# Algebra seminars, 2021-22

## In 'Past algebra seminars'

### Defining an affine partition algebra

#### Maud De Visscher, City, University of London

**Tuesday 28 September 2021, 15:00-16:00 Lecture Theatre C, Watson Building**

The partition algebra was originally defined independently by Martin and Jones in the 1990s. It is a diagram algebra and satisfies a double centraliser property with the symmetric group. In this talk, I will define an affine version of the partition algebra by generators and relations and describe some of its properties. I will also relate it to the affine partition category recently defined by Brundan and Vargas. This is joint work with Samuel Creedon.

### Highest weight categories in the modular representation theory of Lie algebras

#### Matthew Westaway, University of Birmingham

**Tuesday 5 October 2021, 15:00-16:00 Lecture Theatre C, Watson Building**

Highest weight categories are ubiquitous in Lie theoretic representation theory. They are of particular importance in the study of representations of Lie algebras in characteristic zero, and of algebraic groups in all characteristics. For Lie algebras in positive characteristic, however, the story is more complicated. The ideas of highest weight theory are still present, but a slightly different framework is needed to capture all their nuances. This talk will explain this in more detail, utilising ideas from a 2018 paper of Brundan and Stroppel. The work is joint with Simon Goodwin.

_{2}-triples in classical Lie algebras over fields of positive characteristic

#### Rachel Pengelly, University of Birmingham

**Tuesday 12 October 2021, 15:00-16:00 Lecture Theatre C, Watson Building**

Let *K* be an algebraically closed field. Given three elements of some Lie algebra over *K*, we say that these elements form an _{2}-triple if they generate a subalgebra which is a homomorphic image of _{2}(K). In characteristic 0, the Jacobson-Morozov theorem provides a bijection between the orbits of nilpotent elements of the Lie algebra and the orbits of _{2}-triples. In this talk I will discuss the progress made in extending this result to fields of characteristic *p*. In particular, I will focus on the results in classical Lie algebras, which can be found as subsets of _{n}(K).

### Enumerating transitive groups, bounding generator numbers, and complexity of algorithms in Computational Group Theory

#### Derek Holt, University of Warwick

**Tuesday 19 October 2021, 15:00-16:00 Lecture Theatre C, Watson Building**

Computer databases that contain information on various types of groups can play a vital role in research. Examples include finite groups up to order 2000, primitive permutation groups up to degree 4095, and transitive permutation groups, recently extended to degree 48. In the first part of the talk, we provide some details on the recent successful lengthy computer calculations involved in the enumeration of the 195,826,352 transitive groups of degree 48 (i.e. conjugacy classes in the symmetric group). For a finitely generated group *G*, let *d*(*G*) be the smallest number of elements required to generate *G*. In the second part of the talk, we survey results bounding *d*(*G*) for various types of finite permutation and matrix groups of a given degree, and describe how a knowledge of the transitive groups of degree 48 can be used to improve a result of Gareth Tracey bounding *d*(*G*) for transitive permutation groups. Finally we describe recent results obtained jointly with Gareth, which bound *d*(*G*) log |*G*|, and are partly motivated by attempts to estimate the complexity of algorithms to compute the automorphism group of *G*.

### Unipotent representations in the local Langlands correspondence

#### Beth Romano, University of Oxford

**Tuesday 26 October 2021, 15:00-16:00 Lecture Theatre C, Watson Building**

The local Langlands correspondence (LLC) is a kaleidoscope of conjectures relating local Galois theory, complex Lie theory, and representations of *p*-adic groups. This talk will give an introduction to the part of the LLC involving unipotent representations. Reducing modulo *p*, we can move from representations of *p*-adic groups to representations of finite reductive groups, which have a rich structure developed by Deligne-Lusztig. I will talk about joint work with Anne-Marie Aubert and Dan Ciubotaru in which we lift some of this structure to *p*-adic groups. I will not assume previous familiarity with these topics; instead I'll give an introduction to these ideas via examples.

### Maximal irredundant base size and relational complexity

#### Veronica Kelsey, University of Manchester

**Tuesday 2 November 2021, 15:00-16:00 Lecture Theatre C, Watson Building**

We begin with definitions, examples and motivation for various numerical invariants of permutation groups, such as base size, maximal irredundant base size and relational complexity. We then give upper bounds on these numerical invariants for certain families of groups.

