This term’s seminars

Spring 2021

In Spring Term 2021, the LAC will be hosted (online) by Queen Mary University of London. Talks will be on Thursdays, usually 3-4pm, via Zoom. Owing to the late start and the fact that I seem to have a good supply of speakers, we will probably continue into the Summer term.
I will send a weekly email to the LAC mailing list (usually on Tuesdays) with details of the Zoom meeting. If you’re not already on the mailing list, you can sign up here. You can also request details from me (Matt Fayers at QMUL – you can figure out my email address) by email.

4 March Sam Corson (Bristol)
Groups with finitary behaviour

Abstract: This talk will be a discussion of infinite groups which share properties with finite groups, either in their actions (strongly bounded groups) or in their relationship to proper subgroups (Jonsson groups). There will be a historical review and exposition of some recent constructions of such groups. Includes joint work with Saharon Shelah.

11 March Emily Norton (Clermont-Ferrand)
The problem of decomposition numbers of finite classical groups

Abstract: A basic problem in modular representation theory of finite groups is to understand decomposition numbers, that is, how an irreducible representation of a group in characteristic 0 decomposes into irreducible representations over a field of positive characteristic. This problem is open even for symmetric groups. I will discuss the case of a finite group of Lie type B or C in non-defining characteristic. The combinatorics of higher-level Fock spaces plays an important role in this setting, as in the representation theory of type B Hecke algebras at roots of unity. This allowed Olivier Dudas and I to determine some new decomposition numbers of these groups. Based on recent and ongoing joint work with Olivier Dudas.

18 March David Craven (Birmingham)
The maximal subgroups of E8(q)

Abstract: The last two talks I gave at various places were on the maximal subgroups of 2E6(q) and the maximal subgroups of E7(q), so this is the next obvious step. In this talk I will discuss the programme to classify the maximal subgroups of the finite simple groups E8(q), and the progress so far made. If time permits, some indications of the new difficulties that present themselves with E8, rather than smaller groups, will be discussed.

25 March Lucia Morotti (Hannover)
Decomposition matrices for spin representations of symmetric groups

Abstract: When studying decomposition matrices for spin representations of symmetric groups a problem, which does not arise for the non-spin case, is given by pairs of irreducible representations labeled by the same partition. This problem can be avoided by considering generalised decomposition matrices instead. Even for generalised decomposition matrices however not much is known. For example not even the form of the generalised decomposition matrix is known in general. In this talk I will present some results on such matrices.

1 April Noriyuki Abe (Tokyo)
On Soergel bimodules

Abstract: The Hecke category recently plays very important role in modular representation theory. Here the Hecke category means a categorification of the Hecke algebra. There are several realizations of the Hecke category. In this talk, I will explain a new realization. The realization is motivated by the theory of Soergel bimodules. I will also explain some applications of this realization.

8 April Liron Speyer (Okinawa)
Semisimple Specht modules indexed by bihooks

Abstract: I will first give a brief survey of some previous results with Louise Sutton, in which we found a large family of decomposable Specht modules for the Hecke algebra of type $B$ indexed by `bihooks’. We conjectured that outside of some degenerate cases, our family gave all decomposable Specht modules indexed by bihooks. There, our methods largely relied on some hands-on computation with Specht modules, working in the framework of cyclotomic KLR algebras.

I will then move on to discussing a recent project with Rob Muth and Louise Sutton, in which we have studied the structure of these Specht modules. By transporting the problem to one for Schur algebras via a Morita equivalence of Kleshchev and Muth, we are able to give all composition factors (including their grading shifts), and show that in most characteristics, these Specht modules are in fact semisimple. In some other small characteristics, we can explicitly determine their structures, including some in which the modules are `almost semisimple’. I will present this story, with some running examples that will help the audience keep track of what’s going on.

15 April Ivan Tomašić (QMUL)
Difference Galois Theory

Abstract: A difference ring is a ring with a distinguished endomorphism. Such objects can be associated with recurrence relations/difference equations, recursively defined sequences, dynamical systems, functional equations and many other contexts.

We develop a Galois theory of difference ring extensions modelled on Janelidze’s categorical theory, where the relevant extensions are classified in terms of difference Galois groupoids. Given that the space of connected components of a difference ring can be a profinite space with a continuous self-map, the considerations take on a topological dynamics flavour, and we discuss some connections with symbolic dynamics.

Disclaimer: this theory is unrelated to Picard-Vessiot style Galois theory of linear difference equations.

