# Algebra seminars, 2023-24

## In 'Algebra'

Back to 'Mathematics Research'### 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 (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 (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.

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.