# Algebra seminars, 2013-14

## In 'Past algebra seminars'

### Maximal Subgroups of Classical Groups

#### Kay Magaard, University of Birmingham

**Thursday 3 October 2013, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available

### Reading endotrivial modules on the Brauer tree

#### Caroline Lassueur, TU Kaiserslautern

**Wednesday 16 October 2013, 16:00-17:00 Watson Building, Lecture Room C**

If *G* is a finite group and *k* a field of prime characteristic *p*, a *kG*-module is termed 'endo-trivial' if its *k*-endomorphism ring decomposes as a direct sum of a copy of the trivial module and projective summands. In this talk we explain how to determine whether or not a simple *kG*-module is endo-trivial by looking at its position on the Brauer tree. We will explain why this is an important special case of the classification of simple endo-trivial modules (for quasi-simple groups). This is joint work with G. Malle and E. Schulte.

### Geometric invariant theory and spherical buildings

#### Ben Martin, The University of Auckland

**Thursday 24 October 2013, 16:00-17:00 Watson Building, Lecture Room A**

A spherical building is a special kind of simplicial complex with a large symmetry group. Spherical buildings were introduced by Jacques Tits; his original motivation was to study algebraic groups, but building theory is now a large and active branch of mathematics in its own right.

Given a reductive algebraic group *G* - such as a special linear group or a symplectic group - one can construct a spherical building *X*(*G*) on which *G* acts by automorphisms. I will explain some ideas from geometric invariant theory which give rise to interesting subsets of *X*(*G*), and discuss applications to Tits's Centre Conjecture.

### The geometries of the Freudenthal-Tits magic square

#### Jeroen Schillewaert, Imperial College, London

**Thursday 7 November 2013, 16:00-17:00 Watson Building, Lecture Room A**

I will discuss an ongoing project (joint with H. Van Maldeghem) to give a uniform axiomatic description of the embeddings in projective space of the varieties corresponding with the geometries of exceptional Lie type over arbitrary fields.

In particular, I will focus on the second row of the Freudenthal-Tits Magic Square.

I will mainly focus on the split case, and provide a uniform (incidence) geometric characterization of the Severi varieties over arbitrary fields. This can be regarded as a counterpart over arbitrary fields of the classification of smooth complex algebraic Severi varieties. The proofs just use projective geometry.

In the remaining time, I will discuss a geometric characterization of projective planes over quadratic alternative division algebras.

### Krull dimension of affinoid enveloping algebras

#### Konstantin Ardakov, University of Oxford

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

Abstract not available

### New approaches to black box groups

#### Alexandre Borovik, University of Manchester

**Thursday 21 November 2013, 16:00-17:00 Watson Building, Lecture Room A**

We propose a new and systematic approach to the so-called black box group methods in computational group theory. Instead of a single black box we are looking at categories of black boxes and their morphisms. This makes accessible new classes of black box problems. For example, we can deal with black box groups of Lie type over very large fields of odd characteristic.

### Curtis–Tits Groups of simply-laced type

#### Rieuwert Blok, Bowling Green State University

**Thursday 5 December 2013, 16:00-17:00 Watson Building, Lecture Room A**

The classification of Curtis-Tits amalgams with triangle-free, simply-laced diagram over a field of size at least 4 divides them into two major classes: Orientable amalgams are those arising from applying the Curtis-Tits theorem to groups of Kac-Moody type, and indeed, their universal completions are central extensions of those groups of Kac-Moody type. In the case of a Ã_{n−1}, all completions can be described as central extensions of concrete (matrix) groups. For non-orientable amalgams these groups are symmetry groups of certain unitary forms over a ring of skew Laurent polynomials. It was conjectured that in fact all amalgams arising from the classification have nontrivial completions. In this talk I will discuss this conjecture and describe some properties of the resulting groups.

### Combinatorial fun with Kac-Moody-like groups

#### Corneliu Hoffman, University of Birmingham

**Thursday 12 December 2013, 16:00-17:00 Watson Building, Lecture Room C**

Abstract not available

### Braid groups and quiver mutation

#### Joseph Grant, University of Leeds

**Thursday 16 January 2014, 16:00-17:00 Watson Building, Lecture Room A**

I will report on some joint work with Robert Marsh which is currently in progress. First I will talk about how quiver mutation can give us new presentations of braid groups. Then I will talk about some of the underlying homological algebra that explains this connection.

