# Algebra seminars, 2012-13

## In 'Past algebra seminars'

### On local symmetric algebras and finite group algebras

#### Radha Kessar, City, University of London

**Thursday 4 October 2012, 16:00-17:00 Watson Building, Lecture Room C**

Abstract not available

### Normal subgroups of compact groups

#### Nikolay Nikolov, University of Oxford

**Thursday 11 October 2012, 16:00-17:00 Watson Building, Lecture Room C**

Among compact Hausdorff groups *G* whose maximal profinite quotient is finitely generated, we characterize those that possess a proper dense normal subgroup. This relies on asymptotic results on the fixed points of automorphisms of finite simple groups and Lie groups. We also prove that the abstract commutator subgroup [*H*,*G*] is closed for every closed normal subgroup *H* of *G*. Joint work with Dan Segal, Oxford.

### What can you say about a finite group if you know something about the conjugacy class sizes?

#### Alan Camina, University of East Anglia

**Thursday 1 November 2012, 16:00-17:00 Watson Building, Lecture Room C**

The influence of the sizes of conjugacy classes of a finite group.

Let *G* be a finite group.

**Question 1** Given information about the sizes of the conjugacy classes of the elements of a group what can we say about the group?

As stated the question is a little vague but let me give some very early answers:

**Answer 1** (Sylow (1870)) If all the sizes are a power of a prime *p* then *G* has a non-trivial centre.

**Answer 2** (Burnside (1904)) If at least one size is the power of a prime then *G* is not simple.

So we have some quite old answers; although the first answer is straightforward to prove the second is quite hard.

There has been a lot of activity on these problems in recent years and in this lecture I would like to touch on what, I think, are some of the more interesting.

Here is one result I will mention.

**Theorem 1** (Camina and Camina) Let *G* be a finite group with the property that given any three distinct conjugacy class sizes greater than 1 there is a pair which is coprime. Then *G* has at most three conjugacy class sizes greater than 1, and *G* is soluble.

In this talk both the background to the result and some interesting open questions that arise will be discussed. This will be based on the survey recently written jointly with Rachel Camina.

### Localization theory for quantum Hamiltonian reductions

#### Kevin McGerty, University of Oxford

**Thursday 8 November 2012, 16:00-17:00 Watson Building, Lecture Room C**

I will give an overview the classical story of localisation and explain the more recent 'symplectic' point of view. I will then discuss some results of T. Nevins and myself which hold in the case of quantum Hamiltonian reductions.

### Coverings of groups by subgroups

#### Martino Garonzi, University of Padova

**Friday 9 November 2012, 11:00-12:00 52 Pritchatts Rd, SR16**

Let *G* be a finite group. A covering of *G* is a family of proper subgroups of *G* whose union is *G*. Such a family exists if and only if *G* is non-cyclic. If *G* is non-cyclic define *s*(*G*) to be the smallest cardinality of a covering of *G*, and set *s*(*G*) equal to infinity if *G* is cyclic. The function *s* was introduced by J.H.E. Cohn and has been studied by several authors. Let *N* be a normal subgroup of *G*. It is clear that any covering of *G*/*N* can be lifted to a covering of *G* of the same cardinality, and this implies that *s*(*G*) is bounded from above by *s*(*G*/*N*). If there exists *N* such that *s*(*G*) = *s*(*G*/*N*) then we are reduced to study the quotient *G*/*N*, and for this reason we are very much interested in the groups *G* with the property that *s*(*G*) < *s*(*G*/*N*) for any non-trivial normal subgroup *N* of *G* (these groups are traditionally called 'sigma-primitive').

Lucchini and Detomi conjectured that such groups either are abelian or admit exactly one minimal normal subgroup. Proving this would improve very much our understanding of the function *s*. In this talk we present this conjecture and the progress made so far about it.

### Remarks on Amalgams and Fusion

#### Geoffrey Robinson, University of Bristol

**Thursday 15 November 2012, 16:00-17:00 Watson Building, Lecture Room C**

Abstract not available

### Decision problems for profinite groups

#### Henry Wilton, University College, London

**Thursday 29 November 2012, 16:00-17:00 Watson Building, Lecture Room C**

There is no algorithm to determine if a finitely presented profinite group is non-trivial. I will discuss the proof of this result and some of the consequences. This is joint work with Martin Bridson.

