# Algebra seminars, 2022-23

## In 'Past algebra seminars'

### Category *O* for truncated current Lie algebras

#### Matthew Chaffe, University of Birmingham

**Wednesday 28 September 2022, 15:00-16:00 Lecture Theatre C, Watson Building**

The problem of computing the composition multiplicities of Verma modules for a semisimple Lie algebra was famously solved by the proof of the Kazhdan-Lusztig conjecture. In this talk, I will discuss this problem for a related class of Lie algebras, the truncated current Lie algebras. I will also discuss the BGG category *O* of modules for a semisimple Lie algebra and an analogue of this category for the truncated current Lie algebras.

### Fixed point ratios for primitive groups and applications

#### Tim Burness, University of Bristol

**Wednesday 5 October 2022, 15:00-16:00 Lecture Theatre C, Watson Building**

Let *G* be a finite permutation group. The fixed point ratio of an element *x* in *G*, denoted fpr(*x*), is the proportion of points fixed by *x*. Fixed point ratios for finite primitive groups have been studied for many decades, finding a wide range of applications. In this talk, I will discuss some recent joint work with Bob Guralnick where we determine the triples (*G*,*x*,*r*) such that *G* is primitive, *x* has prime order *r* and fpr(*x*)>1/(*r*+1). This turns out to have some interesting applications and we can use it to obtain new results on the minimal degree and minimal index of primitive groups. Another application arises in joint work with Moreto and Navarro on the commuting probability of *p*-elements in finite groups.

### Objective Partial Groups with Cyclic Automorphism Group

#### Jamie Mason, University of Birmingham

**Wednesday 12 October 2022, 15:00-16:00 Lecture Theatre C, Watson Building**

Partial groups were introduced by Chermak in 2013 and are closely related to the theory of fusion systems. Since then a general theory of partial groups has begun to crystallise. I will introduce some of this theory and then go on to talk about automorphism groups of partial groups. Then I will finish with a discussion of whether one can construct objective partial groups with cyclic automorphism group of odd order.

### Springer Fibres - Geometrical and Combinatorial Applications

#### Neil Saunders, University of Greenwich

**Wednesday 19 October 2022, 15:00-16:00 Lecture Theatre C, Watson Building**

Fibres coming from the Springer resolution on the nilpotent cone are incredibly rich algebraic varieties that have many applications in representation theory and combinatorics. Though their geometry can be very difficult to describe in general, in type *A* at least, their irreducible components can be described using standard Young tableaux, and this can help describe their geometry in small dimensions. In this talk I will report on recent and ongoing work with Lewis Topley and separately Daniele Rosso on geometrical and combinatorial applications of the classical 'type *A*' Springer fibres and the 'exotic' type *C* Springer fibres coming from Kato's exotic Springer correspondence.

### Why are physicists interested in E_{8}?

#### Rob Wilson, Queen Mary, London

**Wednesday 26 October 2022, 15:00-16:00 Lecture Theatre C, Watson Building**

The short answer is "I don't know". But they certainly seem to see structures in *E*_{8} that remind them of structures they see in the physics of elementary particles. My best guess is that these are mirages, and that behind it all is the representation theory of the binary icosahedral group (i.e., the double cover of Alt(5)), linked to *E*_{8} via the McKay correspondence.

### Rouquier blocks for Ariki-Koike algebras

#### Sinead Lyle, University of East Anglia

**Wednesday 2 November 2022, 15:00-16:00 Lecture Theatre C, Watson Building**

The Rouquier blocks, also known as the RoCK blocks, are important blocks of the symmetric groups algebras and the Hecke algebras of type A, with the partitions labelling the Specht modules that belong to these blocks having a particular abacus configuration. We generalise the definition of Rouquier blocks to the Ariki-Koike algebras, where the Specht modules are indexed by multipartitions, and explore the properties of these blocks.