### Representations of the alternating group that remain irreducible in characteristic *p*

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

**Thursday 23 January 2014, 16:00-17:00 Watson Building, Lecture Room C**

Given any finite group *G* and any prime *p*, it is interesting to ask which ordinary irreducible representations of *G* remain irreducible in characteristic *p*. This question was solved for the symmetric groups several years ago, by the speaker and others. Here we address the case of the alternating group when *p* is odd. We'll translate the question to one about representations of the symmetric group, and then explain how to answer it.

### The partition algebra and the Kronecker coefficients

#### Chris Bowman, City, University of London

**Thursday 30 January 2014, 16:00-17:00 Watson Building, Lecture Room A**

The Kronecker problem asks for an understanding of the tensor products of simple modules for the symmetric group. We shall introduce the partition algebra as a natural setting in which to study this problem and discuss new results on its representation theory in characteristic *p* ≥ 0.

### When Artin groups are sufficiently large...

#### Sarah Rees, University of Newcastle

**Thursday 20 February 2014, 16:00-17:00 Watson Building, Lecture Room C**

An Artin group is a group with a presentation with generators *x*_{1},*x*_{2},...,*x _{n}*, and relations that

*x*... and

_{i}x_{j}x_{i}*x*... are equal, where there are

_{j}x_{i}x_{j}*m*terms in the first expression and

_{i,j}*m*in the second, for

_{j,i}*m*∈ ℕ ∪ {∞},

_{i,j}*m*≥ 2, which can be described naturally by a Coxeter matrix or graph.

_{i,j}This family of groups contains a wide range of groups, including braid groups, free groups, free abelian groups and much else, and its members exhibit a wide range of behaviour. Many problems remain open for the family as a whole, including the word problem, but are solved for particular subfamilies. The groups of finite type (mapping onto finite Coxeter groups), right-angled type (with each *m _{i,j}* ∈ {2,∞}), large and extra-large type (with each

*m*≥ 3 or 4), FC type (every complete subgraph of the Coxeter graph corresponds to a finite type subgroup) have been particularly studied.

_{i,j}After introducing Artin groups and surveying what is known, I will describe recent work with Derek Holt and (sometimes) Laura Ciobanu, dealing with a big collection of Artin groups, containing all the large groups, which we call 'sufficiently large'. For those Artin groups we have elementary descriptions of the sets of geodesic and shortlex geodesic words, and can reduce any input word to either form. So we can solve the word problem, and prove the groups shortlex automatic. And, following Appel and Schupp we can solve the conjugacy problem in extra-large groups in cubic time.

For many of the large Artin groups, including all extra-large groups, we can deduce the rapid decay property and verify the Baum-Connes conjecture. And although our methods are quite different from those of Godelle and Dehornoy for spherical-type groups, we can pool our resources and derive a weak form of hyperbolicity for many, many Artin groups.

I'll explain some background for the problems we attach, and outline their solution.

### Koszulity for Brauer graph algebras

#### Sibylle Schroll, University of Leicester

**Thursday 27 February 2014, 16:00-17:00 Watson Building, Lecture Room A**

We define coverings of Brauer graph algebras and show how they can simplify the calculation of the Yoneda algebras of Brauer graph algebras leading to a Koszul classification. This talk is based on joint work with E. Green, N. Snashall and R. Taillefer.

### Brauer relations in finite groups

#### Alex Bartel, University of Warwick

**Thursday 6 March 2014, 16:00-17:00 Watson Building, Lecture Room C**

If *G* is a finite group, a Brauer relation is a pair of finite *G*-sets *X*, *Y* such that the complex permutation representations ℂ[*X*] and ℂ[*Y*] are isomorphic. Brauer noticed in the early 1950s that such pairs give rise to number fields that share many properties, but are, in general, not isomorphic. Later Brauer relations were used by Sunada to produce non-isometric isospectral manifolds (drums, whose shape you cannot hear), and most recently by the Dokchitser brothers in the theory of elliptic curves. Often, in order to apply Brauer relations in any of the above contexts, one needs to explicitly produce a suitable Brauer relation for a suitable group *G*. In joint work with Tim Dokchitser, we have completely classified all Brauer relations for all finite groups. I will explain what this classification looks like, giving lots of concrete examples along the way.

