All talks will take place Lecture Theatre A on the ground floor of the Watson Building, which is R15 on the Campus Map. Tea and coffee breaks will be held in the Maths Learning Centre on the first floor of the Watson Building.

Schedule

Monday 11 April

11:30 – 12:30: Career development seminar given by Lewis Topley

12:30 – 13:25: Registration

13:30 – 14:10: Dmitriy Rumynin (Warwick): Enriched Categories in Representation Theory

14:15 – 14:55: Beth Romano (KCL): Invariant theory of graded Lie algebras in arbitrary characteristic

15:00 – 15:30: Coffee Break

15:30 – 16:10: Rudolf Tange (Leeds): Diagram combinatorics for representations of the general linear and symplectic group

16:15 – 16:55: Paul Levy (Lancaster): A new family of symplectic singularities

17:00 – Drinks Reception starts

18:00 – Very short talks from Early Career Researchers

18:00 – 18:10: David Cushing (Newcastle): New simple Lie algebras, SATs the way to do it

18:10 – 18:20: David Brown (Kent): The Ramified Partition Algebra

18:20 – 18:30: Rachel Pengelly (Birmingham): sl_2-triples over fields of positive characteristic

18:30 – 18:40: Aura-Cristiana Radu (Newcastle): On the cohomology of the Ree groups and kernels of exceptional isogenies

18:40 – 18:50: Stefano Scalese (Manchester): The first Kac-Weisfeiler conjecture for Lie algebras of algebraic groups

Tuesday 12 April

09:00 – 09:40: Alison Parker (Leeds): Tilting modules for the blob algebra.

09:45 – 10:25: Jay Taylor (Manchester): Rationality Questions for Representations of Finite Reductive Groups

10:30 – 11:00: Coffee Break

11:00 – 11:40: Amit Hazi (City, London): A path presentation of the principal block of the Temperley-Lieb algebra

11:45 – 12:25: Emily Norton (Kent): Finite-dimensional rational Cherednik algebra modules and their applications

12:30 – 14:30: Lunch

14:30 – 15:10: Vanessa Miemietz (East Anglia): Past and future of 2-representation theory

15:15 – 15:55: Haralampos Geranios (York): Injective, projective and tilting modules for general linear groups: Three in one.

16:00 – 16:30: Coffee Break

16:30 – 17:10: Sue Sierra (Edinburgh): Ideals in enveloping algebras of affine Lie algebras

18:30 – Conference Dinner

Wednesday 13 April

10:15 – 10:55: Michael Bate (York): Pseudo-reductive groups of type $BC_n$

11:00 – 11:30: Coffee Break

11:30 – 12:10: Ben Martin (Aberdeen): Uniform boundedness for algebraic groups

12:15 – 12:55: David Stewart (Newcastle): A bagatelle on the geometric unipotent radicals of pseudo-reductive groups

13:00 – End/Lunch

Abstracts

Dmitriy Rumynin (Warwick):

Title: Enriched Categories in Representation Theory

Abstract: An enriched category is not a category. Given two objects X and Y, it has a hom-object hom(X,Y), which is an object of another category rather that a set. Such categories are ubiquitous in Representation Theory. We start with a motivational example, then review the formalism, then give other examples, where the formalism of enriched categories help us to solve representation theoretic problems.

Beth Romano (KCL):

Title: Invariant theory of graded Lie algebras in arbitrary characteristic

Abstract: The invariant theory for representations coming from graded semisimple Lie algebras has been well studied except over fields of very small characteristic. I will talk about new results related to the property of stability in these representations: in these examples, we can describe stable vectors in a uniform way that is independent of the field of definition. These results are motivated by a construction of representations of p-adic groups by Reeder--Yu, and if time permits I will talk about this motivation. I will not assume familiarity with graded Lie algebras, and will review the necessary definitions from invariant theory.

Rudolf Tange (Leeds):

Title: Diagram combinatorics for representations of the general linear and symplectic group

Abstract: Let G be the general linear or symplectic group over an algebraically closed field of characteristic p>0. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for G in terms of cap and cap-curl diagrams under the assumption that p is bigger than the greatest hook length in the partitions involved. As a corollary we obtain the decomposition numbers for the Brauer algebra under the same assumptions. Our work combines ideas from work of Cox and De Visscher and work of Shalile with techniques from the representation theory of reductive groups. This is partly joint work with my former PhD student Henry Li.

Paul Levy (Lancaster):

Title: A new family of symplectic singularities

Abstract: Symplectic singularities were defined by Beauville more than 20 years ago. The main classes of known examples are quotient singularities \C^{2n}/\Gamma (where \Gamma is a finite subgroup of the symplectic group) and (normalisations of) nilpotent orbit closures. Until very recently, it was suspected that these exhausted all possible isolated symplectic singularities.

In recent joint work with Bellamy, Bonnafé, Fu, Juteau and Sommers, a new family of 4-dimensional isolated symplectic singularities was constructed. Each singularity \chi_n in the family can be obtained in three different ways: as the blow-up of the quotient of \C^4 by a dihedral group; as a deformation of the same quotient, coming from the corresponding Calogero-Moser space; as a certain slice in the (affinization of the) universal cover of the regular nilpotent orbit in \sl_n.

In this talk I will try to explain the three ways of constructing \chi_n, and (time permitting) will outline some related future research questions.

Alison Parker (Leeds):

Title: Tilting modules for the blob algebra.

Abstract: I’ll talk about joint work with Amit Hazi and Paul Martin. We determine many tilting module characters for the Type B Temperley-Lieb algebra (aka the blob algebra).