22 April Carolina Vallejo (Madrid)
Character tables and generation of Sylow 2-subgroups

Abstract: A main topic in the representation theory of finite groups is to understand how much information about the structure of Sylow subgroups can be obtained from the character table of a group.
I will explain how to detect, after an easy inspection of the character table of a group G, whether or not a Sylow 2-subgroup of G is generated by 2 elements. This talk is based on joint works in collaboration with Gabriel Navarro, Noelia Rizo and Mandi Schaeffer Fry.

13 May Louise Sutton (Manchester)
Tilting modules for SL2

Abstract: The family of tilting modules plays a crucial role in the representation theory of the special linear group SLn and the quantum group of the corresponding Lie algebra, and one ideally aims to understand their structure completely. In this talk, I will discuss recent progress on tilting modules for SL2, where we study them as objects inside a monoidal category governed by well-known Temperley-Lieb diagrammatics. We work in the generalised setting based on two characteristic parameters, namely the characteristic of the underlying field and a root of unity. In this setting, we determine all decompositions of tensor products of simple tilting modules into indecomposable tilting modules. We are then able to explicitly describe the morphisms that project onto these indecomposable summands in some of these cases. These morphisms are known as Jones-Wenzl projectors, which we generalise to this arbitrary setting (and have recently been defined independently by Martin and Spencer). We give a description of the decomposition of the tensor product of these arbitrary Jones-Wenzl projectors into orthogonal, primitive idempotents (in our known cases).

This is joint work with Daniel Tubbenhauer, Paul Wedrich and Jieru Zhu.

20 May 3-4pm Olivier Dudas (Paris)
Macdonald polynomials and decomposition numbers for finite unitary groups

Abstract: (work in progress with R. Rouquier) I will present a computational (yet conjectural) method to determine some decomposition matrices for finite groups of Lie type. These matrices encode how ordinary representations decompose when they are reduced to a field with positive characteristic l. There is an algorithm to compute them for GL(n,q) when l is large enough, but finding these matrices for other groups of Lie type is a very challenging problem.

In this talk I will focus on the finite general unitary group GU(n,q). I will first explain how one can produce a “natural” self-equivalence in the case of GL(n,q) coming from the topology of the Hilbert scheme of the complex plane . The combinatorial part of this equivalence is related to Macdonald’s theory of symmetric functions and gives (q,t)-decomposition numbers. The evidence suggests that the case of finite unitary groups is obtained by taking a suitable square root of that equivalence, which encodes the relation between GU(n,q) and GL(n,–q).

27 May 3-4pm Maud de Visscher (City)
TBA

Abstract: TBA

Spring 2020

In Spring Term 2020, the LAC was hosted by City, University of London.

23 January Brita Nucinkis (Royal Holloway, University of London)
An irrational slope Thompson’s group

Abstract: In this talk I will discuss a relative to Thompson’s group $F$, the group $F_\tau,$ which is the group of piecewise linear homeomorphisms of $[0,1]$ with breakpoints in $\mathbb{Z}[\tau]$ and slopes powers of $\tau,$ where $\tau = \frac{\sqrt5 -1}{2}$ is the small Golden Ratio. This group was first considered by S. Cleary, who showed that the group was finitely presented and of type $F_\infty.$ Here we take a combinatorial approach considering elements as tree-pair diagrams, where the trees are finite binary trees, but with two different kinds of carets. We use this representation to show that the commutator subgroup is simple and give a unique normal form for its elements. The surprising feature is that the $T$- and $V$-versions of these groups are not simple, however. This is joint work with J. Burillo and L. Reeves.

30 January Justin Lynd (University of Louisiana)
The Benson-Solomon fusion systems

Abstract: The fusion system of a finite group G at a prime p is a category whose objects are the subgroups of a fixed Sylow p-subgroup S, and where the morphisms are the conjugation homomorphisms induced by the elements of G. The notion of a saturated fusion system is
abstracted from this standard example, and provides a coarse representation of what is meant by the p-local structure of a finite group. Once the group G is abstracted away, there appear many exotic fusion systems not arising in the above fashion. Exotic fusion systems are prevalent at odd primes, but only a single one-parameter family of “simple” fusion systems at the prime 2 are currently known. These are closely related to the groups Spin_7(q), q odd, and were first considered by Solomon and Benson, although not as fusion systems per se. I’ll explain some “coincidences” that allow the Benson-Solomon systems Sol(q) to exist, and then discuss various results about these systems as time allows. The results are related to the questions: How “close” to a group is Sol(q)? Are there any more exotic systems constructed in some direct fashion from the existence of Sol(q)? How many 2-modular “simple modules” would the principal 2-block of Sol(q) have if it were a group? In various combinations, this is joint work with E. Henke, A. Libman, and J. Semeraro.

