# Algebra seminars, 2023-24

## In 'Past algebra seminars'

### Sylow branching coefficients for symmetric groups

#### Stacey Law (Birmingham)

**Thursday 5 October 2023, 15:00-16:00 LG06 Old Gym**

My work is primarily on the representation theory of finite groups, connections to group structures and algebraic combinatorics. One of the key questions in these areas is to understand the relationship between the characters of a finite group *G* and its local subgroups. Sylow branching coefficients describe the restriction of irreducible characters of *G* to a Sylow subgroup *P* of *G*, and have been recently shown to characterise structural properties such as the normality of *P* in *G*. In this talk, we will discuss and present some new results on Sylow branching coefficients for symmetric groups.

### Progress on two general projectivity questions for infinite groups

#### Rudradip Biswas (Warwick)

**Thursday 12 October 2023, 15:00-16:00 LG06 Old Gym**

Two general projectivity questions were proposed for integral group rings for infinite discrete groups in a 2007 paper by Jang Hyun Jo. In a paper I published last year, I reported on some progress on those questions using different methods made possible by some of my previous work on the behaviour of various cohomological invariants of infinite groups. I will talk about these two questions and if time permits, I will chalk out some related questions for future work.

### Some representation theory of Kadar-Martin-Yu algebras

#### Alison Parker (Leeds)

**Thursday 19 October 2023, 15:00-16:00 LG06 Old Gym**

Kadar-Martin-Yu introduced a new chain of subalgebras of the Brauer algebra. These algebras start with Temperley-Lieb and end with the Brauer algebra and build in representation theoretic intensity. This gives a new tool to tackle the long standing problem of understanding the representation theory of the Brauer algebra. We present an introduction to these new algebras and some results about their representation theory. This is joint work with my PhD student N. M. Alraddadi.

### Decomposition numbers for unipotent blocks with small sl_{2}-weight in finite classical groups

#### Emily Norton (Kent)

**Thursday 26 October 2023, 15:00-16:00 LG06 Old Gym**

There are many familiar module categories admitting a categorical action of a Lie algebra. The combinatorial shadow of such an action often yields answers to module-theoretic questions, for instance via crystals. In proving a conjecture of Gerber, Hiss, and Jacon, it was shown by Dudas, Varagnolo, and Vasserot that the category of unipotent representations of a finite classical group has such a categorical action. In this talk I will explain how we can use the categorical action to deduce closed formulas for certain families of decomposition numbers of these groups. This is joint work in progress with Olivier Dudas.

### Group generation

#### Veronica Kelsey (Manchester)

**Thursday 9 November 2023, 15:00-16:00 LG06 Old Gym**

This talk will be a very gentle stroll through some results on generating groups. A group *G* is 2-generated if there exist elements *x* and *y* in *G* such that <*x*,*y*>=*G*. In 1962 Steinberg proved that the finite simple groups known at the time were 2-generated. In fact all finite simple groups are 2-generated. We consider what conditions we can impose on the elements *x* and *y* such that they still generate *G*, for example insisting *y* lies in a certain conjugacy class or subgroup.

### Qualitative results on the dimensions of irreducible representations of linear groups over local rings

#### Alexander Stasinski (Durham)

**Thursday 16 November 2023, 15:00-16:00 LG06 Old Gym**

Let *G _{r}* = GL

_{n}(

*O*/

*P*), where

^{r}*O*is the ring of integers of a local field with finite residue field

_{q}of characteristic

*p*,

*P*is the maximal ideal and

*r*is a positive integer. It has been conjectured by U. Onn that the dimensions of the irreducible representations of

*G*, as well as the number of irreducible representations of a fixed dimension, are given by evaluating finitely many polynomials (only depending on

_{r}*n*and

*r*) at the residue field cardinality

*q*. In particular, it is conjectured that the two groups GL

_{n}(

_{p}[

*t*]/

*t*) and GL

^{r}_{n}(ℤ/

*p*) have the same number of irreducible representations of dimension

^{r}*d*, for each

*d*. These conjectures can be generalised by allowing other (reductive) group schemes than GL

_{n}.

