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

Organisers: Adam Thomas and Gareth Tracey

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)

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

30th January: Ilaria Colazzo (Leeds)

6th February: Ewan Cassidy (Durham)

13th February: Jakob Moosbauer (Warwick)

20th February: David Guo (Bristol)

27th February: Michael Turner (Birmingham)

6th March: Michael Burkhart (Cambridge)

13th March: Marina Anagnostopoulou-Merkouri (Bristol)

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.