6 February Cheryl Praeger (The University of Western Australia)
Diagonal structures and primitive permutation groups

Abstract: Many maximal subgroups of finite symmetric groups arise as stabilisers of some structure on the point set: for example the maximal intransitive permutation groups are subset stabilisers. The primitive groups of diagonal type for a long time have seemed exceptional in this respect. Csaba Schneider and I have introduced diagonal structures which, for the first time, give a combinatorial interpretation to these primitive groups of simple diagonal type. In further work together also with Peter Cameron and Rosemary Bailey, we’ve exhibited these groups as automorphism groups of `diagonal graphs’.

13 February Dave Benson (University of Aberdeen)
Some exotic tensor categories in prime characteristic

This talk is about joint work with Pavel Etingof and Victor Ostrik. A theorem of Deligne says that in characteristic zero, any symmetric tensor category “of moderate growth” admits a tensor functor to vector spaces or to super (i.e., Z/2-graded) vector spaces. In prime characteristic, this is not true, but one may ask whether there is a good list of “incompressible” symmetric tensor categories to which they they do all map. We construct an infinite ascending chain of finite symmetric tensor categories in characteristic p, all of which are incompressible. The constructions are based on the theory of tilting modules over the algebraic group SL(2). It is possible that this is the complete list, but we have not proved that.

20 February Jay Taylor (University of Southern California)
Unitriangularity of Decomposition Matrices of Unipotent Blocks

Abstract: One of the distinguished features of the representation theory of finite groups is the ability to take a representation in characteristic zero and reduce it to obtain a representation over a fixed field of positive characteristic (a modular representation). If one starts with a representation that is irreducible in characteristic zero then its modular reduction can fail to be irreducible. The decomposition matrix encodes the multiplicities of the modular irreducible representations in this reduction.

In this talk I will present recent joint work with Olivier Brunat and Olivier Dudas establishing a fundamental property of the decomposition matrix for finite reductive groups, namely that it has a unitriangular shape. The solution to this problem involves the interplay between Lusztig’s geometric theory of character sheaves and a family of representations whose construction was originally proposed by Kawanaka.

12 March John Murray (National University of Ireland, Maynooth)
Brauer characters and normal subgroups

Abstract: Clifford’s theorem explores the relationship between the irreducible modules of a group G and those of a normal subgroup N, over an arbitrary field F. In particular it applies to irreducible Brauer characters. Our focus here is on irreducible 2-Brauer characters.

We begin by showing that if \theta is an irreducible 2-Brauer character of N, then G has a real-valued irreducible 2-Brauer character over \theta if and only if \theta is G-conjugate to its complex conjugate.

Now suppose that \theta is real-valued. Then it is a remarkable fact that \theta has a unique real extension to its stabilizer in G. So G has a unique real-valued irreducible 2-Brauer character \mu such that \theta occurs with odd multiplicity in the restriction to N of \mu.

Next let \phi be a real-valued irreducible 2-Brauer character of G. Fong’s Lemma asserts that \phi is the Brauer character of a symplectic representation of G. However it is a delicate question to determine
whether \phi has orthogonal type. Suppose not and also that N is not contained in the kernel of \phi. Then we show that the restriction to N of \phi is a sum of distinct real-valued non-orthogonal irreducible
2-Brauer character of N.

Finally we discuss a consequence for blocks. Recall that a block of G is weakly regular with respect to N if its central character vanishes off N. Now let b be a real 2-block of N. We show that set of 2-blocks of G which lie over b and which are weakly regular with respect to N contains a unique real 2-block.

Autumn 2019

In Autumn Term 2019, the LAC was hosted by Imperial College.

10th October Alastair Litterick (Essex)
Rigidity and representation varieties

Abstract: Let F be a finitely generated group and G be a reductive algebraic group. The study of homomorphic images of F in G has a long and distinguished history, having applications to representation theory, generating sets of finite simple groups, Hurwitz surfaces, regular maps and hypermaps, the inverse Galois problem, differential geometry, and more besides.

The space Hom(F,G) is an algebraic variety with a natural G-action. In joint work with Ben Martin (Aberdeen), using algebraic geometry and geometric invariant theory we are able to prove a ‘rigidity’ result: under natural hypotheses, the G-orbits of certain interesting homomorphisms are both closed and open in an appropriate subvariety of Hom(F,G).

As an application, if F is generated by torsion elements which multiply to 1, if G is defined over the finite field F_q, and if a certain dimension bound holds for conjugacy classes of G, then only finitely many groups of Lie type G(q^e) are quotients of F. This proves and generalises a 2010 conjecture of C. Marion on triangle groups.

