifolds via approximations.