# Algebra seminars, 2015-16

## In 'Past algebra seminars'

### On a general Schur's partition identity

#### Shunsuke Tsuchioka, University of Tokyo

**Thursday 1 October 2015, 16:00-17:00 Watson Building, Lecture Room B**

We will talk on a generalization of Schur’s partition identity (this is a joint work with Masaki Watanabe).

### Modular representations of gl_{n}

#### Simon Goodwin, University of Birmingham

**Thursday 8 October 2015, 16:00-17:00 Watson Building, Lecture Room B**

Abstract not available

### Branching problems and maximal subgroups of classical groups

#### Kay Magaard, University of Birmingham

**Thursday 15 October 2015, 16:00-17:00 Watson Building, Lecture Room B**

Abstract not available

### Minimal and maximal constituents of plethysms

#### Rowena Paget, University of Kent

**Thursday 22 October 2015, 16:00-17:00 Arts Building, LR3**

Abstract not available

### Universal K-matrix for quantum symmetric pairs

#### Stefan Kolb, University of Newcastle

**Thursday 5 November 2015, 16:00-17:00 Watson Building, Lecture Room A**

Quantum groups provide a uniform setting for solutions of the quantum Yang-Baxter equation, which in turn leads to representations of the classical braid group in finitely many strands. Underlying this construction is the fact that the finite dimensional representations of a quantum group form a braided tensor category. In a program to extend this construction to braid groups of type B, the topologist Tammo tom Dieck studied braids in a cylinder with one fixed axis. In the late 90s he introduced the notion of a braided tensor category with a cylinder twist which extends the categorical framework from type A to type B. However, only very few examples were known. In this talk I will explain the above notions. I will then indicate how the theory of quantum symmetric pairs provides a large class of examples for tom Dieck's theory. The construction builds on a program of canonical basis for quantum symmetric pairs initiated by H. Bao and W. Wang and related work by M. Ehrig and C. Stroppel. The new results in this talk are joint work with Martina Balagovic.

### A proof of De Concini-Kac-Procesi conjecture and Lusztig's partition

#### Alexey Sevastyanov, University of Aberdeen

**Thursday 12 November 2015, 16:00-17:00 Watson Building, Lecture Room B**

In 1992 De Concini, Kac and Procesi observed that isomorphism classes of irreducible representations of a quantum group at odd primitive root of unity *m* are parameterized by conjugacy classes in the corresponding algebraic group *G*. They also conjectured that the dimensions of irreducible representations corresponding to a given conjugacy class *O* are divisible by *m*^{1/2 dim O}. In this talk I shall outline a proof of an improved version of this conjecture and derive some important consequences of it related to *q*-*W* algebras.

A key ingredient of the proof are transversal slices *S* to the set of conjugacy classes in *G*. Namely, for every conjugacy class *O* in *G* one can find a special transversal slice *S* such that *O* intersects *S* and dim *O* = codim *S*. The construction of the slice utilizes some new combinatorics related to invariant planes for the action of Weyl group elements in the real reflection representation. The condition dim *O* = codim *S* is checked using some new mysterious results by Lusztig on intersection of conjugacy classes in algebraic groups with Bruhat cells.

### Derangements and extremal permutation groups

#### Tim Burness, University of Bristol

**Thursday 19 November 2015, 16:00-17:00 Watson Building, Lecture Room B**

Let *G* be a transitive permutation group. If *G* is finite, then a classical theorem of Jordan implies the existence of derangements, which are fixed-point-free elements. This result has some interesting and unexpected applications, and it leads to several natural problems on the abundance and order of derangements that have been the focus of recent research. In this talk, I will discuss some of these related problems, and I will report on recent joint work with Hung Tong Viet on primitive permutation groups with extremal derangement properties.

### Broué's perfect isometry conjecture holds for the double covers of the symmetric and alternating groups

#### Michael Livesey, University of Manchester

**Thursday 10 December 2015, 16:00-17:00 Watson Building, Lecture Room B**

O. Brunat and J. Gramain recently proved that any two blocks of double covers of symmetric or alternating groups are Broué perfectly isometric provided they have the same weight and sign. They also proved 'crossover' isometries when they have opposite signs. Using both the results and methods of Brunat and Gramain we prove that when the weight of a block of a double cover of a symmetric or alternating group is less than *p* then the block is Broué perfectly isometric to its Brauer correspondent. This means that Broué's perfect isometry conjecture holds for both these classes of groups.

### Prime ideals in completed group algebras

#### Simon Wadsley, University of Cambridge

**Thursday 14 January 2016, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available

### Rationality of blocks of quasi-simple finite groups

#### Niamh Farrell, City, University of London

**Wednesday 20 January 2016, 14:00-15:00 Watson Building, Lecture Room A**

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.

### Graded decomposition numbers for Specht modules

#### Louise Sutton, Queen Mary, University of London

**Wednesday 20 January 2016, 15:00-16:00 Watson Building, Lecture Room A**

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.