17th October Lewis Topley (Birmingham/Kent)
Yangians and representations of the general linear Lie algebra in positive
characteristic

Abstract: In this talk I will discuss the representation theory of the general linear Lie algebra over a field of positive characteristic. The irreducible representations factor through certain quotients of the enveloping algebra, known as reduced enveloping algebras. It turns out that these reduced enveloping algebras may be described completely by examining a finite collection of such algebras, labelled by the conjugacy classes of nilpotent matrices of rank n. Premet has shown that each of these reduced enveloping algebras is actually Morita equivalent to an algebra known as a restricted finite W-algebra. The main result of this talk is a joint work with Simon Goodwin, in which we show that these restricted finite W-algebras can be described explicitly as certain subquotients of a Yangian.​

24th October Nick Gill (South Wales)
Some interesting statistics concerning finite primitive permutation groups

Abstract: Let G be a finite permutation group on a set X. A base for G is a subset Y of X such that G_(Y), the pointwise-stabilizer of Y in G, is trivial. There has been a long history of studying how small a base can be for different classes of group G. We will discuss some variants of this study, particularly focusing on upper bounds for primitive groups: in particular, we want to know how big a minimal base can be, how big an irredundant base can be, and how big an independent set can be. (The precise definition of these three notions will be given in the seminar.)

Our interest in these statistics stems from their connection to another statistic — the relational complexity of a finite permutation group. This last statistic was introduced in the 1990’s by Greg Cherlin in work applying certain model theoretic ideas of Lachlan. In particular the relational complexity of a permutation group gives an idea of the “efficiency” with which the group can be represented as the automorphism group of a homogeneous relational structure.

31st October Michele Zordan (Imperial)
Zeta functions of groups, model theory and rationality
7th November Anitha Thillaisundaram (Lincoln)
Maximal subgroups of Grigorchuk-Gupta-Sidki (GGS-)groups

Abstract: The GGS-groups were some of the early positive answers to the famous Burnside problem. These groups act on infinite rooted trees and are easy to describe, plus possess interesting properties. A natural aspect of these groups to study is their maximal subgroups, and in particular, whether these groups have maximal subgroups of infinite index. It was proved by Pervova in 2005 that the torsion GGS-groups do not have maximal subgroups of infinite index. In this talk, I will consider the remaining non-torsion GGS-groups. This is joint work with Dominik Francoeur.

14th November Dan Segal (Oxford)
21st November Emmanuel Breuillard (Cambridge)
28th November Peter Cameron (St Andrews)
Diagonal groups, synchronization, and association schemes
5th December Alison Parker (Leeds)
Tilting modules for the blob algebra
12th December Francois Thilmany (Louvain)
Lattices of minimal covolume in SL(n,R)

Abstract: A classical result of Siegel asserts that the (2,3,7)-triangle group attains the smallest covolume among lattices SL(2,R). In general, given a semisimple Lie group G over some local field F, one may ask which lattices in G attain the smallest covolume. A complete answer to this question seems out of reach at the moment; nevertheless, many steps have been made in the last decades. Inspired by Siegel’s result, Lubotzky determined that a lattice of minimal covolume in SL(2,F) with F=F_q((t)) is given by the so-called characteristic p modular group SL(2,F_q[1/t]). He noted that, in contrast with Siegel’s lattice, the quotient by SL(2,F_q[1/t]) was not compact, and asked what the typical situation should be: “for a semisimple Lie group over a local field, is a lattice of minimal covolume a cocompact or nonuniform lattice? “.

In the talk, we will review some of the known results, and then discuss the case of SL(n,R}) for n > 2. It turns out that, up to automorphism, the unique lattice of minimal covolume in SL(n,R) (n > 2) is SL(n,Z). In particular, it is not uniform, giving a partial answer to Lubotzky’s question in this case.

Winter/Spring 2019

In Winter/Spring Term 2019, the LAC was hosted in Queen Mary, University of London.

17th January John R. Parker (Durham)
Constructing fundamental polyhedra for groups generated by complex reflections

A complex reflection is a complex linear map given by a matrix A for which (A-I) has rank 1. In this talk I will describe an algorithm for finding polyhedra associated to certain groups acting on C^3 generated by three complex reflections. In many cases these polyhedra may be geometrised in such a way that they are fundamental polyhedra and the group is discrete.
An application of this algorithm is that it gives fundamental domains for all known (commensurability classes) of non-arithmetic lattices in PU(2,1).