### Whittaker coinvariants in category O for gl(*m*|*n*)

#### Simon Goodwin, University of Birmingham

**Thursday 13 March 2014, 16:00-17:00 Watson Building, Lecture Room A**

A W-algebra *U*(** g**,

*e*) is a certain associative algebras associated a Lie (super)algebra

**and a nilpotent element**

*g**e*in

**. The representation theory of**

*g**U*(

**,**

*g**e*) has close connections with the representation theory of the universal enveloping algebra

*U*(

**) of**

*g***. We'll give some background on the theory of W-algebras before considering the case of the principal W-algebra for the general linear Lie superalgebra gl(**

*g**m*|

*n*). Then we'll explain how the Whittaker coinvariants functor relates category

*O*for gl(

*m*|

*n*) to the category of finite dimensional modules for this W-algebra. This is joint work with J. Brundan and J. Brown.

### Height extensions of the Temperley–Lieb category

#### Shona Yu, University of Leeds

**Thursday 1 May 2014, 17:00-18:00 Watson Building, Lecture Room A**

The Partition, Brauer and Temperley–Lieb categories and various associated diagram algebras are well-studied objects originating from different parts of mathematics and physics. Inspired by geometrical, representation-theoretic and physical reasons, we will look into an 'interpolation' of algebras and categories which lie between the Temperley–Lieb and Brauer. No prior knowledge of these objects required; but a non-disliking of partitions and pictures would be helpful. This is based on joint work with Zoltan Kadar and Paul Martin.

### On Fixed-Point-Free Automorphisms

#### Glen Collins, University of Birmingham

**Thursday 8 May 2014, 16:00-17:00 Watson Building, Lecture Room A**

If a finite group *A* acts on a finite group *G* in such a way that C_{G}(*A*)=1, then one can often say something about the structure of *G* given properties of *A*. This idea has long been a source of strong and useful results within finite group theory. I will begin with a brief historical overview of results of this kind before talking in more detail about Khukhro's recent work in this area, and how a question which I have considered is related to his work. Namely, if we have a group *RF* where *R* is cyclic of prime order and *F* is nilpotent with *F*=[*F*,*R*], which acts on a group *G* such that C_{G}(*F*)=1, then is F(C_{G}(*R*)) ≤ F(*G*)? I will finish by outlining directions for further work and potential obstacles that one might encounter in pursuing these.

### Chebyshev polynomials and tensor diagrams

#### Lisa Lamberti, University of Oxford

**Thursday 15 May 2014, 16:00-17:00 Watson Building, Lecture Room A**

The aim of this talk is to show that a class of elements in the ring of SL(*V*)-invariants of configurations of vectors and linear forms of a three dimensional vector space *V* are determined by Chebyshev polynomials.

### Globally reductive groups

#### Jonathan Elmer, University of Aberdeen

**Thursday 22 May 2014, 16:00-17:00 Watson Building, Lecture Room A**

A linear algebraic group *G* is called 'linearly reductive' if, given any *G*-module *V*, and any fixed point *v* in *V ^{G}*, we can find an invariant linear function

*f*in

*V*such that

^{G}*f*(

*v*) is not zero. It is called 'geometrically reductive' if this condition holds when we allow

*f*to be instead a polynomial invariant of arbitrary finite degree. Note that the maximum degree

*d*required may depend on the choice of

*V*in general. One might reasonably say a group is 'globally reductive' if there is a finite number

*d*which works for any

*G*-module

*V*. This condition lies between the two notions of reductivity. It is straightforward to show that any group whose identity component is linearly reductive is globally reductive; we will report on progress towards proving that these are the only globally reductive groups. Joint work with Martin Kohls (TU Munich).