I will report on some recent progress on the polynomiality of the representation dimensions in joint work with A. Jackson as well as some independent related work by I. Hadas. The latter proved that for any affine group scheme *G* of finite type over ℤ, all *r* and all large enough *p* (depending on *G* and *r*), the groups *G*(_{p}[*t*]/*t ^{r}*) and

*G*(ℤ/

*p*) have the same number of irreducible representations of dimension

^{r}*d*, for each

*d*. A crucial intermediate result is that the stabilisers of representations of certain finite groups are the

_{q}-points of algebraic groups with boundedly many geometric connected components.

### All Kronecker coefficients are reduced Kronecker coefficients

#### Christian Ikenmeyer (Warwick)

**Thursday 23 November 2023, 15:00-16:00 LG06 Old Gym**

We settle the question of where exactly the reduced Kronecker coefficients lie on the spectrum between the Littlewood-Richardson and Kronecker coefficients by showing that every Kronecker coefficient of the symmetric group is equal to a reduced Kronecker coefficient by an explicit construction. This implies the equivalence of a question by Stanley from 2000 and a question by Kirillov from 2004 about combinatorial interpretations of these two families of coefficients. This is joint work with Greta Panova, arXiv:2305.03003.

### Bases for permutation groups

#### Hongyi Huang, University of Bristol

**Thursday 30 November 2023, 15:00-16:00 LG06 Old Gym**

Let *G*≤Sym(Ω) be a permutation group on a finite set Ω. A base for *G* is a subset of Ω with trivial pointwise stabiliser, and the base size of *G*, denoted *b*(*G*), is the minimal size of a base for *G*. This classical concept has been studied since the early years of permutation group theory in the nineteenth century, finding a wide range of applications.

Recall that *G* is called primitive if it is transitive and its point stabiliser is a maximal subgroup. Primitive groups can be viewed as the basic building blocks of all finite permutation groups, and much work has been done in recent years in bounding or determining the base sizes of primitive groups. In this talk, I will report on recent progress of this study. In particular, I will give the first family of primitive groups arising in the O'Nan-Scott theorem for which the exact base size has been computed in all cases.

### An introduction to τ-exceptional sequences

#### Bethany Marsh, University of Leeds

**Thursday 7 December 2023, 15:00-16:00 LG06 Old Gym**

Joint work with Aslak Bakke Buan.

Exceptional sequences in module categories over hereditary algebras (e.g. path algebras of quivers) were introduced and studied by W. Crawley-Boevey and C. M. Ringel in the early 1990s, as a way of understanding the structure of such categories. They were motivated by the consideration of exceptional sequences in algebraic geometry by A. I. Bondal, A. L. Gorodontsev and A. N. Rudakov.

Exceptional sequences can also be considered over arbitrary finite-dimensional algebras, but their behaviour is not so good in general: for example, complete exceptional sequences may not exist. We look at different ways of generalising to the hereditary case, with a focus on τ-exceptional sequences, recently introduced in joint work with A. B. Buan (NTNU), motivated by the τ-tilting theory of T. Adachi, O. Iyama and I. Reiten, and signed exceptional sequences in the hereditary case defined by K. Igusa and G. Todorov.

### Coarse geometry of groups and spaces

#### David Hume, University of Birmingham

**Thursday 18 January 2024, 11:00-12:00Arts LR6**

In the study of countably infinite groups, it is typical (because finite group theory is hard) to consider two groups as "equivalent" if they are commensurable: i.e. they admit finite-index subgroups which are isomorphic. As a result, properties of groups which are invariant under commensurability are desirable. For finitely generated groups, there are a wealth of such properties coming from many different areas of mathematics (combinatorics, topology, algebra, geometry...) and a large and active area of research dedicated to further understanding these properties.

Perhaps surprisingly, the corresponding notion for subgroup "inclusion" – the first group admits a finite-index subgroup which is isomorphic to some subgroup of the second – has received comparatively little attention. I will motivate the problem in a more general geometric setting and describe recent work of myself and collaborators to address this.

