Algebra Seminar

Seminars are held on Thursdays at 12:00 in B3.02

Organisers: Adam Thomas and Gareth Tracey

Term 3:

1st May: Dmitriy Rumynin (Warwick)

Title: Brauer 14 and Dyson 10

Abstract: Brauer’s 14th Problem, solved in 2023 by Murray and Sambale, was to count the number of (complex irreducible) characters of real type of a finite group using “group theory”. In the context of Dyson 10-way, the irreducible characters could be of 10 possible types. After reviewing Brauer's 14 and Dyson's 10, we will discuss how to count the characters of each type.

8th May: Rudradip Biswas (Warwick)

Title: Gorenstein analogues of a projectivity criterion over group algebras

Abstract: I will talk about a recent preprint (arXiv:2502.00868; joint with Dimitra-Dionysia Stergiopoulou) where we formulated and answered Gorenstein projective, flat, and injective analogues of a classical projectivity question for group rings under some mild additional assumptions. Although the original question, that was proposed by Jang-Hyun Jo in 2007, was for integral group rings, in our paper, we dealt with more general commutative base rings. We made use of the developments that have happened in the field of Gorenstein homological algebra over group rings in recent years, and we managed to improve and generalize several existing results from this area along the way.

22nd May: Daniele Garzoni (USC)

Title: Derangements in non-Frobenius groups

Abstract: An old lemma of Jordan states that a finite transitive permutation group contains a derangement, i.e., an element fixing no point. I will explain why this is interesting and I will present a recent lower bound on the proportion of derangements in a non-Frobenius permutation group.

12th June: Melissa Lee (Monash)

Title: Graphs on finite groups

Abstract: In recent years, there has been accelerating interest in encoding various properties of groups into graphs. In this talk, I will discuss some of these encodings, show how graph-theoretic properties can be used to prove deep results about the underlying group, and discuss some open problems in the area.

Term 2:

9th January: Adam Thomas (Warwick)

Title: Searching for 3-dimensional subalgebras

Abstract: Let g be the Lie algebra of a simple algebraic group over an algebraically closed field of characteristic p. When p=0 the celebrated Jacobson-Morozov Theorem promises that every non-zero nilpotent element of g is contained in a simple 3-dimensional subalgebra of g (an sl₂). This has been extended to odd primes but what about p=2? There is still a unique 3-dimensional simple Lie algebra, known colloquially as fake sl₂, but there are other very sensible candidates like sl₂ and pgl₂. In this talk we will discuss recent joint work with David Stewart determining which nilpotent elements of g live in subalgebras isomorphic to one of these three Lie algebras. There will be an abundance of concrete examples, calculations with small matrices and even some combinatorics.

16th January: Coen del Valle (St Andrews)

Title: It's (probably) okay to be greedy

Abstract: Let G be a group of permutations of a finite set Omega. A base for G is a subset of Omega whose pointwise stabiliser is trivial; call the size of a smallest base for G the 'base size' of G, denoted b(G). There is a natural greedy algorithm for finding a base of relatively small size. In 1999, Peter Cameron conjectured that for G primitive, this algorithm produces a base of size at most some absolute constant multiple of b(G). In this talk we will explore recent progress towards settling this conjecture.

23rd January: Paula Macedo Lins de Araujo (Lincoln)

Title: Zeta functions and applications to the Reidemeister Spectrum

Abstract: In this talk, I will introduce zeta functions that count group-theoretical objects, specifically focusing on the number of twisted conjugacy classes within a group. Twisted conjugacy is a generalization of the usual conjugacy, incorporating a twist by an endomorphism. Given a group G and an automorphism f, we consider the action gx = gx f(g)^-1. The orbits of this action are known as twisted conjugacy classes, or Reidemeister classes. Recent years have seen intensive investigation into the sizes of these classes. A major goal in this area is to classify groups where all automorphisms give infinitely many classes. For groups that do not possess this property, the focus shifts to understanding the possible sizes of the classes across all automorphisms, known as the Reidemeister spectrum. In this talk, we will explore how the zeta functions mentioned above can be used to understand twisted conjugacy classes of certain nilpotent groups.

30th January: Ilaria Colazzo (Leeds)

Title: Matched pairs of groups and the Pentagon Equation

Abstract: In this talk, I will present a complete classification of finite bijective set-theoretic solutions to the Pentagon Equation, uncovering a surprising connection with matched pairs of groups. We will introduce all necessary definitions. Next, we will focus on the fundamental components: the irretractable solutions, and we will examine how these solutions relate to matched pairs of groups. Finally, we will show how each irretractable solution can be lifted to classify all bijective solutions.

6th February: Ewan Cassidy (Durham)

Title: Asymptotic character bounds for w--random permutations

Abstract: Given a word w in the free group F_r, a w-random permutation is obtained by choosing \phi_n in Hom(F_r,S_n) uniformly at random and evaluating \phi_n(w). I will discuss new asymptotic bounds for the expected character of a w-random permutation for a natural family of irreducible representations of the symmetric group, some details of the proofs using `combinatorial integration' and, if time permits, an application of these results towards proving expansion properties of random Schreier graphs.