### Decomposing Generalised Gelfand–Graev Representations (GGGRs)

#### Jay Taylor, TU Kaiserslautern

**Thursday 29 May 2014, 16:00-17:00 Watson Building, Lecture Room A**

Assume *G* is a finite reductive group defined over a field of good characteristic. In 1986 Kawanaka introduced a family of (ordinary) representations of *G* called GGGRs. These are indexed by the unipotent conjugacy classes of G and elucidate an interesting, and somewhat surprising, relationship between the unipotent conjugacy classes and the irreducible characters of *G*. In this talk we discuss a programme concerned with determining the decomposition of the GGGRs into irreducible characters. Such information should help us to further levy the interesting relationship between conjugacy classes and characters.

### Decomposition numbers from tilings

#### Joseph Chuang, City, University of London

**Monday 14 July 2014, 14:00-15:00 Watson Building, Lecture Room A**

I'll describe how incidence matrices of parallelepiped tilings turn up in decomposition matrices of *q*-Schur algebras at complex roots of unity. This is joint work with Hyohe Miyachi and Kai Meng Tan.

### Modular representations of symmetric groups and KLR algebras

#### Alexander Kleshchev, University of Oregon

**Monday 14 July 2014, 15:30-16:30 Watson Building, Lecture Room A**

We describe the standard module theory for KLR algebras of affine type and describe connections to modular representation theory of symmetric groups.

### Blocks of KLR algebras of small defect

#### Sinéad Lyle, University of East Anglia

**Monday 14 July 2014, 16:30-17:30 Watson Building, Lecture Room A**

We look at some (graded) decomposition numbers and adjustment matrices for blocks of KLR algebras of small defect, with particular interest in cases where the adjustment matrix is the identity matrix. We consider the combinatorics of blocks in which multipartition indexing each Specht module is a multicore and present some results (joint work with Oliver Ruff) on such blocks of weight 2 and 3. We speculate on some other blocks where we believe the adjustment matrix may be the identity matrix.

### Garnir relations and Dyck tilings

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

**Tuesday 15 July 2014, 09:30-10:30 Watson Building, Lecture Room A**

Kleshchev, Mathas and Ram have given a homogeneous presentation for any Specht module for a KLR algebra in type A. The relations include 'homogeneous Garnir relations', which appear in a slightly nebulous (and inefficient) form. I'll describe what happens when the Garnir relations are re-written in terms of standard generators; as we'll see, some interesting combinatorics related to Dyck tilings comes into play.

### Jantzen filtrations for cyclotomic Hecke algebras

#### Andrew Mathas, University of Sydney

**Tuesday 15 July 2014, 11:00-12:00 Watson Building, Lecture Room A**

The Specht modules of the cyclotomic Hecke algebras have a nice filtration known as the Jantzen filtration. The Jantzen sum formula describes the sum of the terms of this filtration as a ℤ-linear combination of Specht modules in the Grothendieck group. This formula is very useful, both theoretically and computationally, however the formula is 'wrong' in the sense that any of the terms in the formula cancel. I will talk about a cancellation free sum formula and discuss what happens in the graded case. This is joint work with Jun Hu.

### Graded column removal for homomorphisms between Specht modules

#### Liron Speyer, Queen Mary, University of London

**Tuesday 15 July 2014, 12:00-12:30 Watson Building, Lecture Room A**

We will discuss a generalisation of Lyle and Mathas's results on row and column removal for homomorphisms between Specht modules for the Iwahori–Hecke algebra. Our result is in the context of KLR algebras and applies to arbitrary levels, as well as incorporating the graded structure. This is joint work with Matthew Fayers.

### Transitive 2-representation of finitary 2-categories

#### Vanessa Miemietz, University of East Anglia

**Tuesday 15 July 2014, 14:00-15:00 Watson Building, Lecture Room A**

I will give an introduction to the 2-representation theory of finitary 2-categories, explain and (in some cases classify simple) transitive 2-representations, and explain how this proves uniqueness of categorification of *U* for simple finite dimensional Lie algebras (via quiver Hecke algebras).