Jay Taylor (Manchester):

Title: Rationality Questions for Representations of Finite Reductive Groups

Abstract: Each (complex) irreducible character of a finite group determines a subfield of the complex numbers called its character field. This is the smallest subfield containing all character values. One can then ask whether there exists a representation of the group affording the character all of whose matrix entries are contained in the character field. Such a representation need not exist in general and the failure of such a representation to exist is measured by the Schur index.

In this talk we will discuss character fields and Schur indices in the case where the group is a finite reductive group. We will present some old and new results about these invariants and highlight a few difficulties on the road ahead.

Amit Hazi (City, London):

Title: A path presentation of the principal block of the Temperley-Lieb algebra

Abstract: The Temperley-Lieb algebra is a particularly well-understood quotient of the Hecke algebra of the symmetric group. In this talk I will construct an explicit characteristic-free monoidal presentation of the principal block of the Temperley-Lieb algebra at a root of unity. This presentation is closely related to the diagrammatic category of Soergel bimodules, and is a special case of a much more general isomorphism. Based on joint work with Chris Bowman and Anton Cox.

Emily Norton (Kent):

Title: Finite-dimensional rational Cherednik algebra modules and their applications

Abstract: Rational Cherednik algebras associated to complex reflection groups W are the most studied type of symplectic reflection algebras, defined by Etingof and Ginzburg. They have a category O of representations which contains all finite-dimensional representations. The finite-dimensional representations show up in knot theory, cohomology of affine Springer fibers, and diagonal coinvariant rings. In this talk I will discuss, for W a Coxeter group of exceptional type, potential applications of the finite-dimensional representations to understanding the ring of diagonal coinvariants of type W (work in progress with Stephen Griffeth).

Vanessa Miemietz (East Anglia):

Title: Past and future of 2-representation theory

Abstract: I will describe the background and some of the motivating examples for the development of 2-representation theory and explain what we can and what we cannot do so far. In particular, I will explain some of the features of the classification of simple 2-representations for Soergel bimodules associated to finite Coxeter groups (categorifying finite-dimensional Hecke algebras) and what the challenges are in extending these to infinite Coxeter groups (categorifying affine Hecke algebras).

Haralampos Geranios (York):

Title: Injective, projective and tilting modules for general linear groups: Three in one.

Abstract: We will first analyse some of the main objects in the category of the polynomial representations of the general linear groups, over fields of positive characteristic. Then we will focus on a nice conjecture that aims to describe how these objects interact with each other.

Sue Sierra (Edinburgh):

Title: Ideals in enveloping algebras of affine Lie algebras

Abstract: Although the representation theory of affine Lie algebras is well-understood, the (two-sided) ideal structure of their enveloping algebras remains mysterious: for example, primitive ideals have not, in general, been computed beyond Chari's well-known 1985 result that the annihilator ideal of a Verma module is generated by a maximal ideal of the centre.

Let L be an affine Lie algebra. We prove a just-infinite growth result for the universal enveloping algebra U(L). More precisely, if m is any maximal ideal of the centre of U(L), then any proper quotient of U(L)/(m) has polynomial growth. As an application, we show that the annihilator of any nontrivial integrable irreducible highest weight representation of L is centrally generated.

Our growth result and other recent developments provide strong evidence that the ideal structure of enveloping algebras of affine Lie algebras is rather sparse, in spite of the non-noetherianity of these algebras and the presence of large abelian subalgebras.

Michael Bate (York):

Title: Pseudo-reductive groups of type $BC_n$

Abstract: Pseudo-reductive groups have come to the fore in recent years thanks primarily to the work of Conrad-Gabber-Prasad, which allowed some spectacular applications by Conrad in number theory. There is a beautiful uniform classification of these groups away from characteristics 2 and 3 via the so-called "standard construction", but in small characteristics the classification is incomplete (and what is known is somewhat ad hoc). In this talk I will explain a new and relatively straightforward construction of one of the more interesting known families of examples in characteristic 2, with non-reduced root systems of type $BC_n$. I will also touch on some other interesting points of the theory, open questions, future directions, etc.

Ben Martin (Aberdeen):

Title: Uniform boundedness for algebraic groups

Abstract: A group $G$ is said to be {\em finitely normally generated} if it is generated by a subset $S$ that is a finite union of conjugacy classes, and {\em bounded} if the diameter of the Cayley graph of $G$ is bounded for some such $S$. We say that $G$ is {\em uniformly bounded} if there is a bound for the diameter of the Cayley graph that does not depend on $S$. A straightforward dimension-counting argument shows that if $G$ is a semisimple linear algebraic group over an algebraically closed field then $G$ is uniformly bounded. K\k{e}dra-Libman-Martin showed that a split semisimple real Lie group is uniformly bounded; on the other hand, a compact semisimple Lie group is bounded but not uniformly bounded. I will discuss the boundedness properties of semisimple algebraic groups $G$ over an arbitrary field $k; the key factor determining the behaviour is whether or not $G$ is isotropic. If time permits then I will also discuss recent results of Alex Trost on boundedness of split semisimple arithmetic groups. Here another version of boundedness (``strong boundedness’’) comes into play.

David Stewart (Newcastle):

Title: A bagatelle on the geometric unipotent radicals of pseudo-reductive groups

Abstract: Pseudo-reductive groups occur naturally in representation theory or number theory when one is working with algebraic groups defined over arbitrary fields. Pseudo-reductive groups differ from reductive groups in that they exhibit non-trivial unipotent radicals over extension fields. We mention a few features of these unipotent radicals.