24th January Eleonore Faber (Leeds)
Reflections, rotations, and singularities via the McKay correspondence

The classification of finite subgroups of SO(3) is well known: these are either cyclic or dihedral groups or one of the symmetry groups of the Platonic solids. In the 19th century, Felix Klein investigated the orbit spaces of those groups and their double covers, the so-called binary polyhedral groups. This investigation is at the origin of singularity theory.
Quite surprisingly, in 1979, John McKay found a direct relationship between the resolution of the singularities of the orbit spaces and the representation theory of the finite group one starts from. This “classical McKay correspondence” is manifested, in particular, by the ubiquitious Coxeter-Dynkin diagrams.
In this talk I will first review the history of this fascinating result, and then give an outlook on recent joint work with Ragnar-Olaf Buchweitz and Colin Ingalls about a McKay correspondence for finite reflection groups in GL(n,C).

7th February Vladimir Dotsenko (Trinity College Dublin)
Three guises of toric varieties of Loday’s associahedra and related algebraic structures

Associahedra are remarkable CW-complexes introduced by Stasheff in 1960s to encode a homotopically coherent notion of associativity. They have been realised as polytopes with integer coordinates in several different ways over the past few decades. I shall explain that the realisations of associahedra due to Loday lead to toric varieties of particular merit. These varieties have been already identified with “brick manifolds” arising when studying subword complexes for Coxeter groups (Escobar, 2014). It turns out that they also arise as “wonderful models” in the sense of de Concini and Procesi for certain subspace arrangements. Guided by that geometric picture, I shall argue that in some sense these varieties give a “noncommutative version” of Deligne-Mumford compactifications of moduli spaces of genus zero curves with marked points, in that they give rise to remarkable algebraic structures resembling cohomological field theories of Kontsevich and Manin. This is a joint work with Sergey Shadrin and Bruno Vallette.

14th February Zeinab Toghani (QMUL)
Tropical Differential Algebra

Let I be an ideal of the ring of Laurent polynomials with coefficients in a real-valued field. The fundamental theorem of tropical algebraic geometry states the equality between the tropicalisation of the variety V (I) and the tropical variety associated to the tropicalisation of the ideal I.
In this talk I show this result for a differential ideal J of the ring of differential polynomials
K[[t]]{x_{1} ,…, x_{n} }, where K is an uncountable algebraically closed field of characteristic zero.
I show the equality between the tropicalisation of the set of solutions of J, and the set of solutions of tropicalisation of J.

28th February Felipe Rincón (QMUL)
CSM cycles of matroids

Abstract: I will introduce Chern-Schwartz-MacPherson cycles of an arbitrary matroid M, which are a special collection of balanced polyhedral fans associated to M. These CSM cycles are of special significance in tropical geometry, and they satisfy very interesting combinatorics. In the case the matroid M arises from a complex hyperplane arrangement A, these cycles naturally represent the CSM class of the complement of A. This is joint work with Lucía López de Medrano and Kristin Shaw.

Autumn 2018

In Autumn Term 2018, the LAC was hosted in Birkbeck, University of London.

Oct 11th Sibylle Schroll (Leicester)
On the geometric model for the bounded derived category of gentle algebras

Abstract: In recent years, gentle algebras have been connected to many different areas of mathematics such as cluster theory, nodal stacky curves and homological mirror symmetry. In this talk we will give a geometric model of the bounded derived category of gentles algebras developed in joint work with Pierre-Guy Plamondon and Sebastian Opper. Our model is based on the representation theory of gentle algebras. By work of Haiden-Katzarkov-Kontsevich and Lekili-Polishchuk this gives a model of the partially wrapped Fukaya category of surfaces with stops.

Oct 18th Katerina Hristova (Warwick)
Frobenius Reciprocity for Topological Groups

Abstract: Given a representation of an abstract group G, one can always define a representation of a subgroup H of G, by simply restricting the action of the group to the subgroup. This procedure yields a functor called restriction. In the other direction, given a representation of a subgroup H of G, there is a recipe for defining a representation of G from the representation of H. This also gives a functor called induction. A classic result in the representation theory of abstract groups is the adjunction relation between induction and restriction known as Frobenius reciprocity. The aim of this talk is to explain under what conditions we have an analogue of Frobenius reciprocity in the setting of continuous represention for a topological group G and a closed subgroup H in three different categories: discrete representations, linear complete representation and linearly compact representations.

Oct 25th Noah Arbesfeld (Imperial)
Virasoro algebras and the Yang-Baxter equation

Abstract: Generalizing work of Maulik and Okounkov, we explain how to use certain intertwiners of highest-weight modules for Virasoro algebras to produce solutions to the Yang-Baxter equation. The proof uses the geometry of the Hilbert scheme of points on a surface.