### On relational complexity and other statistics for permutation groups

#### Nick Gill, Open University

**Tuesday 9 November 2021, 15:00-16:00 Lecture Theatre C, Watson Building**

I will discuss some recent work studying various statistics to do with finite permutation groups. These include *b*(*G*), minimum base size and *RC*(*G*), relational complexity. I will focus mainly on open questions concerning these statistics but will also discuss various bits of joint work with Hudson, Liebeck, Loda and Spiga.

### Decomposition of spin representations of symmetric groups in characteristic 2

#### Lucia Morotti, Hannover University

**Tuesday 16 November 2021, 15:00-16:00Zoom**

Any representation of a double cover of a symmetric group *S̃ _{n}* can also be viewed as a representation of

*S*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.

_{n}### Structure properties of nilpotent elements in simple Lie superalgebras

#### Leyu Han, University of Birmingham

**Tuesday 23 November 2021, 15:00-16:00 Lecture Theatre C, Watson Building**

Lie superalgebras, can be viewed as a generalization of Lie algebras including a ℤ_{2}-grading, has become a topic of intense research interest both in mathematics and physics since mid 1970s. In this talk, I will mainly focus on introducing some background of Lie superalgebras, for example classification of simple Lie superalgebras, nilptent orbits and so on. I will also give examples of Lie superalgebras which are simple. Then I will mention my research on the centralizer of nilpotent elements in simple Lie superalgebras and its centre and state my main result on the above structures.

### Class 2 nilpotent and twisted Heisenberg groups

#### Dávid Szabó, Rényi Institute, Budapest

**Tuesday 30 November 2021, 15:00-16:00 Zoom**

Every finitely generated nilpotent group *G* of class at most 2 can be obtained from 2-generated such groups using central and subdirect products. As a corollary, *G* embeds to a generalisation of the 3×3 Heisenberg matrix group with entries coming from suitable abelian groups depending on *G*. In this talk, we present the key ideas of these statements and briefly mention how they emerged from investigating the so-called Jordan property of various transformation groups.

### The dual approach to Coxeter and Artin groups

#### Barbara Baumeister, Bielefeld University

**Tuesday 7 December 2021, 15:00-16:00 Zoom**

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.

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

#### Ralf Koehl, Giessen University

**Tuesday 14 December 2021, 15:00-16:00 Zoom**

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.

### The quaternionic x dihedral group of order 32 quotient singularity is also a quiver variety, as are its 81 crepant resolutions.

#### Travis Schedler, Imperial College London

**Wednesday 2 February 2022, 15:30-16:30 Zoom**

I will consider the order-32 central product of the quaternionic and dihedral groups of order eight, which naturally acts via symplectic four-by-four matrices. The quotient is a fascinating singular cone which was predicted in 2015 by physicists to be isomorphic to a quite different object, a Nakajima quiver variety. We will prove this, using basic representation theory and geometry. This allows us to give a new description of all 81 crepant resolutions of the singularity, which are all given as hyperpentagon spaces (a hyperkaehler version of the moduli of pentagons in R3). Moving beyond this, we prove that all crepant resolutions of the analogous quiver cone for the n-pointed star are also hyperpolygon spaces. For example, there are precisely 1684 hyperhexagon spaces. The count uses the combinatorics of hyperplane arrangements.

### Cluster structures for Grassmannians

#### Karin Baur, University of Leeds

**Wednesday 9 February 2022, 15:30-16:30 Zoom**

The category of Cohen Macaulay modules over a quotient of a preprojective algebra is a cluster category associated to the coordinate ring of the Grassmannian Gr(k,n). We study this category and describe some of its indecomposable modules. We also explain how one can associate frieze patterns to them.

### Groups, languages, and automata

#### Marialaura Noce, University of Gottingen

**Wednesday 16 February 2022, 15:30-16:30 Zoom**

In this talk we will give an introduction to automata groups, explaining the connections between the latter, groups of automorphisms of rooted trees, and formal languages. Then, we present examples, important recent developments, and open problems.

### Generation versus invariable generation

#### Daniele Garzoni, Tel Aviv University

**Wednesday 23 February 2022, 15:30-16:30 Zoom**

I will first define the concept of invariable generation of finite groups, and give some motivation. There are some striking differences with respect to the usual generation. We will experience this by taking some known results for the usual generation, and seeing what happens in the invariable setting. We will discuss minimal generating sets, generating graphs, random generation, and we will see some open questions. We will mainly consider groups at the extremes: soluble groups, at one end; nonabelian simple groups, at the other end.