### You need 27 tickets to guarantee a win on the UK National Lottery

#### David Stewart, University of Manchester

**Thursday 25 January 2024, 11:00-12:00Arts LR6**

(Joint with David Cushing.) The authors came across the problem of finding minimal lottery design numbers *j*=*L*(*n*,*k*,*p*,*t*); that is, a set *B*_{1}, ..., *B _{j}* subsets of {1,..,

*n*} each of size

*k*, such that for any subset

*D*of {1,..,

*n*} of size

*p*, one finds an intersection

*D*∩

*B*with at least

_{i}*t*elements. In the context of a lottery,

*n*represents the number of balls,

*k*the number of choices of balls on a ticket,

*p*the size of a draw. For the UK national lottery,

*n*=59,

*k*=

*p*=6 and one gets a (rather meagre) prize as long as

*t*is at least 2. Using the constraint solving library in Prolog, we calculated

*j*for

*k*=

*p*=6,

*t*=2 and

*n*all the way up to 70. For example

*L*(59,6,6,2)=27. This is the second paper where we have aimed to show the value of Prolog and constraint programming in pure mathematics.

I'll give an overview of constraint programming, logic programming in Prolog, and describe how we used these tools to solve the problem described in the title.

### Unitary units of modular group algebras

#### Victor Bovdi, UAE University, Al Ain

**Thursday 1 February 2024, 11:00-12:00Arts LR6**

Let *FG* be the group algebra of a group *G* over the field *F*. The subset V*(*FG*) of unitary units, under the classical involution *, of the group of normalised units of the algebra *FG* forms a group called the unitary subgroup of the group algebra *FG*. In the talk, we present some recent results about the structure of the unitary subgroup V*(*FG*), such as nilpotency, locally nilpotency and others. We will also discuss the connections of the structure of V*(*FG*) with other parts of mathematics.

### Non-singular identities for finite groups

#### Henry Bradford, University of Cambridge

**Thursday 8 February 2024, 11:00-12:00Arts LR6**

A law (respectively an identity with constants) for a group *G* is an equation in one or more variables (respectively variables and constants from G) which is satisfied by all tuples of elements from *G*. Every finite group *G* satisfies some law, and the length of the shortest law, or the shortest identity with constants, is a natural invariant of *G*. We survey some of what is known about the asymptotic behaviour of these lengths in various families of finite groups. Motivated by a conjecture of Larsen and Shalev concerning the class of profinite groups satisfying laws, we draw particular attention to the class of 'non-singular' identities with constants. Joint work with Jakob Schneider and Andreas Thom.

### Modular represenations for Yangian *Y*_{2}

#### Hao Chang, Central China Normal University

**Thursday 15 February 2024, 11:00-12:00Arts LR6**

The connection between Yangians and finite *W*-algebras of type *A* was first noticed by mathematical physicists Briot, Ragoucy and Sorba, and then constructed in general cases by Brundan and Kleshchev. This provides a useful tool for the study of representation theory of finite *W*-algebras. Over an algebraically closed field of positive characteristic, Brundan-Kleshchev’s theory was established by Goodwin and Topley. I will talk about the modular representations of the Yangian *Y*_{2}. From this, we may understand the representations of certain reduced enveloping algebras of type *A* Lie algebras.

### Spin representations of symmetric groups in characteristic 2

#### Matt Fayers, Queen Mary, University of London

**Thursday 22 February 2024, 11:00-12:00Arts LR6**

Let *G* be a finite group and *p* a prime. Then there is a well-defined (at the level of composition factors) process of *p*-modular reduction for representations of *G*. It sometimes happens that two different irreducible modules in characteristic 0 can become the same when reduced modulo *p*, and it is interesting to determine exactly when this happens. For example, if *G* is the symmetric group, and two ordinary irreducibles are obtained from each other by tensoring with the sign representation, then their reductions modulo 2 will be the same.

In this talk we consider this problem for the double covers of the symmetric groups in characteristic 2; in fact, we solve the more general problem of when the 2-modular reductions of two modules are proportional to each other. I will give the result, and explain some of the techniques used to prove it.