### Chevalley restriction theorem for vector-valued functions on quantum groups

#### Martina Balagović, University of York

**Thursday 6 December 2012, 16:00-17:00 Watson Building, Lecture Room C**

Abstract not available

### Donovan's conjecture

#### Charles Eaton, University of Manchester

**Thursday 10 January 2013, 16:00-17:00 Watson Building, R17/18**

Abstract not available

### Character deflations, wreath products and Foulkes' Conjecture

#### Mark Wildon, Royal Holloway, University of London

**Thursday 17 January 2013, 17:00-18:00 Watson Building, Lecture Room C**

Foulkes’ Conjecture is one of the main open problems in algebraic combinatorics: it states that if *a* ≤ *b* then the permutation character of the symmetric group *S _{ab}* acting on set partitions of a set of size

*ab*into

*b*sets each of size

*a*contains the permutation character of

*S*acting on set partitions of the same set into

_{ab}*a*sets each of size

*b*. In my talk I will present a new approach to Foulkes' Conjecture based on a deflation map that sends characters of the wreath product of

*S*with

_{a}*S*to characters of

_{b}*S*. Our main result is a combinatorial rule for the values taken by these deflated character. This rule generalizes the Murnaghan–Nakayama rule and Young’s rule, and leads to a new algorithm for computing the irreducible constituents of Foulkes characters. Using this algorithm we have verified Foulkes' Conjecture in several new cases, including all

_{b}*a*,

*b*such that

*a*+

*b*≤ 18.

This talk is on joint work with Anton Evseev and Rowena Paget.

### A Majorana theory?

#### Sergey Shpectorov, University of Birmingham

**Thursday 24 January 2013, 17:00-18:00 Watson Building, Lecture Room C**

Majorana algebras were introduced several years ago as a generalization of the Griess–Norton algebra for the Monster sporadic simple group. One feature of this developing area is that there are now many examples of such algebras while there have been very few general theoretical results.

In the lecture I will attempt to lay some groundwork and present outlines of a possible theory.

### Simple groups and magic words

#### Martin Liebeck, Imperial College, London

**Thursday 21 February 2013, 16:00-17:00 Watson Building, Lecture Room C**

Abstract not available

### Permutation groups where non-trivial elements have few fixed points

#### Rebecca Waldecker, Martin Luther University Halle-Wittenberg

**Thursday 28 February 2013, 16:00-17:00 Watson Building, Lecture Room C**

Abstract not available

*Z**-theorem for odd primes and for source algebras in blocks of finite groups

#### Shigeo Koshitani, Chiba University

**Thursday 7 March 2013, 16:00-17:00 Watson Building, Lecture Room C**

Abstract not available

### Derived subalgebras of centralisers and primitive ideals

#### Alexander Premet, University of Manchester

**Thursday 14 March 2013, 16:00-17:00 Watson Building, Lecture Room C**

Abstract not available

### Some algebraic properties of compact topological groups

#### Dan Segal, University of Oxford

**Thursday 21 March 2013, 16:00-17:00 Watson Building, Lecture Room C**

Abstract not available

### Overgroups of regular elements

#### Donna Testerman, EPFL

**Thursday 25 April 2013, 16:00-17:00 Watson Building, Lecture Room A**

We report on recent work with A. Zalesski concerning the study of reductive overgroups of regular elements in reductive algebraic groups.

In the case where the regular element is a regular unipotent element, this work complements the work of Saxl and Seitz, where they classified the maximal positive-dimensional closed subgroups *H* of a simple algebraic group *G*, such that *H* contains a regular unipotent element of *G*. Our main result on unipotent elements shows that a connected reductive subgroup *H* containing a regular unipotent element of *G* does not lie in a proper parabolic subgroup of *G*.

We will describe the proof of the above result and its application to the study of overgroups of regular unipotent elements. Then we will discuss our current work on overgroups of general regular elements in simple algebraic groups.