### On the character degree sets of *S*_{n}, *A*_{n} and related groups

_{n}

_{n}

#### Christine Bessenrodt, Leibniz Universität Hannover

**Tuesday 15 July 2014, 15:30-16:30 Watson Building, Lecture Room A**

It is a long standing question due to Brauer to find out what the complex group algebra ℂ*G* tells about the finite group *G*, i.e., what can be deduced on the structure of *G* from the multiset of degrees of its irreducible characters. Concluding the work by Nguyen and Tong-Viet on quasisimple groups, in joint work with them and Olsson, the double covers of the symmetric and alternating groups were considered. For nonabelian simple groups *G*, Huppert conjectured that just the character degree set (without multiplicities) determines G up to an abelian factor. For alternating groups *A _{n}*, this was shown by Huppert up to

*n*=11, and by Nguyen, Tong-Viet and Wakefield for

*n*=12,13. I will report on progress on this obtained in recent joint work with Tong-Viet and Jiping Zhang.

### Foulkes characters: deflations, twists and algorithms

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

**Tuesday 15 July 2014, 16:30-17:30 Watson Building, Lecture Room A**

Let φ^{(mn)} be the permutation character of the symmetric group *S _{mn}* acting on the set partitions of {1,...,

*mn*} into

*n*sets each of size

*m*. A fundamental problem in algebraic combinatorics asks for the decomposition of φ

^{(mn)}into irreducible characters. Closely related is Foulkes' Conjecture, which states that φ

^{(mn)}contains φ

^{(nm)}whenever

*m*≤

*n*. My talk will begin with a deflation map that sends characters of

*S*to characters of

_{mn}*S*, via characters of the wreath product of

_{n}*S*by

_{m}*S*. This deflation map leads to a new algorithm for decomposing Foulkes characters that has been used to verify Foulkes' Conjecture when

_{n}*m*+

*n*≤19. I will then state a combinatorial rule giving the values of the deflations of irreducible characters of

*S*and show that this rule generalises the Murnaghan–Nakayama rule and Young's rule. This is joint work with Anton Evseev and Rowena Paget.

_{mn}For each partition ν of *n* there is a twisted analogue of the Foulkes character φ^{(mn)} corresponding to the plethysm of the Schur functors Δ^{ν} and Sym^{m}. These characters may also be decomposed using the deflation maps. I will end with a recent result, obtained in joint work with Rowena Paget, that uses the colexicographic order on sets to determine all minimal irreducible constituents of twisted Foulkes characters.

### Categorical actions and Cherednik algebras

#### Eric Vasserot, Université de Paris 7

**Wednesday 16 July 2014, 09:30-10:30 Watson Building, Lecture Room A**

We'll review a few facts on categorical actions and explain how it proves Kazhdan-Lusztig type character formulas and Koszul duality for modules of rational Cherednik algebras.

### A central extension of the Hecke algebra at *q* = −1

#### Ivan Marin, Université de Picardie Jules Verne

**Wednesday 16 July 2014, 11:00-12:00 Watson Building, Lecture Room A**

In a joint work with E. Wagner, we introduced a central extension of the Iwahori-Hecke algebra at *q* = −1, attached to any Coxeter system (*W*,*S*). This extension also admits a uniform presentation as a quotient of the attached Artin–Tits group. In type *A*, we define a Markov trace on this algebra that provides a 'new' link invariant.

### Vertices of simple modules of symmetric groups

#### Eugenio Giannelli, (Royal Holloway, University of London

**Wednesday 16 July 2014, 13:30-14:00 Watson Building, Lecture Room A**

Introduced by J.A. Green in 1959, vertices of indecomposable modules over modular group algebras have proved to be important invariants linking the global and local representation theory of finite groups over fields of positive characteristic. In this talk I will describe the problem of determining vertices for simple modules of the symmetric groups. In particular I will present the recently completed classification of the vertices of simple modules labelled by hook partitions. This is joint work with Susanne Danz.

### Lusztig symbols and the representation theory in type Bn

#### Nicolas Jacon, Université de Reims Champagne-Ardenne

**Wednesday 16 July 2014, 14:00-15:00 Watson Building, Lecture Room A**

It is well-known that the representation theory of the symmetric group is related to the combinatorics of partitions in both the ordinary and the modular case. The aim of this talk is to show that, in the context of the Weyl group of type *B* and its Hecke algebra, the notion of Lusztig symbols play a similar role

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.