# Algebra seminars, 2018-19

## In 'Algebra'

Back to 'Mathematics Research'### The Jacobson-Morozov Theorem and complete reducibility of Lie subalgebras

#### Adam Thomas (University of Birmingham)

**Wednesday 3 October 2018, 15:00-16:00 Watson Building, Lecture Room A**

The well-known Jacobson-Morozov Theorem states that every nilpotent element of a complex semisimple Lie algebra Lie(*G*) can be embedded in an _{2}-subalgebra. Moreover, a result of Kostant shows that this can be done uniquely, up to *G*-conjugacy. Much work has been done on extending these fundamental results to the modular case when G is a reductive algebraic group over an algebraically closed field of characteristic *p* > 0. I will discuss joint work with David Stewart, proving that the uniqueness statement of the theorem holds in the modular case precisely when *p* > *h*(*G*), the Coxeter number of *G*. In doing so, we consider complete reducibility of subalgebras of Lie(*G*) in the sense of Serre/McNinch.

### Simple fusion systems over *p*-groups with unique abelian subgroup of index *p*

#### Bob Oliver Université Paris 13

**Wednesday 10 October 2018, 15:00-16:00 Watson Building, Lecture Room A**

Fix a prime *p*. The fusion system of a finite group *G* with respect to a Sylow *p*-subgroup *S* of *G* is the category $\mathcal{F}_S(G)$ whose objects are the subgroups of *S*, and whose morphisms are the homomorphisms induced by conjugation in *G*. More generally, an abstract fusion system over a *p*-group *S* is a category $\mathcal{F}$ whose objects are the subgroups of *S* and whose morphisms are injective homomorphisms between the subgroups that satisfy certain axioms formulated by Lluis Puig and motivated by the Sylow theorems for finite groups. Normal fusion subsystems of $\mathcal{F}$ are defined by analogy with normal subgroups of a group, and $\mathcal{F}$ is simple if it has no nontrivial proper normal subsystems. One special case which yields many interesting examples is that where $\mathcal{F}$ is simple and *S* contains a (unique) abelian subgroup *A*<*S* of index *p*. When *p *= 2, each such $\mathcal{F}$ is isomorphic to the fusion system of $L_2(q)$ or $L_3(q)$ for some odd *q*. However, when *p* is odd (especially *p* ≥ 5), there is a remarkably large variety of such fusion systems, many of which are 'exotic' (not fusion systems of finite groups), and this provides a rich source of interesting examples. When *A* is elementary abelian, such fusion systems were classified in my joint work with David Craven and Jason Semeraro. In this talk, I want to focus on my more recent work with Albert Ruiz, where we described simple fusion systems $\mathcal{F}$ over nonabelian *p*-groups that contain an abelian subgroup *A* of index *p* that is not elementary abelian. It turns out that for such $\mathcal{F}$, *A* is homocyclic except in a few cases where dim(Ω_{1}(*A*))=*p*-1. Also, when *A* is homocyclic, $\mathcal{F}$ is determined by $\textup{Aut}_{\mathcal{F}}(A)$, its action on Ω_{1}(*A*), the exponent of *A*, and the $\mathcal{F}$-essential subgroups. We also looked at the 'limiting' case where *S* is infinite, and *A* is a product of copies of $\Z/p^\infty$.

### Maximal subalgebras of simple Lie algebras over fields of good characteristic

#### Alexander Premet, Manchester

**Wednesday 17 October 2018, 15:00-16:00 Watson Building, Lecture Room A**

Abstract not available

### Endotrivial modules, local group theory, and homotopy theory

#### Jesper Grodal Copenhagen

**Wednesday 7 November 2018, 15:00-16:00 Watson Building, Lecture Room A**

For G a finite group and k a field of characteristic p, an endotrivial module is a kG-module M such that End(M) is isomorphic to a trivial module plus a projective module. Equivalence classes of such modules form an abelian group under tensor product. It has has long been a quest to calculate this abelian group, for any finite group G, starting with work of Dade in the 70s. In my talk I'll explain how this question via homotopy theory reduces to the calculation of a certain finite p'-group, describable in terms of the p-local structure of G. The talk will then explain a number of examples where this can be calculated, such as for the Monster.

### Dirac operators and Hecke algebras

#### Dan Ciubotaru Oxford

**Wednesday 14 November 2018, 16:00-17:00 Watson Building, Lecture Room A**

I will explain the construction and main properties of Dirac operators for representations of various Hecke-type algebras (e.g., Lusztig's graded Hecke algebra for p-adic groups, Drinfeld's Hecke algebras, rational Cherednik algebras). The approach is motivated by the classical Dirac operator which acts on sections of spinor bundles over Riemannian symmetric spaces, and by its algebraic version for Harish-Chandra modules of real reductive groups. The algebraic Dirac theory developed for these Hecke algebras turns out to lead to interesting applications: e.g., a Springer parameterization of projective representations of finite Weyl groups (in terms of the geometry of the nilpotent cone of complex semisimple Lie algebras), spectral gaps for unitary representations of reductive p-adic groups, connections between the Calogero-Moser space and Kazhdan-Lusztig double cells. I will present some of these applications in the talk.