Nov 1st Maura Paterson (Birkbeck)
Reciprocally-Weighted External Difference Families and the Bimodal Property

Abstract: Let G be a finite abelian group of order n. An (n,k,λ) m-External Difference Family (EDF)is a collection of m disjoint subsets of G each of size k, with the property that each nonzero group element occurs precisely λ times as a difference between group elements in two different subsets from the collection.  Motivated by an application to the construction of weak algebraic manipulation detection codes, a reciprocally-weight EDF (RWEDF) is defined to be a generalisation of an EDF in which the subsets may have different sizes, and the differences are counted with a weighting given by the reciprocal of the set sizes.

In this talk I will discuss some interesting structural properties of RWEDFs with certain parameters, and describe a construction of an infinite families of nontrivial RWEDFs.

Nov 8th Gerald Williams (Essex)
Generalized graph groups with balanced presentations

Abstract: A balanced presentation of a group is one with an equal number of generators and relators. Since presentations with more generators than relators define infinite groups, balanced presentations present a borderline situation where both finite and infinite groups can be found. It is of interest to find which balanced presentations can define finite groups, and what groups can arise. We consider groups defined by balanced presentations with the property that each relator is of the form R(x,y) where R is some fixed word in two generators. Examples of such groups include Right Angled Artin Groups, Higman groups, and cyclically presented groups in which the relators involve exactly two generators. To each such presentation we associate a directed graph whose vertices correspond to the generators and whose arcs correspond to the relators. Extending work of Pride, we show that if the graph is triangle-free then the corresponding group cannot be trivial or finite of rank greater than 2. This is joint work with Johannes Cuno.

Nov 15th Brendan Masterson (Middlesex)
On the table of marks of a direct product of finite groups

Abstract: The table of marks of a finite group G characterises the actions of G on the transitive G-sets, which are in bijection to the conjugacy classes of subgroups of G. Thus the table of marks provides a complete classification of the permutation representations of a finite group G up to equivalence.
In contrast to the character table of a direct product of two finite groups, its table of marks is not simply the Kronecker product of the tables of marks of the two groups. Based on a decomposition of the inclusion order on the subgroup lattice of a direct product as a relation product of three smaller partial orders, we describe the table of marks of the direct product essentially as a matrix product of three class incidence matrices. Each of these matrices is in turn described as a sparse block diagonal matrix.
This is joint work with Goetz Pfeiffer.

Nov 22nd Radha Kessar (City)
Weight conjectures for fusion systems

Abstract: I will present joint work with Markus Linckelmann, Justin Lynd, and Jason Semeraro connecting local-global relationships (known and conjectural) in the modular representation theory of finite groups to the theory of fusion systems.

Nov 29th Derek Holt (Warwick)
Polynomial time computation in matrix groups over finite fields

Abstract: The new results described in this talk were proved jointly work with Charles Leedham-Green and Eamonn O’Brien.
Over the past 30 years, an algorithm CompositionTree has been developed for enabling practical computation in large matrix groups over finite fields. The principal aim is to find a composition series and membership test for an input group GGL(d,q). This has been implemented in Magma and performs well in practice.
In 2009, Babai, Beals and Seress published a polynomial time algorithm (assuming oracles for integer factorization and discrete logs) with the same aims for odd q. But this is not suitable for implementation.
We have been asked whether it is feasible to show that CompositionTree can be easily adapted to run in polynomial time, and we can now prove that this is possible with a few provisos.
The main new idea is that we can modify our black box algorithms for constructive recogniton of the finite nonabelian simple groups so that, if the input group is not simple, then a nontrivial element in a proper normal subgroup is output.

Dec 6th Haralampos Geranios (York)
New families of decomposable Specht modules

Abstract: The Specht modules are the key players in the representation theory of the symmetric groups. If the characteristic of the field is different than 2, it is well-known that these modules are indecomposable. In characteristic 2 there exist decomposable Specht modules and the first example of such a module was found by Gordon James in the 70s. Surprisingly enough, only a few other examples of such modules have been discovered since then. In this talk I will present many new families of decomposable Specht modules and describe explicitly their indecomposable summands. This is a joint work with Stephen Donkin.

Summer 2018

In Summer Term 2018, the following LAC talks took place at City University of London.

19th June Sigiswald Barbier (Gent)
A minimal representation of the orthosymplectic Lie superalgebra

Abstract: Minimal representations are an important class of “small” infinite dimensional unitary representations of Lie groups. They are characterised by the fact that their annihilator ideal is equal to the Joseph ideal. Two prominent examples are the metaplectic representations of Mp(2n) (a double cover of Sp(2n)) and the minimal representation of the indefinite orthogonal group O(p,q).
There exists a unified framework to construct the minimal representation of a Lie group associated to a simple Jordan algebra.
In this talk I will construct a generalisation of the minimal representation of so(p,q) to the orthosymplectic Lie superalgebra osp(p,q|2n) using Jordan superalgebras. This representation also has an annihilator ideal equal to a Joseph-like ideal. I will also mention the obstacles which prevent a straightforward generalisation to other Lie superalgebras.

