# 8th meeting: Permutation Groups

University of Bristol, Wednesday June 1st, 2016

School of Mathematics, University Walk, lectures will take place in room SM1.

After the last talk we will go for drinks and dinner.

2:00-3:00 Gareth Tracey (Warwick) Generating minimally transitive permutation groups: a conjecture of Pyber

Suppose that G is a transitive permutation group, of degree n, but that G needs a large number of generators (in terms of n). If possible, we would like to “reduce” the number of generators, whilst keeping our group transitive. More precisely, we would like to take a subset X of G, minimal with the property that <X> is transitive. The question is: can we find a good upper bound for |X|, in terms of n? In this talk, we discuss the history of this question, including an old conjecture of Pyber, and some new results. We will also briefly describe some of the applications of the question.

3:00-3:30 Break

3:30-4:30 Peter Cameron (St Andrews) Road colouring and road closures: synchronization and idempotent generation

In recent years it has been noticed that knowledge about permutation groups can be applied to many old problems about transformation semigroups, especially considering regularity and idempotent generation. I have been working on this with Joao Araujo (Lisbon) and others. I will talk about two aspects of this.

A common type of problem asks what properties of a permutation group G force the semigroup generated by G and a non-permutation \$a\$ to have a particular property, for all or some specified elements a. Recently, we have considered the question of which permutation groups G have the property that, for any rank 2 map a, the semigroup <G,a> is idempotent generated. This condition forces G to be primitive, and is equivalent to the following property of G: for any orbit O of G on 2-sets, and any block B of imprimitivity for G acting on O, the graph with edge set O\B is connected. Such a group must be basic. It fails if G has an imprimitive subgroup of index 2, and it also fails for a class of primitive groups arising from triality. We conjecture, backed by substantial computational evidence, that it is true for all other primitive groups.

The other part of my talk will be an update on the ongoing synchronization project, which asks: which permutation groups G have the property that, for any non-permutation a, the semigroup <G,a> contains an element of rank 1.

4:30-5:30 Barbara Baumeister (Bielefeld) Covering a group by conjugates of a coset

For every doubly transitive permutation group G the conjugates
of a non-⁠trivial coset of a point stabilizer H cover the group.
This implies that every non-trivial conjugacy class of elements
of G that contains an element of H does contain a transversal
of H in G and that every other non-trivial conjugacy class
contains a transversal for the set of cosets of H in G different
from H.

We study the finite groups satisfying this property; more precisely
the class of primitive permutation groups, called
CCI groups, that in fact properly contains the class of 2-⁠transitive
permutation groups.

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# 7th meeting: Finite Groups

University of Manchester, Wednesday March 9th, 2016

All talks will be held in room G108 of the the Alan Turing building. The talks will be followed by “the usual pilgramage to the pub and curry place.”

1:30-2:20 Peter Rowley (Manchester) An algorithm for the Thompson subgroup of a p-group

2:30-3:20 Madeleine Whybrow (Imperial) Majorana Representations of Triangle-Point Groups

Majorana Theory was introduced by A.A. Ivanov in 2009 as the axiomatisation of certain properties of the 2A-axial vectors of the 196,884-dimensional Monster Algebra. Ivanov’s work was inspired by a result of S. Sakuma which reproved certain important properties of the Monster Algebra in the context of Vertex Operator Algebras. Majorana Theory takes the key hypotheses of Sakuma’s result to provide a powerful framework, independent of Vertex Operator Algebras, in which to study the Monster Algebra and other related objects.

In this talk, I will discuss the history and motivation behind Majorana Theory before presenting my own work on Majorana Representations of Triangle-Point Groups. These are 6-transposition groups which are generated by 3 involutions, two of whom commute. They play an important role in the proof of the uniqueness of the Monster Group and in the study of the Monster Graph.

3:30-4:00 Break

4:00-4:50 Gernot Stroth (Halle) Groups which are almost groups of Lie type in characteristic p

# 6th meeting: Representations of Groups

University of Birmingham, Wednesday January 20th, 2016

All talks will be held in LRA of the Watson building on the main campus of the University of Birmingham, whilst welcome and registration will be held in the Common Room on the Second Floor of the same building. Directions here.

14:00-15:00 Niamh Farrell (City) Rationality of blocks of quasi-simple finite groups

The Morita Frobenius number of an algebra is the number of Morita equivalence classes of its Frobenius twists. They were first introduced by Kessar in 2004 the context of Donovan’s conjecture. I will present results in ongoing work in which we aim to calculate the Morita Frobenius numbers of all blocks of group algebras of quasi-simple finite groups.

15:00-16:00 Louise Sutton (QMUL) Graded decomposition numbers for Specht modules

In 2008, Brundan, Kleshchev and Wang showed that Specht modules over cyclotomic Hecke algebras are gradable. Firstly, I will discuss the grading on these modules together with results on graded dimensions of certain Specht modules, in particular, those indexed by hook partitions. Using these formulae, I will then discuss an alternative proof of Chuang, Miyachi and Tan’s result on the graded decomposition numbers of these particular Specht modules in level 1. Finally, I will give an overview of my current work, in which I am using an analogous approach to obtain the (graded) decomposition numbers of a particular set of Specht modules in level 2.

16:00-16:30 Coffee

16:30-17:30 Charles Eaton (Manchester) Classifying Morita equivalence classes of blocks

Advances in our understanding of blocks of finite groups of Lie type raise the possibility of classifying blocks with certain defect groups for the prime two. I will survey some of the methods involved, as well as some recent results.