### The loop space homology of a small category

#### Carles Broto Universitat Autònoma de Barcelona

**Wednesday 21 November 2018, 15:00-16:00 Watson Building, Lecture Room A**

We generalise some results of Dave Benson on the homology of the loop space of a p-completed classifying space of a finite group. In particular, we show that the loop space homology of a plus construction (in the sense of Quillen) of the nerve of a small category can be computed in purely algebraic terms as the homology of a chain complex of projective RC-modules satisfying a certain list of algebraic conditions that determine it uniquely up to chain homotopy. This is joint work with Ran Levi and Bob Oliver.

### large centralizers in finite groups.

#### Geoff Robinson Lancaster

**Wednesday 5 December 2018, 16:00-17:00 Watson Building, Lecture Room A**

We discuss recent joint work with Bob Guralnick, partly inspired by the famous Brauer-Fowler paper of 1956. We prove a number of results ( some using the classification of finite simple groups (CFSG), some not) which bound the index of the Fitting subgroup of a finite group G in terms of centralizers of certain elements of G, especially involutions. We also provide a CFSG-free bound on the maximum dimension of the fixed-point space of an involution in a finite linear group.

### On Cherlin's conjecture for finite primitive binary permutation groups.

#### Nick Gill South Wales

**Wednesday 12 December 2018, 16:00-17:00 TBA**

This talk concerns the connection between the "local symmetry" and "global symmetry" of a mathematical object.

### First cohomology of Frobenius kernels and truncated invariants

#### Rudolf Tange

**Thursday 17 January 2019, 15:00-16:00 Nuffield G13**

Let *k* be an algebraically closed field of characteristic *p* > 0 and let *G* be a connected reductive group over *k* with Lie algebra . Consider the rings *k*[*G*] and *k*[] of regular functions on *G* and as *G*-modules via the conjugation action. They have been studied extensively, for example in Kostant’s 1963 paper. I will discuss the result that, under some mild assumptions, the first restricted cohomology of these modules is zero. After this I will discuss the problem of describing the invariants in a certain finite dimensional quotient of *k*[_{n}]. This is related to the centre of the restricted enveloping algebra of _{n}.

### From blocks to surface algebras

#### Karin Erdmann (University of Oxford)

**Thursday 24 January 2019, 15:00-16:00 Nuffield G13**

This is joint work with Andrzej Skowroński. We investigate tame symmetric algebras constructed from surface triangulations, which generalize tame blocks of group algebras. Tame blocks are the ones whose defect groups are dihedral, or quaternion, or semidihedral. I will focus on the extended version (work in progress) which includes new algebras, and also includes some of the algebras from the block setting which we did not have in the first attempt.

### Crowns in finite groups: Theory and applications

#### Gareth Tracey

**Thursday 31 January 2019, 15:00-16:00 Arts LR3**

The notion of a *crown* in a finite group was introduced by Gaschütz in the 1960s, and later developed by Dalla Volta and Lucchini as a tool to study minimal generation and the presentation rank in finite and profinite groups. In this talk, we will give an introduction to the theory, and detail some recent applications to generation and enumeration problems in certain classes of finite groups. One application is ongoing joint work with Colva Roney-Dougal, while another is joint work with Andrea Lucchini and Claude Marion.

### Towards a generic representation theory.

#### David Craven (University of Birmingham)

**Thursday 7 February 2019, 15:00-16:00 Nuffield G13**

In combinatorics, the 'nicest' way to prove that two sets have the same size is to find a bijection between them, giving more structure to the seeming numerical coincidences. In representation theory, many of the outstanding conjectures seem to imply that the characteristic *p* of the ground field can be allowed to vary, and we can relate different groups and different primes, to say that they have 'the same' representation theory. In this talk I will try to make precise what we could mean by this.

### Irreducible modules for algebraic groups

#### Michael Bate

**Thursday 14 February 2019, 16:00-17:00 TBA**

In this talk I'll report on some current work with David Stewart. The main results are a high weight classification of irreducible modules for linear algebraic groups which works over any field, and a result which reduces the calculation of the dimension of such modules to the reductive case and the commutative case. Whilst a classification by highest weight might not be so surprising, one feature of interest is that over imperfect fields the commutative case is rather more involved than one might expect. I'll spend most of the talk giving a detailed example of the sort of behaviour one sees before sketching the main steps in the classification, and some open problems.