### Minimum eigenspace codimension in irreducible representations of simple classical linear algebraic groups

#### Ana M. Retegan, University of Birmingham

**Wednesday 9 November 2022, 15:00-16:00 Lecture Theatre C, Watson Building**

Let *k* be an algebraically closed field of characteristic *p *≥ 0, let *G* be a simple simply connected classical linear algebraic group of rank *l* and let *T* be a maximal torus in *G* with rational character group *X*(*T*). For a non-zero *p*-restricted dominant weight λ ∈ *X*(*T*), let *V* be the associated irreducible *kG*-module. Let *V _{g}*(μ) denote the eigenspace corresponding to the eigenvalue μ ∈

*k*

^{∗}of

*g*∈

*G*on

*V*and define

*ν _{G}*(

*V*)=min{dim(

*V*) − dim(

*V*(μ)) | g ∈

_{g}*G*\ Z(

*G*), μ ∈ k

^{∗}}

to be the minimum eigenspace codimension on *V*. In this talk, we determine *ν _{G}*(

*V*) for

*G*of type

*A*,

_{l}*l*≥ 16 and dim(

*V*) ≤

*l*

^{3}/2; for G of type

*B*,

_{l}*C*,

_{l}*l*≥ 14 and dim(

*V*) ≤ 4

*l*

^{3}; and for G of type

*D*,

_{l}*l*≥ 16 and dim(

*V*) ≤ 4

*l*

^{3}. Moreover, for the groups of smaller rank and their corresponding irreducible modules with dimension satisfying the above bounds, we determine lower bounds for

*ν*(

_{G}*V*).

### Grassmannian cluster structures and line singularities

#### Eleonore Faber, University of Leeds

**Wednesday 16 November 2022, 15:00-16:00 Lecture Theatre C, Watson Building**

This talk is about a categorification of the coordinate rings of Grassmannians of infinite rank in terms of graded maximal Cohen-Macaulay modules over the commutative ring ℂ[*x*,*y*]/(*x ^{k}*). This yields an infinite rank analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. In the special case

*k*=2, Spec(ℂ[

*x*,

*y*]/(

*x*

^{2})) is a type A

_{∞}-curve singularity and the ungraded version of our category has been studied by Buchweitz, Greuel, and Schreyer in the 1980s. We show that this Frobenius category has infinite type A cluster combinatorics, in particular, that it has cluster-tilting subcategories modelled by certain triangulations of the (completed) infinity-gon. We use the Frobenius structure to extend this further to consider maximal almost rigid subcategories, and show that these subcategories and their mutations exhibit the combinatorics of the completed infinity-gon. This is joint work with Jenny August, Man-Wai Cheung, Sira Gratz, and Sibylle Schroll.

### Twist conjugacy and Zeta functions

#### Paula Macedo Lins de Araujo, University of Lincoln

**Wednesday 23 November 2022, 15:00-16:00 Lecture Theatre C, Watson Building**

A group automorphism φ: Γ → Γ induces the action *g* ⋅ *x* = *gx*φ(*g*)^{-1} on Γ. The orbits of such action are called twisted conjugacy classes (also known as Reidemeister classes). In the past few years, the sizes of such classes have been intensively investigated. One of the main goals in the area is to classify groups where all classes are infinite. Such groups are said to satisfy the R_{∞} property. For groups not having such property, one is interested in the possible sizes of the classes. In this talk, we will discuss how zeta functions of groups can be used to determine these sizes for certain nilpotent groups.

### Module tensor categories and the Landau-Ginzburg/conformal field theory correspondence

#### Ana Ros Camacho, Cardiff University

**Wednesday 7 December 2022, 15:00-16:00 online via zoom**

The Landau-Ginzburg/conformal field theory correspondence is a physics result from the late 80s and early 90s predicting some relation between categories of representations of vertex operator algebras and categories of matrix factorizations. At present we lack an explicit mathematical statement for this result, yet we have examples available. The only example of a tensor equivalence in this context was proved back in 2014 by Davydov-Runkel-RC, for representations of the *N*=2 unitary minimal model with central charge 3(1-2/*d*) (where *d*>2 is an integer) and matrix factorizations of the potential *x ^{d}*-