2nd July Michael Batanin (Macquarie University)
Deformation complex of a tensor category is an E_3-algebra.

Abstract: Famous Deligne’s conjecture, which is now a theorem, claims that Hochschild complex of an associative algebra admits an action an operad weakly equivalent to the little 2-cubes operad.
Davydov-Yetter deformation complex of a tensor category is, in a sense, a categorification of Hochschild complex. It is natural to ask if there is an analogue of Deligne’s statement on this context.
We show that a similar action exists but instead of little 2-cubes we get little 3-cubes action. The proof is combinatorial and relies on liftings of certain paths on a commutative lattice to paths of restricted complexity on a noncommutative lattice.
This is a joint work with Alexei Davydov.

Spring 2018

In Spring Term 2018, the LAC was held at City University.

18th January Simon Peacock (Bristol)
Representation dimension and separable equivalences

Abstract: The representation dimension of an algebra is a finite integer that is supposed to indicate how complicated an algebra’s module category is. This dimension was first introduce by Auslander in 1971 and is, in general, notoriously hard to compute. This measure is related to the representation type of an algebra: an algebra has finite representation type if and only if it’s representation dimension is less than 3.
Separable equivalence is an equivalence relation on finite dimensional algebras. Over a field of a characteristic p, a group algebra is separably equivalent to the group algebra of its Sylow p-subgroup. We use this relationship between a group and its Sylows to put an upper bound on the representation dimension of a group algebra for any finite group with a elementary-abelian Sylow subgroup.

25th January Ivan Tomašić (Queen Mary)
Cohomology of difference algebraic groups

Abstract: Difference algebra studies algebraic structures equipped with an endomorphism/difference operator, and difference algebraic varieties are defined by systems of difference polynomial equations over difference rings and fields. In this talk, we will:

  • argue that twisted groups of Lie type are best viewed as difference algebraic groups;
  • develop the cohomology theory of difference algebraic groups;
  • compute the cohomology in a number of interesting cases, and discuss its applications.
1st February Joseph Karmazyn (Sheffield)
Equivalences of singularity categories via noncommutative algebras

Abstract: Singularity categories are triangulated categories occurring as invariants associated to singular algebras. For hypersurface singularities these categories can be realised via matrix factorisations, and in this case Knorrer periodicity constructs equivalences between the singularity categories of many different hypersurfaces.
I will discuss these ideas, and talk about how equivalences of singularity categories in the non-hypersurface (and non-Gorenstein) setting can be constructed by considering quasi-hereditary noncommutative resolutions produced from certain geometric situations. In addition, Ringel duality has a very explicit description and interpretation for these quasi-hereditary algebras.

8th February Eugenio Giannelli (Cambridge)
Restriction of characters to Sylow p-subgroups

Abstract: The relevance of the McKay conjecture in the representation theory of finite groups led to the study of the decomposition into irreducible constituents of the restriction of characters to Sylow p-subgroups.
I will present some recent results on the topic.

15th February Ivo Dell’Ambrogio (Lille)
A categorification of the representation theory of finite groups

Abstract: Dress’s theory of Mackey functors is a successful axiomatization of the representation theory of finite groups, capturing the formal aspects of such classical invariants as the character ring or group (co)homology. But, typically, each such invariant is only a partial shadow (consisting of abelian groups and homomorphisms) of a richer structure (consisting of additive, abelian or triangulated categories and suitable functors between them).
In joint work with Paul Balmer, we develop a theory of “Mackey 2-functors” in order to study this higher structure, thus explaining certain phenomena which, though invisible to classical Mackey functors, occur throughout equivariant mathematics.
In this talk I will provide examples of Mackey 2-functors, such as derived and stable module categories in representation theory or equivariant stable homotopy categories in topology, I will motivate our axioms and explain the first results of the theory.

22nd March David Pauksztello (Lancaster)
Silting theory and stability spaces

Abstract: In this talk I will introduce the notion of silting objects and mutation of silting objects. I will then show how the combinatorics of silting mutation can give one information regarding the structure of the space of stability conditions. In particular, I will show how a certain discreteness of this mutation theory enables one to employ techniques of Qiu and Woolf to obtain the contractibility of the space of stability conditions for a class of mainstream algebraic examples, the so-called silting-discrete algebras. This talk will be a discussion of joint work with Nathan Broomhead, David Ploog, Manuel Saorin and Alexandra Zvonareva.