### Cohomology of PSL(2,q)

#### Jack Saunders

**Thursday 21 February 2019, 15:00-16:00 Nuffield G13**

We investigate the cohomology of finite Chevalley groups in cross characteristic and use results of Guralnick and Tiep from 2011 on the first cohomology group to obtain a general result for the *n*th cohomology group. In this talk, we will use these results to completely the determine the cohomology of all irreducible modules for PSL (2,*q*) when the characteristic of the representation is not 2.

### The Structure of Induced Simple Modules for 0-Hecke Algebras

#### Imen Belmokhtar (QMUL)

**Wednesday 27 February 2019, 13:30-14:30 Watson Building, Lecture Room C**

We shall be concerned with the 0-Hecke algebra; its irreducible representations were classified and shown to be one-dimensional by Norton in 1979. The structure of a finite-dimensional module can be fully described by computing its submodule lattice. We will discuss how this can be encoded in a generally much smaller poset given certain conditions; this allows us to obtain branching rules which remarkably describe the full structure of an induced simple module in types *B* and *D*.

### Picard groups of blocks and Donovan’s conjecture

#### Florian Eisele (Glasgow)

**Wednesday 27 February 2019, 14:30-15:30 Watson Building, Lecture Room C**

While the outer automorphism group of a block over an algebraically closed field of characteristic p is usually an infinite group, it was recently observed by Boltje, Kessar and Linckelmann that the outer automorphism group of a block defined over a suitable discrete valuation ring is finite in all known examples. As of yet there is no general explanation for the finiteness of these outer automorphism groups, and the closely related Picard groups of blocks. I will talk about recent results on the structure of these groups, and an application to Donovan’s conjecture (which is joint work with C. Eaton and M. Livesey).

### Why the Galois-McKay conjecture and new consequences

#### Gabriel Navarro (Valencia)

**Wednesday 27 February 2019, 16:15-17:15 Watson Building, Lecture Room C**

While the celebrated McKay conjecture asserts that two numbers, one local, the other global, are in fact the same, the Galois version of this conjecture is strong enough to relate certain global and local structures. We will give a survey on this and offer new recent consequences.

### Equiangular lines, Incoherent sets and the Mathieu Group *M*_{23}

#### Neil Gillespie (Bristol)

**Thursday 14 March 2019, 15:00-16:00 Nuffield G13**

The problem of finding the maximum number of equiangular lines in *d*-dimensional Euclidean space has been studied extensively over the past 80 years. The absolute upper bound on the number of equiangular lines that can be found in ℝ^{d} is *d*(*d*+1)/2. However, examples of sets of lines that saturate this bound are only known to exist in dimensions *d* = 2,3,7 or 23, and it is an open question whether this bound is achieved in any other dimension. The known examples of equiangular lines that saturate the absolute bound are related to highly symmetrical objects, such as the regular hexagon, the icosahedron, and the *E*_{8} and Leech lattices. By considering the additional property of incoherence, we prove that there exists a set of equiangular lines that saturates the absolute bound and the incoherence bound if and only if *d* = 2,3,7 or 23. We also show that many of the maximal sets of equiangular lines in small dimensions can be realised as subsets of the 276 equiangular lines in dimension 23. We do this by looking at the involutions of the the Mathieu Group *M*_{23}. In particular, we show how the involution structure of *M*_{23} can be used to describe the roots of *E*_{8}. This has the effect of providing what we believe is a new way of relating the *E*_{8} and Leech lattices. It also leads us to observe a correspondence between sets of equiangular lines in small dimensions and the exceptional curves of del Pezzo surfaces, which in turn leads us to speculate a possible connection to the Mysterious Duality of string theory.

### Almost Engel compact groups

#### Evgeny Khukhro (Charlotte Scott Research Centre for Algebra, University of Lincoln, UK)

**Thursday 28 March 2019, 15:00-16:00 Nuffield G13**

We say that a group *G* is almost Engel if for every *g*\in *G* there is a finite set {\mathscr E}(*g*) such that for every *x*\in *G* all sufficiently long commutators [...[[*x*,*g*],*g*],...,*g*] belong to {\mathscr E}(*g*), that is, for every *x*\in *G* there is a positive integer *n*(*x*,*g*) such that [...[[*x*,*g*],*g*],...,*g*]\in {\mathscr E}(*g*) if *g* is repeated at least *n*(*x*,*g*) times. (Thus, Engel groups are precisely the almost Engel groups for which we can choose {\mathscr E}(*g*) = {1} for all *g*\in *G*.)

We prove that if a compact (Hausdorff) group *G* is almost Engel, then *G* has a finite normal subgroup *N* such that *G*/*N* is locally nilpotent. If in addition there is a uniform bound |{\mathscr E}(*g*)| ≤ *m* for the orders of the corresponding sets, then the subgroup *N* can be chosen of order bounded in terms of *m*. The proofs use the Wilson-Zelmanov theorem saying that Engel profinite groups are locally nilpotent.

This is joint work with Pavel Shumyatsky.

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