### Lie group symmetries of crystals with defects

#### Rachel Nicks, University of Birmingham

**Thursday 2 May 2013, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available

### Finite Group Actions on Compact Riemann Surfaces

#### Aaron Wootton, University of Portland

**Thursday 9 May 2013, 16:00-17:00 Watson Building, Lecture Room A**

The problem of determining the distinct finite group actions on a compact Riemann surface of a fixed genus is a classical problem in complex analysis. Though there has been tremendous progress in the last few years toward solving this problem, it is widely accepted that a complete answer is intractable due to its computational complexity. Therefore, much current research in this area focuses on easier but more amenable problems which provide insight into the more general problem.

In this talk, we introduce a new and simpler way to consider this problem. Though this new approach ignores much of the geometric structure of an automorphism group, it does provide a new and elegant way to represent group actions which we shall illustrate through a number of classical examples. We shall finish by showing that this very elementary way of describing group actions is in fact sufficient to prove that for a fixed genus σ ≥ 6, there is minimally a quadratic lower bound in the genus σ for the number of distinct group actions on a surface of genus σ.

### Maximal subgroups of free idempotent generated semigroups

#### Vicky Gould, University of York

**Tuesday 14 May 2013, 14:00-15:00 Watson Building, Lecture Room C**

Abstract not available

### Primitive triangle free strongly regular graphs, a survey: from Dale Mesner to recent results

#### Mikhail Klin, Ben-Gurion University of the Negev

**Thursday 16 May 2013, 16:00-17:00 Watson Building, Lecture Room A**

In this survey lecture we will try to cover a few topics.

First we wish to recall an intriguing hidden history about the unique strongly regular graph *NL*_{2}(10) with the parameters (100, 22, 0, 6), specifically as it relates to the work of Dale Mesner during the years 1956-1964. We will mention the later independent construction of this graph by D.G. Higman and C.C. Sims, which led them to discover a new sporadic simple group.

We also will discuss in some detail our own computer aided efforts to better understand the graph *NL*_{2}(10). In fact, this graph contains all known 7 primitive triangle-free strongly graphs (those on 5,10,16,50,56,77 and 100 vertices).

New models for some of these graphs will be presented, which rely on knowledge of a relatively small subgroup of a graph in consideration.

We will finish with a brief discussion of recent attempts and ongoing perspectives in search for new primitive triangle free strongly regular graphs.

This is a joint project together with Matan Ziv-Av (BGU) and Andy Woldar (Villanova, USA).

### How do you decide if a group is an amalgam?

#### Aditi Kar, University of Oxford

**Thursday 23 May 2013, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available

### Local subgroups of topological Kac-Moody groups

#### Inna Capdeboscq, University of Warwick

**Thursday 6 June 2013, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available

### Classifying superconformal field theories via chiral rings

#### Oliver Gray, University of Bristol

**Thursday 13 June 2013, 16:00-17:00 Watson Building, Lecture Room A**

I will introduce the main mathematical ideas behind conformal field theory, using the *N*=2 minimal models as a concrete example. We will see how the so-called 'chiral rings' arise from basic Lie algebra representation theory plus ideas from quantum field theory. Then we'll use the chiral rings to classify a special class of *N*=2 minimal models (those with 'space-time supersymmetry') into the famous ADE pattern (as seen in finite subgroups of SU(2), surface quotient singularities, simple finite-dimensional simply-laced Lie algebras, simply-laced Dynkin diagrams, etc.) Finally I'll present the chiral rings of the entire class of *N*=2 minimal models (even those without the extra supersymmetry) and leave as an open question the classification of these theories.

### Classical and non-classical subgroups of Schottky groups

#### James Anderson, University of Southampton

**Friday 14 June 2013, 14:00-15:00 Watson Building, Lecture Room A**

Schottky groups are among the earliest examples given of Kleinian groups acting on hyperbolic 3-space. The proof by Marden of the existence of nonclassical Schottky groups, and the subsequent construction of explicit examples in rank 2 by Yamamoto, led to the question of whether every Riemann surface can be uniformed by a classical Schottky group. Consideration of this difficult question quickly leads to the conclusion that in order to make any significant progress, we need to have some clear understanding of the differences between classical and non-classical Schottky groups. The purpose of this talk is to define what we mean by classical and non-classical Schottky groups and to present results describing the possibilities for classical or non-classical subgroups in both classical and non-classical Schottky groups.

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.