# 5th meeting: Groups and Geometry

Birkbeck, University of London, Wednesday December 9th, 2015

All talks will be held in room 402 in Birkbeck, University of London

If you wish to attend, please contact b.fairbairn [at] bbk.ac.uk

13:30-14:30 Robert Kropholler (Oxford) Bestvina-Brady Morse Theory and Subgroups of Hyperbolic Groups

I will introduce the key concepts from Bestvina-Brady Morse theory showing how this can be used to give groups with exotic finiteness properties. I will then show how this is used to construct finitely presented subgroups of hyperbolic groups which are not hyperbolic. Finally, I will discuss the limitations of the methods used in this construction.

14:30-15:30 Charles Cox (Southampton) An introduction to the R property

There has been much work on the R property: finding families of groups which satisfy it, and groups which do not. In this talk we will consider the following question. Let G be a group which acts faithfully on an infinite set X. Let FSym(X) denote the group of all finitely supported permutations of X (those which move finitely many points). Then does <G, FSym(X)> have the R property? There will be some nice arguments using cycle type along the way.

15:30-16:00 Coffee

16:00-17:00 Alina Vdovina (Newcastle) Quadratic equations in hyperbolic groups are NP-complete

Using the geometry of closed surfaces we solve equations in hyperbolic and toral relatively hyperbolic graphs. We also show that the tasks of solving quadratic equations in hyperbolic and toral relatively hyperbolic groups are NP-complete. The main results are based joint work with O. Kharlampovich, A. Mohajeri and A. Taam.

# 4th meeting: Algebraic Groups

University of Bristol, Wednesday May 20th, 2015

All talks will take place in Room SM2 in the School of Mathematics on University Walk. The School’s common room is located on the first floor.

After the last talk we will go for drinks and dinner.

If you wish to attend, please contact b.fairbairn [at] bbk.ac.uk

2:00-3:00 Adam Thomas (Cambridge) Complete reducibility in exceptional algebraic groups

The notion of complete reducibility was introduced by J.-P. Serre in 1998. It generalises the notion of a completely reducible module in classical representation theory. After giving an introduction to complete reducibility for algebraic groups, we discuss recent work with A. Litterick on classifying the subgroups of exceptional algebraic groups that are not completely reducible. The techniques used are a mix of standard representation theory, non-abelian cohomology and computational group theory.

3:00-3:30 Break

3:30-4:30 Michael Bate (York) Representation varieties, reductive pairs and a question of Kulshammer

Let Γ be a finite group. A natural and interesting way to think about the representation theory of Γ over an algebraically closed field k is to study the representation varieties Hom(Γ,GL(n,k)) for different n. These sets carry the structure of an affine variety, and the algebraic group GL(n,k) acts naturally. Understanding the orbit structure of this action allows one to use techniques from Geometric Invariant Theory to study representation-theoretic questions. For example, the closed orbits correspond to isomorphism classes of n-dimensional semisimple representations of Γ.

More generally, one can study the variety Hom(Γ, G) for any reductive group G. In this talk I’ll give a survey of some of the basic results in this area, including results of Richardson and Slodowy which allow one to relate the structure of Hom(Γ, G) to that of Hom(Γ, GL(n,k)), given an embedding of G in GL(n,k). Some of my own work has explored the limitations of Richardson’s techniques; in this vein I’ll also present an example which answers a question of Kulshammer.

4:30-5:30 Martin Liebeck (Imperial) Multiplicity-free representations of algebraic groups

A finite-dimensional representation of a group is said to be multiplicity-free if every irreducible representation appears at most once as a composition factor. I shall talk about the problem of classifying irreducible representations of simple algebraic groups for which the restriction to some proper subgroup is multiplicity-free. There are many interesting examples of such representations, and under suitable assumptions there is some hope of classifying them. There are also some nice connections with other aspects of algebraic group theory – for example, invariant theory and unipotent classes.

# 3rd meeting: Finite groups

University of Manchester, Wednesday April 15th, 2015

If you wish to attend, please contact b.fairbairn [at] bbk.ac.uk

2:00-3:00 Nadia Mazza (Lancaster) Quillen category and soft subgroups

3:00-3:30 Coffee

3:30-4:30 John Ballantyne (Manchester) Products of conjugate involutions in SLn(2a)

4:30-5:30 Derek Holt (Warwick) Generator numbers in finite permutation and matrix groups

# 2nd meeting: Representations of Groups

University of Birmingham, Wednesday January 14th, 2015

If you wish to attend, please contact b.fairbairn [at] bbk.ac.uk

2:00-3:00 David Stewart (Cambridge) Cohomology for finite and algebraic groups

The representation theory for finite groups of Lie type in defining characteristic is rather complicated, but owing to the link with algebraic groups, there are quite a few tools. I want to give an overview of some of the theory and show how some of these can be used to bound the cohomology of finite groups with coefficients in simple modules, motivated by an old conjecture of Guralnick.

3:00-4:00 Melanie de Boeck (Kent) Foulkes modules for the symmetric group

The action of the symmetric group \mathfrak{S}_{mn} on set partitions of sets of size mn into n sets of size m gives rise to a permutation module called the Foulkes module.  Structurally, very little is known about Foulkes modules, even over \mathbb{C}.  In this talk, we will introduce Foulkes modules and their twisted analogues before presenting some results which shed light on the irreducible constituents of the ordinary characters of twisted Foulkes modules.

4:00-4:30 Coffee

4:30-5:30 Gunter Malle (Kaiserslautern) Local-global conjectures in the representation theory of finite groups

More than 60 years ago Richard Brauer developed the theory of representations of finite groups over arbitrary fields. It showed a strong connection between the representation theory of a finite group and that of its p-local subgroups, for p a prime. Many more such connections have been observed in the meantime, but most of these are still conjectural.

Recently, a new reduction approach has offered the hope to solve all of these fundamental conjectures by using the classification of finite simple groups.  In our talk we will try and explain the nature of these problems and will report on recent progress which might eventually lead to a solution of these long standing fundamental questions.