### Skew-power series over prime rings (joint with William Woods)

#### Adam Jones, University of Manchester

**Wednesday 9 March 2022, 15:30-16:30 Zoom**

Let *R* be a ring carrying an automorphism σ and a σ-derivation δ. We are interested in the skew-power series ring *R*[[x;σ,δ]], in the cases when it is well defined. Specifically, we want to prove analogues of properties of the well studied skew-polynomial ring *R*[x;σ,δ] to the skew power series case. We will focus on the question: if *R* is a prime, Noetherian ring, is *R*[[x;σ,δ]] also prime? We will partially answer this question in the case where *R* carries an appropriate topology such that (σ,δ) are continuous, focusing particularly on the case where δ=σ-1.

### Irreducible Representations of Rational Cherednik Algebras in Positive Characteristic

#### Martina Balagovic, University of Newcastle

**Wednesday 16 March 2022, 15:30-16:30 310 Watson**

Rational Cherednik algebras are a class of associative non-commutative infinite dimensional algebras depending on a reflection group and several parameters. In this talk I will consider such an algebra over a field of positive characteristic and its Category *O*, explain how it differs from the corresponding category over complex numbers, and talk about some methods for finding explicit descriptions of irreducible representations in this category in terms of singular vectors and characters. The first half of the talk (setup) is old joint work with Harrison Chen, and the second half (type A computations) is recent joint work with Jordan Barnes.

### The Orbit Method for Complex Groups

#### Lucas Mason-Brown, University of Oxford

**Wednesday 23 March 2022, 15:30-16:30 Venue to be confirmed**

The classification of all irreducible unitary representations of a reductive Lie group *G* is one of the fundamental unsolved problems in representation theory. In the 1960s, Kostant and Kirillov proved that the irreducible unitary representations of a *solvable* Lie group are (approximately) classified by co-adjoint orbits of *G*. In the 1980s, David Vogan conjectured that a version of this result should hold for semisimple Lie groups. This set of theorems (in the solvable case) and conjectures (in the reductive case) is referred to as the "Orbit Method". In recent joint work with Ivan Losev and Dmitryo Matvieievskyi, we define an orbit method correspondence for complex reductive algebraic groups (regarded as real groups by restriction of scalars). I will report on this work and discuss possible extensions to the case of real groups.

### A new family of symplectic singularities

#### Paul Levy, Lancaster University

**Wednesday 30 March 2022, 14:30-15:30 Lecture Theatre B, Watson Building**

Symplectic singularities were defined by Beauville more than 20 years ago. The main classes of known examples are symplectic quotient singularities ℂ^{2n}/Γ (where Γ is a finite subgroup of the symplectic group) and (normalisations of) nilpotent orbit closures. Until very recently, it was suspected that these exhausted all possible isolated symplectic singularities. In this talk, I will explain three markedly different constructions of a completely new family of isolated symplectic singularities χ_{n} (*n*≥ 5): as partial resolutions of quotient singularities for the dihedral group of order 2*n*; as deformations arising via the corresponding Calogero-Moser space; in the universal cover of the nilpotent cone of gl_{n}. The special case *n*=5 had earlier appeared in relation to a particular Slodowy slice in type E_{8}.

### Powerfully nilpotent, solvable and simple groups

#### Gunnar Traustason, University of Bath

**Wednesday 27 April 2022, 15:30-16:30 Lecture Theatre B, Watson Building**

In this talk we discuss a special subclass of powerful groups called powerfully nilpotent groups. These are finite p-groups that possess a central series of a special kind. We will describe some structure theory and a 'classification' in terms of an ancestry tree and powerful coclass. One can view powerfully nilpotent groups as the powerful analogue of nilpotent groups. There is likewise a natural powerful analogue of solvable groups, "powerfully solvable groups", that we will also discuss briefly. For a special situation one can also introduce "powerfully simple groups".

### Frobenius algebras and fractional Calabi-Yau categories

#### Joseph Grant, University of East Anglia

**Wednesday 11 May 2022, 15:30-16:30 Lecture Theatre B, Watson Building**

Given a quiver we consider two algebras: its path algebra and its preprojective algebra. If the quiver is Dynkin, i.e., its underlying graph is a simply laced Dynkin diagram, then both algebras have nice properties: the derived category of the path algebra is fractionally Calabi-Yau, and the preprojective algebra is Frobenius with a Nakayama automorphism of finite order. One can show that, if stated carefully, these properties are equivalent. I will give an introduction to the concepts above and, time permitting, will describe some of the ingredients of the proof of this equivalence.

We address the challenges facing society and the economy, from shedding light on the refugee crisis, to character education in schools, through to developing leaders in the NHS.