# Algebra seminars, 2016-17

## In 'Past algebra seminars'

### Maximal subgroups of the Monster

#### Rob Wilson, Queen Mary, University of London

**Wednesday 5 October 2016, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available.

### Some open problems on restricted enveloping algebras

#### Matthew Towers, University of Leicester

**Thursday 13 October 2016, 16:00-17:00 Physics West Seminar Room 2 (Room 106)**

Abstract not available.

### Algebras of Jordan type

#### Sergey Shpectorov, University of Birmingham

**Wednesday 19 October 2016, 16:00-17:00 Watson Building, Lecture Room C**

Algebras of Jordan type η are a new interesting class of commutative non-associative algebras arising within the paradigm of axial algebras. After reviewing the general definitions, relation to groups, and examples, we will focus on the recent results concerning the classification of such algebras.

### The regular representations of GL_{N} over finite local principal ideal rings

#### Alexander Stasinski, Durham University

**Thursday 27 October 2016, 16:00-17:00 Physics West Seminar Room 2 (Room 106)**

Let *F* be a non-Archimedean local field with ring of integers *O* and maximal ideal *p*. T. Shintani and G. Hill independently introduced a large class of smooth representations of GL_{N}(*O*), called regular representations. Roughly speaking they correspond to elements in the Lie algebra M_{N}(*O*) which are regular mod *p* (i.e, having centraliser of dimension *N*). The study of regular representations of GL_{N}(*O*) goes back to Shintani in the 1960s, and independently and later, Hill, who both constructed the regular representations with even conductor, but left the much harder case of odd conductor open. In recent simultaneous and independent work, Krakovski, Onn and Singla gave a construction of the regular representations of GL_{N}(*O*) when the residue characteristic of *O* is not 2. In this talk I will present a complete construction of all the regular representations of GL_{N}(*O*).

The approach is analogous to, and motivated by, the construction of supercuspidal representations of GL_{N}(*F*) due to Bushnell and Kutzko. This is joint work with Shaun Stevens.

### Poisson homology of the nilpotent cone

#### Gwyn Bellamy, University of Glasgow

**Wednesday 2 November 2016, 16:00-17:00 Watson Building, Lecture Room A**

Poisson homology is an important invariant of Poisson varieties. In the case of conic symplectic singularities (such as those occurring in representation theory), this is a bigraded vector space. I will describe how to calculate this bigrading in the case where the Poisson variety is the nilpotent cone inside a semi-simple Lie algebra. This answers a question of Lusztig, and is joint work with T. Schedler.

### Modular invariant theory and Galois ring extensions

#### Peter Fleischmann, University of Kent

**Thursday 10 November 2016, 16:00-17:00 Physics West Seminar Room 2 (Room 106)**

This is joint work with C.F. Woodcock (Kent). We investigate trace surjective commutative *G*-algebras in characteristic *p* >0, defined by non-linear actions of finite *p*-groups. These arise in the analysis of dehomogenised modular invariant rings and related localisations. We describe criteria for such a dehomogenised invariant ring to be a polynomial ring or stably polynomial. Among other things it turns out that every finite *p*-group has a faithful (non-linear) representation on a polynomial ring with invariant ring being again a polynomial ring. This is in contrast to homogeneous linear actions, where, due to a result of Serre, the graded invariants can only form a polynomial ring, if *G* is generated by pseudo-reflections.

### Generalisations of Koszul algebras and the Ext algebra

#### Nicole Snashall, University of Leicester

**Thursday 17 November 2016, 16:00-17:00 Physics West Seminar Room 2 (Room 106)**

Koszul algebras occur in many areas of algebra and algebraic topology. A classical result is that the Ext algebra of a Koszul algebra is finitely generated as an algebra and is again a Koszul algebra. We discuss several generalisations of Koszul algebras and introduce a more general class of graded algebras, called (*D*,*A*)-stacked algebras. We show that the Ext algebra is finitely generated and give a regrading on the Ext algebra so that the regraded algebra is again a Koszul algebra.

We then consider Brauer graph algebras and classify those which are Koszul, as well as giving more general results on the Ext algebra of a Brauer graph algebra. Brauer graph algebras play a major role in the classification of finite-dimensional self-injective algebras of tame representation type, in addition to the role of Brauer tree algebras in studying blocks of group algebras with cyclic defect groups.

This talk is based on joint work with J Leader, and with EL Green, S Schroll and R Taillefer.

### The Generalised Nilradical of a Lie Algebra

#### David Towers, Lancaster University

**Wednesday 30 November 2016, 16:00-17:00 Watson Building, Lecture Room B**

A solvable Lie algebra *L* has the property that its nilradical *N* contains its own centraliser. This is interesting because it gives a representation of *L* as a subalgebra of the derivation algebra of its nilradical, with kernel equal to the centre of *N*. I will consider several possible generalisations of the nilradical for which this property holds in any Lie algebra. The main result states that for every Lie algebra *L*, *L*/Z(*N*), where Z(*N*) is the centre of the nilradical of *L*, is isomorphic to a subalgebra of Der(*N**) where *N** is an ideal of *L* such that *N**/*N* is the socle of a semisimple Lie algebra.

### Parking spaces and Catalan combinatorics for complex reflection groups

#### Martina Lanini, University of Edinburgh

**Wednesday 7 December 2016, 16:00-17:00 Watson Building, Lecture Room B**

Recently, Armstrong, Reiner and Rhoades associated with any (well-generated) complex reflection group two parking spaces, and conjectured their isomorphism. This has to be seen as a generalisation of the bijection between non-crossing and non-nesting partitions, both counted by the Catalan numbers. In this talk, I will review the conjecture and discuss a new approach towards its proof, based on the geometry of the discriminant of a complex reflection group. This is an ongoing joint project with Iain Gordon.