13th February: Jakob Moosbauer (Warwick)

Title: The Orbit Rank of Tensors and Complexity of Matrix Multiplication

Abstract: The computational complexity of matrix multiplication is governed by the rank of the matrix multiplication tensor. The rank of a tensor is the size of a minimal decomposition into simple tensors, called rank-one tensors. Let G be a subgroup of the symmetry group of a tensor. In this talk, we introduce the concept of G-orbit rank of a tensor, which measures the size of a decomposition that is invariant under the action of G. We will introduce techniques for finding orbit rank decompositions by computer aided search and use them to derive bounds on the rank of the matrix multiplication tensor.

20th February: David Guo (Bristol)

Title: Residually finite groups with uniformly almost flat quotients.

Abstract: Given a residually finite group, there is considerable interest in understanding what information is encoded in the finite quotients. Khukhro and Valette investigated what algebraic properties we can deduce from the diameter of finite quotients of the group. They showed that the finite quotients have diameter bounded uniformly below by a constant multiple of their size if and only if the group is virtually cyclic, and that if the diameter of finite quotients is bounded uniformly below by a polynomial in their size then the group virtually maps onto Z. We conjecture that this latter assumption in fact implies that the group is virtually nilpotent, and we prove our conjecture or variants of it for various specific classes of groups. Joint work with Matthew Tointon.

27th February: Michael Turner (Birmingham)

Title: New Approaches to 6-Transposition Groups

Abstract: A finite group along with a normal set of involutions, D, is called a k-transposition group if D generates the group and the product of any two distinct elements in D has order at most k. Around 1970, Fischer (while at Warwick) investigated 3-transposition groups and found three new sporadic simple groups. He also predicted the Baby Monster and Monster simple groups by looking at larger transposition groups. We will start that talk with some examples before discussing the history of transposition groups. Next, we will look at basic observations and results. The rest of the talk will be focused on the progress so far on simple, almost simple, and quasisimple groups with this 6-transposition group property. This is joint work with Chris Parker.

6th March: Michael Burkhart (Cambridge)

Title: Local conjugacy and primary-type decompositions in nonabelian cohomology

Abstract: In 1979, Losey and Stonehewer introduced local conjugacy and provided some conditions under which locally conjugate complements J & J' to a normal nilpotent subgroup N must be conjugate. We will interpret their results in the language of nonabelian group cohomology and relate them to the existence of a primary-type decomposition of the first cohomology set H^1(J,N). We will cover the cases: N is abelian, J is nilpotent, and N⋊J is supersoluble. This decomposition will also provide us with an inclusion-based version of their result. We will motivate our discussion in terms of fixed point results in the style of Glauberman.

13th March: Marina Anagnostopoulou-Merkouri (Bristol)

Title: The regularity number of a finite group

Abstract: Let G be a transitive permutation group on a finite set \Omega with point stabiliser H. Recall that a base for G is a subset of \Omega whose pointwise stabiliser in G is trivial. The minimal size of a base is called the base size of G, denoted b(G), and this classical invariant has been widely studied for many years, finding many applications.

Observe that b(G) coincides with the smallest positive integer k such that G has a regular orbit on the Cartesian product (G/H)^k. As a natural generalisation, we can consider the minimal number r such that G has a regular orbit on (G/H₁) x ... x (G/H_r) for any core-free subgroups H₁, ..., H_r of G. We call this invariant the regularity number of G.

In this talk, we will explain how a combination of probabilistic, combinatorial and computational methods can be used to study the regularity number of almost simple groups, as well as several other related problems. In particular, we will report on some recent progress towards extensions of well studied base size conjectures of Cameron and Vdovin. This is joint work with Tim Burness.

Term 1:

3rd October: Diego Martin Duro (Warwick)

Title: Global Representation Ring and Knutson Index

Abstract: This talk introduces the global representation ring and table of a finite group G. They encapsulate a lot of information about the group including the Burnside table of marks and the character table. We will also discuss additional properties that can be recovered. We then generalise the notion of Knutson Index for a general commutative ring and explore this index for the global representation ring. This talk is based on the paper ``Global Representation Ring and Knutson Index" (arXiv 2403.18498), which is joint work with Dylan and Dmitriy.

10th October: Dylan Johnston (Warwick)

Title: Classification of disconnected reductive algebraic groups

Abstract: We say a disconnected algebraic group is reductive if its connected component is a reductive group in the usual sense. Even if one is only interested in connected reductive groups, disconnected ones enter the picture as subgroups, so gaining an understanding of them is a fruitful endeavour. In this talk, I will discuss the following question: Given a connected reductive algebraic group N and a finite group H, which algebraic groups G fit into the short exact sequence 1 ---> N ---> G ---> H ---> 1? If time permits, I will also briefly discuss some representation-theoretic results for such groups, including a bound on their Knutson Index. This talk is based on the paper ``Disconnected Reductive Groups: Classification and Representations" (arXiv 2409.06375), which is joint work with Diego and Dmitriy.