*y*. This equivalence was proved back in the day only for

^{d}*d*odd, and in this talk we explain how to generalize this result for any

*d*, realising these categories as module tensor categories enriched over ℤ

*-graded vector spaces. Joint work with T. Wasserman (University of Oxford).*

_{d}### Some composition multiplicities for tensor products of irreducible representations of GL(*n*)

#### Miriam Norris, University of Manchester

**Tuesday 31 January 2023, 11:00-12:00 Room R17/18, Watson building**

Understanding the composition factors of tensor products is an important question in representation theory. The classical Littlewood-Richardson coefficients describe the composition factors of both the tensor products of simple ℂ*GL _{n}*(ℂ)-modules and the restriction of simple ℂ

*GL*(ℂ)-modules to some Levi subgroup. Now let F denote an algebraically closed field of characteristic p > 0. In comparison very little is known about composition factors of tensor products of simple F

_{n}*GL*(F)-modules but it is thought that there may still be a relationship with the restriction of simple F

_{n}*GL*(F)-modules to some Levi subgroup. In this talk we will explore an explicit relationship of this kind for tensor products of simple F

_{n}*GL*(F)-modules with the wedge square of the dual natural module and see how this might be used to find composition factors.

_{n}### sl^_{2} and its integrable, admissible, weight, and logarithmic modules

#### Juan Villarreal, University of Bath

**Wednesday 8 February 2023, 15:00-16:00 Lecture Theatre C, Watson Building**

In the first part of the talk, I will make an introduction to the Lie algebra sl^_{2} I will put particular emphasis on the infinite-dimensional properties. Then, I will introduce the admissible modules and the problems of the Verlinde formula. Finally, I will talk about weight and logarithmic modules.

### (Informal): Highest weight theory and wall-crossing functors for reduced enveloping algebras

#### Matthew Westaway, University of Birmingham

**Wednesday 1 March 2023, 15:00-16:00 Lecture Theatre C, Watson Building**

In the last few years, major advances have been made in our understanding of the representation theory of reductive algebraic groups over algebraically closed fields of positive characteristic. Four key tools which are central to this progress are highest weight theory, reduction to the principal block, wall-crossing functors and tilting modules. When considering instead the representation theory of the Lie algebras of these algebraic groups, more subtleties arise. If we look at those modules whose p-character is in so-called standard Levi form we are able to recover the four tools mentioned above, but they have been less-well-studied in this setting. In this talk, we will explore the similarities and differences which arise when employing these tools for the Lie algebras rather than the algebraic groups. This research is funded by a research fellowship from the Royal Commission for the Exhibition of 1851.

### CANCELLED: An overview of Non-Reductive Geometric Invariant Theory and its applications

#### Eloise Hamilton, University of Cambridge

**Wednesday 29 March 2023, 15:00-16:00 Lecture Theatre B, Watson Building**

Geometric Invariant Theory (GIT) is a powerful theory for constructing and studying the geometry of moduli spaces in algebraic geometry. In this talk I will give an overview of a recent generalisation of GIT called Non-Reductive GIT, and explain how it can be used to construct and study the geometry of new moduli spaces. These include moduli spaces of unstable objects, hypersurfaces in weighted projective space, *k*-jets of curves in ℂ^{n} and curve singularities.

### Extending the Lax type operator for finite W-algebras

#### Jonathan Brown, SUNY Oneonta

**Wednesday 19 April 2023, 15:00-16:00 Lecture Theatre C, Watson Building**

Finite W-algebras are intimately related to the geometry of nilpotent orbits and the infinite dimensional representation theory of Lie algebras. They are defined in terms of a nilpotent orbit, and they are an invariant subalgebra of a left ideal in universal enveloping algebra of a reductive Lie algebra. Outside of type A there is no known formula for calculating generators of these algebras (apart from specific examples). Recent work by Kac, De Sole, and Valeri has resulted in a Lax type operator which produces generators of an important subalgebra of a finite W-algebra, and in talk I will explain how their results can be extended. This results in a formula for generators of important classes of classical f finite W-algebras.