### Classifying Morita equivalence classes of blocks

#### Charles Eaton, University of Manchester

**Wednesday 20 January 2016, 16:30-17:30 Watson Building, Lecture Room A**

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 2. I will survey some of the methods involved, as well as some recent results.

### Invertible modules for discrete groups

#### Nadia Mazza, Lancaster University

**Thursday 21 January 2016, 16:00-17:00 Watson Building, Lecture Room B**

Given a commutative noetherian ring *k* of finite global dimension and a discrete group *G*, subject to certain conditions, we introduce the concept of an invertible *kG*-module in an attempt to generalise the endotrivial modules for finite groups. In the first part of the talk we will present the categorical framework which we (want to) use, before giving the definition and a few results. (Joint with Peter Symonds.)

### On the automorphisms of quantum Weyl algebras

#### Andrew Kitchin, University of Kent

**Wednesday 27 January 2016, 16:00-17:00 Watson Building, Lecture Room A**

Motivated by noncommutative analogues of the Jacobian conjecture and the Tame Generators problem, we discuss quantum versions of these problems for a family of analogues to the Weyl algebras. This work is joint with Stephane Launois and will cover the results in our recent preprint.

### Automorphism groups of edge-transitive maps

#### Gareth Jones, University of Southampton

**Wednesday 3 February 2016, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available

### Torsion growth for Bianchi groups

#### Haluk Sengun, University of Sheffield

**Thursday 11 February 2016, 16:00-17:00 Watson Building, Lecture Room A**

Bianchi groups, e.g. SL_{2}(ℤ[i]), are natural generalizations of the classical modular group SL_{2}(ℤ). Motivated by recent developments in the Langlands programme, we consider the size of torsion in the abelianization of finite index subgroups of Bianchi groups. After giving some background and motivation, we shall discuss joint work with Akshay Venkatesh (Stanford) and Nicolas Bergeron (Paris 6) which shows that under suitable hypotheses the size of torsion grows exponentially with respect to the index. The main tool is the celebrated Cheeger-Mueller theorem from global analysis which essentially asserts the equality of topological and analytical torsions.

### The structure of twisted Foulkes modules

#### Melanie de Boeck, University of Birmingham

**Thursday 18 February 2016, 16:00-17:00 Watson Building, Lecture Room A**

In this talk, we discuss the modular structure of twisted Foulkes modules. In particular, we describe some indecomposable summands in blocks of small weight.

### Blobs and towers of recollement in affine Hecke algebras

#### Paul Martin, University of Leeds

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

A blob is a decoration on a Temperley-Lieb diagram. A ToR is a device for organising functorial machinery in towers of algebras. A Hecke algebra is a certain tower of algebras with interesting finite-dimensional quotients. We use blobs and ToRs to determine decomposition matrices for certain such quotients—an approach inspired by statistical mechanics.

### Uncountably many groups of type FP

#### Ian Leary, University of Southampton

**Thursday 24 March 2016, 16:00-17:00 Watson Building, Lecture Room A**

FP_{2} is the homological algebra version of 'finitely presented'. For around 30 years it was open whether FP_{2} is equivalent to 'finitely presented'. Bestvina and Brady gave examples of non-finitely presented FP_{2} groups in the late 1990s. Their examples are subgroups of finitely presented groups; in particular there are only countably many of them. I will discuss their construction and my new version which constructs a lot more groups.

### Coprime action and Brauer characters

#### Carolina Vallejo Rodríguez, University of Valencia

**Wednesday 4 May 2016, 16:00-17:00 Watson Building, Lecture Room A**

Let *A* and *G* be finite groups such that the orders of *A* and *G* are coprime numbers. If *A* acts on *G*, then it is well-known that there is a one-to-one correspondence between the irreducible *A*-invariant characters of *G* and the irreducible characters of the subgroup C_{G}(*A*) of fixed points under the action. In the modular case, it is an open conjecture that the number of irreducible *A*-invariant Brauer characters of *G* equals the number of irreducible Brauer characters of C_{G}(*A*). We present a reduction of this conjecture to a question on finite simple groups.

### Dessins d'enfants and finite groups

#### George Shabat, Moscow University

**Tuesday 17 May 2016, 16:00-17:00 Watson Building, Lecture Room B**

The theory of dessins d'enfants is based on the equivalence of a certain combinatorial-topological category (of graphs on compact oriented surfaces) and an arithmetico-geometric one (of algebraic curves over number fields together with special rational functions on them). The objects of both categories admit a group-theoretic description as well.

The talk will start with a brief introduction to the theory. After that the group-theoretic aspects of it will be discussed, basically related to a certain graphic images of 2-generated groups and to the Inverse Galois Problem. Some examples will be presented and the lines of research suggested.

### Maximal subgroups and irreducible restrictions

#### Amanda Schaeffer Fry, MSU Denver

**Thursday 2 June 2016, 16:00-17:00 Watson Building, Lecture Room B**

Abstract not available

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.