17th October: Jack Saunders (Bristol)

Title: Linear groups acting 4-arc-transitively on cubic graphs

Abstract: In this talk, we give a brief overview of s-arc-transitive graphs and show how their study in the case of cubic (3-regular) graphs reduces to a generation problem for finite almost simple groups. We then discuss current progress towards solving this generation problem for PSL(n,q) when n is sufficiently large and q is coprime to 6.

24th October: Alice Dell'Arciprete (York)

Title: Quiver presentations for Hecke categories and KLR algebras

Abstract: We discuss the algebraic structure of KLR algebras by way of the diagrammatic Hecke categories of maximal parabolics of finite symmetric groups. Combinatorics (in the shape of Dyck tableaux) plays a huge role in understanding the structure of these algebras. Instead of looking only at the sets of Dyck tableaux (which enumerate the q-decomposition numbers) we look at the relationships for passing between these Dyck tableaux. In fact, this “meta-Kazhdan-Lusztig combinatorics” is sufficiently rich as to completely determine the complete Ext-quiver and relations of these algebras.

31st October: Evgeny Khukhro (Lincoln)

Title: Engel sinks in finite, profinite, and compact groups

Abstract: Using Zelmanov's deep results on Engel Lie algebras, Wilson and Zelmanov proved that any profinite Engel group is locally nilpotent, and Medvedev extended this result to Engel compact groups. We state generalizations of the Engel condition as restrictions on the so-called Engel sinks of group elements. For example, a group can be considered to be `almost Engel' if all Engel sinks are finite. We proved that almost Engel compact groups are almost locally nilpotent (in certain precise terms). Similar results for finite groups have quantitative nature, with almost nilpotency expressed as a function of sizes of Engel sinks. Our most recent results concern imposing restrictions on Engel sinks of commutators (rather than all elements). This is joint work with Pavel Shumyatsky.

7th November: Chris Bowman (York)

Title: A combinatorial introduction to Hecke Categories

Abstract: Hecke categories control the representation theory of symmetric and algebraic groups, and generalise this theory from Weyl groups to all parabolic Coxeter systems. We give an introductory survey of some of the recent results in this area from a concrete combinatorial point of view.

14th November: Charley Cummings (Aarhus)

Title: Metric completions of cluster categories

Abstract: The completion of a metric space is a classical method for generating new mathematical structures from old. Recently, Neeman emulated this idea to define a metric completion of a triangulated category, thereby providing a novel way to construct new triangulated categories. However, computing these completions often requires using the properties of an already completed ambient category, like the derived category. In this talk, based on joint work with Sira Gratz, we present an example from cluster theory that avoids this requirement by focusing on categories that have combinatorial models, and show that their categorial completions can be viewed as topological completions of the associated models.

21st November: Luca Sabatini (Warwick)

Title: Abelian subgroups and sections of finite groups

Abstract: Let G be a group of order n, where n is a large integer. In 1976, Erdős and Straus used a simple argument to show that G contains an abelian subgroup of order roughly log n. Twenty years later, Pyber used the classification of the finite simple groups to improve this result up to 2^{\sqrt{\log n}}. This is best possible, because of certain wild p-groups of class 2 that were obtained with probabilistic methods by Ol'shanskii. On the other hand, it can be seen that G always contains an abelian section of size at least n^{1/ log log n}, which is much bigger. In this talk, we present these questions and some of the methods used in the proofs. We also introduce new probabilistic constructions of wild p-groups, which is joint work with S. Eberhard.

28th November: Neil Saunders (City)

Title: Exotic Spaltenstein Varieties

Abstract: We define a new class of varieties that extends the notion of exotic Springer fibres to the case of symplectic partial flags. In previous work, we established that there was a bijection between the irreducible components of the exotic Springer fibres with standard Young bitableaux. In this new setting of exotic Spaltenstein varieties, we prove an analogous result that, for nilpotent elements of order two, the top-dimensional irreducible components can be described combinatorically using semistandard Young bitableaux, and conjecture that this should hold for nilpotent elements of arbitrary order. This is joint work with Daniele Rosso.

5th December: Patricia Medina Capilla (Warwick)

Title: The second maximal subgroups of the almost simple groups

Abstract: In 2018, Lucchini, Marion, and Tracey showed that every maximal subgroup of an almost simple group is 5-generated, lowering the previously known bound of 6. Naturally, one can ask the same about second maximal subgroups of almost simple groups. Burness, Liebeck and Shalev looked into this question in 2016, determining that almost all such subgroups were 70-generated. In this talk, we will present recent work aiming to lower this bound, and survey some of the key techniques involved. In particular, we will discuss the method of crowns, developed by Lucchini and Dalla Volta and used by Lucchini, Marion and Tracey to great effect, as well as an improved classification of the second maximal subgroups of almost simple groups with alternating or classical socle.