### Detecting local properties in the character table.

#### Carolina Vallejo Rodríguez, Instituto de Ciencias Matemáticas

**Wednesday 25 January 2017, 14:00-15:00 Watson Building, Lecture Room B**

Let *G* be a finite group and let *p* be a prime number. In this talk, we discuss local properties of *G* that can be read off from its character table. More precisely, we characterize globally when the principal block of the normalizer of a Sylow *p*-subgroup has one simple module for *p* odd. We also talk about the *p* = 2 case of this problem, which remains open.

This is joint work with G.Navarro and P.H.Tiep.

### Characters of odd degree of symmetric groups

#### Eugenio Gianelli, University of Cambridge

**Wednesday 25 January 2017, 15:00-16:00 Watson Building, Lecture Room B**

Let *G* be a finite group and let *P* be a Sylow *p*-subgroup of *G*. Denote by Irr_{p'}(*G*) the set consisting of all irreducible characters of *G* of degree prime to *p*. The McKay Conjecture asserts that |Irr_{p'}(*G*)|=|Irr_{p'}(N_{G}(*P*))|.

Sometimes, we do not only have the above equality, but it is also possible to determine explicit natural bijections (McKay bijections) between Irr_{p'}(*G*) and Irr_{p'}(N_{G}(*P*)).

In the first part of this talk I will describe the construction of McKay bijections for symmetric groups at the prime *p*=2. In the second part of the talk I will present recent joint work with Kleshchev, Navarro and Tiep, concerning the construction of natural bijections between Irr_{p'}(*G*) and Irr_{p'}(*H*) for various classes of finite groups *G* and corresponding subgroups *H* of odd index. This includes the case *G*=*S*_{n} and *H* any maximal subgroup of odd index in *S*_{n}, as well as the construction of McKay bijections for solvable and general linear groups.

### On a subgroup introduced by J.Grodal

#### Geoff Robinson, University of Aberdeen/Lancaster

**Wednesday 25 January 2017, 16:30-17:30 Watson Building, Lecture Room B**

(Report on ongoing joint work with J. Grodal.) We will discuss the structure of the (normal) subgroup of a finite group *G* generated by the elements whose centraliser has order divisible by the prime *p*. This leads quickly to a study of an interesting generalisation of Frobenius complements.

The abelianization of the associated quotient group plays a role in J. Grodal's work on endotrivial modules.

### 3-spherical Curtis-Tits groups

#### Rieuwert Blok, University of Birmingham

**Wednesday 1 February 2017, 16:00-17:00 Watson Building, Lecture Room A**

Curtis-Tits groups are groups defined as non-trivial completions of Curtis-Tits amalgams. Via the Curtis-Tits theorem they generalize groups of Lie type and groups of Kac-Moody type. We'll describe all Curtis-Tits groups with 3-spherical diagram and explore some geometric and algebraic properties.

### Fusion systems and sporadic J-components

#### Justin Lynd, University of Aberdeen

**Wednesday 15 February 2017, 16:00-17:00 Watson Building, Lecture Room A**

Properly interpreted, a *J*-component in a 2-fusion system *F* is a component *K* in the centralizer *C* of some involution with the property that the 2-rank of *C* is the 2-rank of *F*. I will explain how these fit into Michael Aschbacher's framework for a classification of simple 2-fusion systems of odd type, and then describe a classification of such *F* when *K* is the 2-fusion system of one of several sporadic groups, under the assumption that the "centralizer" of *K* in *F* is cyclic. Of particular interest is the case in which *K* is the 2-fusion system of the Hall-Janko group.

This is joint work with Julianne Rainbolt.

### Partial linear spaces with symmetry

#### Joanna Fawcett, University of Cambridge

**Wednesday 15 March 2017, 16:00-17:00 Watson Building, Lecture Room A**

A partial linear space consists of a non-empty set of points *P* and a collection of subsets of *P* called lines such that each pair of points lies on at most one line, and each line contains at least two points. A partial linear space is proper if it is not a linear space or a graph. In this talk, we will consider some recent progress on classifying the finite proper partial linear spaces whose automorphism groups are transitive on pairs of collinear points, as well as pairs of non-collinear points.

### The Exotic Robinson-Schensted Correspondence

#### Neil Saunders, City, University of London

**Wednesday 22 March 2017, 16:00-17:00 Watson Building, Lecture Room A**

The classical Robinson-Schensted correspondence is an algorithm that describes a bijection between elements of the symmetric group and pairs of standard Young tableaux. This algorithm is combinatorially rich and has many applications to the representation theory of the symmetric group and the general linear group. In 1988, Steinberg discovered a geometric setting for this algorithm coming from the nilpotent cone of the adjoint representation for the general linear group. In this talk, I will describe an exotic Robinson-Schensted algorithm for the Weyl group of type *C* coming from Kato's exotic Springer correspondence for the symplectic group.

### Growth of homology torsion

#### Nikolay Nikolov, University of Oxford

**Wednesday 24 May 2017, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available

### Computing with matrix groups over infinite fields: methods, algorithms and applications

#### Alla Detinko, University of St Andrews

**Wednesday 31 May 2017, 16:00-17:00 Watson Building, Lecture Room A**

In the talk we will survey a novel domain of computational group theory: computing with linear groups over infinite fields. We will provide an introduction to the area, and will discuss available methods and algorithms. Special consideration will be given to the most recent developments in computing with Zariski dense groups and its applications. This talk is aimed at a general algebraic audience.

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.