### Pseudo-finite semigroups and diameter

#### Victoria Gould, University of York

**Wednesday 26 April 2023, 15:00-16:00 Lecture Theatre C, Watson Building**

A semigroup *S* is said to be (right) pseudo-finite if the universal right congruence ω* _{S}* :=

*S*×

*S*can be generated by a finite set

*U*⊆

*S*×

*S*, and there is a bound on the length of derivations for an arbitrary pair (

*s*,

*t*) ∈

*S*×

*S*as a consequence of those in

*U*. The diameter of a pseudo-finite semigroup is the smallest such bound taken over all finite generating sets. The notion of being pseudo-finite was introduced by White in the language of ancestry, motivated by a conjecture of Dales and Zelazko for Banach algebras. The property also arises from a number of other sources. Without assuming any prior knowledge, this talk investigates the somewhat unpredictable notion of pseudo-finiteness. Some well-known uncountable semigroups have diameter 1; on the other hand, a pseudo-finite group is forced to be finite. Actions, presentations, Rees matrix constructions and some good old-fashioned semigroup tools all play a part. This research sits in the wider framework of a study of finitary conditions for semigroups.

### A local-global principle for unipotent characters

#### Damiano Rossi, Mathematisches Forschungsinstitut Oberwolfach

**Wednesday 3 May 2023, 15:00-16:00 Lecture Theatre C, Watson Building**

In this talk, we present some new results on Dade's Conjecture for finite reductive groups. If * l * is a prime number dividing the order of a finite group * G*, then Dade's Conjecture describes the number of irreducible representations of * G * with a fixed * l*-defect in terms of the *l*-local structure of the group. This behaviour is in accordance with a recurring phenomenon known as the local-global principle in representation theory of finite groups. For a finite reductive group *G* with underlying algebraic group * G*, the

*l*-local structure of

*G*can be replaced by a geometric analog: the

*e*-local structure of

*. Using this new setup, we introduce an adaptation of Dade's Conjecture to finite reductive groups and prove it for the unipotent representations of groups of types*

**G***,*

**A***, and*

**B***. If time permits, I will explain how this work fits into a long-term project that aims at solving Dade's Conjecture.*

**C**### The Thompson chain of perfect groups

#### Rob Curtis, University of Birmingham

**Wednesday 17 May 2023, 15:00-16:00 Lecture Theatre B, Watson Building**

When in 1965 Zvonimir Janko discovered a new simple group J_{1} which fitted into no infinite family it caused great excitement. After all, the only known such 'sporadic' simple groups were, at the time, the famous Mathieu groups M_{24}, M_{23}, M_{22}, M_{12} and M_{11} which had been discovered by Émile Mathieu in the second half of the 19th century. In the next few years several fascinating new groups were found in very different ways: through abstract group theoretic considerations, as was the case with J_{1}; or as the group of symmetries of some geometric or combinatorial object. In 1969 John Conway, in a *pièce de résistance*, worked out the group of symmetries of the 24-dimensional lattice which had recently been constructed by John Leech. When this group, which Conway called 'dotto' or ·O, was factored by its centre of order 2 this resulted in a new sporadic simple group Co_{1} which involved many of the recently discovered exceptional groups together with two further new sporadic simple groups Co_{2} and Co_{3}; Robert Griess refers to these as *the first generation*. The group Co_{1} contains subgroups isomorphic to the alternating group A_{9}, and so contains a chain A_{9} ≥ A_{8} ≥ ··· ≥ A_{3}. John Thompson showed that, with one exception, the normalisers in Co_{1} of the subgroups in this chain are maximal in Co_{1}, and so the resulting sequence is known as the *Thompson Chain*. In this talk we consider a small unitary group in a certain way, and show how this leads to all the groups in the chain, up to and including Co_{1} itself, revealing themselves spontaneously. The approach is abstract in terms of generators and relations, and so we conclude by showing how this all appears as automorphisms of the Leech lattice.

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.