### Walls in CAT(0) spaces and beyond

#### Davide Spriano, University of Oxford

**Thursday 29 February 2024, 11:00-12:00Arts LR6**

CAT cube complexes are a particularly well-behaved class of CAT spaces. One of the reasons why we understand them so much better is because they have hyperplanes, combinatorial objects that encode the geometry of the space. The goal of this talk is to discuss generalizations of hyperplane in the setting of CAT spaces and beyond. This is joint work with Harry Petyt and Abdul Zalloum.

### Subgroup structure of exceptional algebraic groups

#### Vanthana Ganeshalingam, University of Warwick

**Thursday 7 March 2024, 11:00-12:00Arts LR6**

This talk will introduce the concept of *G* complete-reducibility (G c-r) originally thought of by Serre in the 90s. This idea has important connections to the open problem of classifying the subgroups of a reductive group *G*. I will explain the methodology of the classification so far and the main obstacle which is understanding the non-G-cr subgroups.

### Loxodromic elements of right-angled Artin groups

#### Alice Kerr, University of Bristol

**Thursday 14 March 2024, 11:00-12:00Arts LR6**

Given a finite subset of a group, we can ask if we can combine a bounded number of elements of that subset to get a group element with a specific property. In our case, the property we are looking for is that it has unbounded orbits in a certain action on a hyperbolic space, which is important for many statements about group growth. Here we will be considering this question for right-angled Artin groups, which are generalisations of both free groups and free abelian groups, and we will show that it can be solved by considering actions on trees. This is joint work with Elia Fioravanti.

### Plethysm via the partition algebra

#### Rowena Paget, University of Kent

**Thursday 21 March 2024, 11:00-12:00Arts LR6**

The symmetric group *S _{mn}* acts naturally on the collection of set partitions of a set of size

*mn*into

*n*sets each of size

*m*. The irreducible constituents of the associated ordinary character are largely unknown; in particular, they are the subject of the longstanding Foulkes Conjecture. There are equivalent reformulations using polynomial representations of infinite general linear groups or using plethysms of symmetric functions. I will review plethysm from these three perspectives then present a new approach to studying plethysm: using the Schur-Weyl duality between the symmetric group and the partition algebra. This is joint work with Chris Bowman (arXiv: 1809.08128) and with Chris Bowman and Mark Wildon (arXiv: 2311.02721).

### A semi-infinite exploration of semi-direct products with the Witt algebra

#### Girish Vishwa, University of Edinburgh

**Wednesday 5 June 2024, 16:00-17:00Watson Lecture Theatre A**

Recently, in both physics and mathematics, there has been an increased interest in special cases of a class of Lie algebras obtained by taking the semi-direct product of the Witt algebra with its tensor density modules. Some well-known examples include the twisted Heisenberg-Virasoro algebra, the *W*(2,2) algebra and Ovsienko-Roger algebra. In this talk, I would like to introduce this class of Lie algebras in general, elucidate their role in recent conformal field theoretic and string theoretic developments and present some preliminary findings on their semi-infinite cohomology (otherwise known as BRST cohomology in physics).

### Swapping runners to find spin representations which reduce modulo 2 to Specht modules

#### Eoghan McDowell, Okinawa Institute of Science and Technology

**Wednesday 24 July 2024, 14:00-15:00Watson Lecture Theatre C**

When do two ordinary irreducible representations of a group have the same p-modular reduction? In this talk I will address this question for the double cover of the symmetric group, giving a necessary and sufficient condition for a spin representation of the symmetric group to reduce modulo 2 to a multiple of a Specht module (in the sense of Brauer characters or in the Grothendieck group). This is joint work with Matthew Fayers. I will discuss different aspects of the problem to those presented by Matt in his recent seminar talk at the University of Birmingham: I will exhibit our "runner swapping function" (a certain combination of induction and restriction functors which has the effect of swapping adjacent runners in an abacus display for the labelling partition of a character), and show how to use it to identify the modules which have equal 2-modular reductions.

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.