29th March Wajid Mannan (Queen Mary)
Non-standard syzygies over quaternion groups

Abstract: For finite balanced presentations of quaternion groups Q_{4n}, n>5, it is unknown if the kernel of the associated matrix is always generated by a single element. A positive answer for any value of n>5 would resolve one of the most fundamental and longstanding questions in topology: Is cohomological dimension the same as geometric dimension for finite cell complexes.

I will explain the background to this, contrast with the situation for dihedral groups which is completely understood, and explain my recent incremental result for the case of two generators and two relators.

5th April Nadia Mazza (Lancaster)
On a pro-p group of upper triangular matrices

Abstract: In this talk, we will discuss a pro-p group G whose finite quotient groups give your “favourite” Sylow p-subgroups of GL_n(q) for all positive integers n, where q is a power of p.

Elaborating on work by Weir in the 50s and recent results by Bier and Holubowski, we will dip into the subgroup structure of G.

Time permitting, we will also discuss field extensions, a p-adic variant of G and Hausdorff dimensions of some closed subgroups.

12th April Sira Gratz (Glasgow)
Homotopy invariants of singularity categories

Abstract: The existence of a grading on a ring often makes computations a lot easier. In particular this is true for the computation of homotopy invariants. For example one can readily compute such invariants for the stable categories of graded modules over connected graded self-injective algebras. Using work of Tabuada, we’ll show how to deduce from this knowledge the homotopy invariants of the ungraded stable categories for such algebras. As another illustration of these ideas we’ll show that cluster categories of Dynkin type A_n, for even n, are “A^1-homotopy phantoms”. All this is based on joint work with Greg Stevenson.

Autumn 2017

The Autumn 2017 seminars were held at Imperial College.

October 12 Chris Bowman (Kent)
Complex reflection groups of type G(l,1,n) and their deformations
October 19 John MacQuarrie (UFMG)
The path algebra as a left adjoint functor
October 26 Alexander Molev (Sydney)
Vinberg’s problem for classical Lie algebras
November 9 Joanna Fawcett (Imperial)
Partial linear spaces with symmetry
November 16 Dan Segal (Oxford)
Small profinite groups
November 23 Jason Semeraro (Leicester)
Representations of Fusion Systems
November 30 Emilio Pierro (LSE)
Finite simple quotients of Mapping Class Groups
December 7 Charlotte Kestner (Imperial)
Strongly Minimal Semigroups
December 7 Dugald MacPherson (Leeds)
Model theory of profinite groups
December 14 Florian Eisele (City)
A counterexample to the first Zassenhaus conjecture

Summer 2017

In Summer Term 2017, the LAC was held at City University, organized by Jorge Vittoria.

8th June Jay Taylor (Arizona)
Harish-Chandra Induction and Lusztig’s Jordan Decomposition of Characters
22nd June Arik Wilbert (Bonn)
Two-block Springer fibers and Springer representations in type D

Abstract: We explain how to construct an explicit topological model for every two-block Springer fiber of type D. These so-called topological Springer fibers are homeomorphic to their corresponding algebro-geometric Springer fiber. They are defined combinatorially using cup diagrams which appear in the context of finding closed formulas for parabolic Kazhdan-Lusztig polynomials of type D with respect to a maximal parabolic of type A. As an application it is discussed how the topological Springer fibers can be used to reconstruct the famous Springer representation in an elementary and combinatorial way.

29th June Benjamin Briggs (Bonn)
The characteristic action of Hochschild cohomology, and Koszul duality
4th July Andrew Mathas (Sydney)
Jantzen filtrations and graded Specht modules

Abstract: The Jantzen sum formula is a classical result in the representation theory of the symmetric and general linear groups that describes a natural filtration of the modular reductions of the simple modules of these groups. Analogues of this result exist for many algebras including the cyclotomic Hecke algebras of type A. Quite remarkably, the cyclotomic Hecke algebras of type A are now know to admit a Z-grading because they are isomorphic to cyclotomic KLR algebras. I will explain how to give an easy proof of the Jantzen sum formula for the Specht modules of the cyclotomic Hecke algebras of type A using the KLR grading. I will discuss some consequences and applications of this approach.

20th July Olaf Schnürer (Bonn)
Geometric applications of conservative descent for semi-orthogonal decompositions

Abstract: Motivated by the local flavor of several well-known semi-orthogonal decompositions in algebraic geometry we introduce a technique called “conservative descent” in order to establish such decompositions locally. The decompositions we have in mind are those for projective bundles, blow-ups and root constructions. Our technique simplifies the proof of these decompositions and establishes them in greater generality. We also discuss semi-orthogonal decompositions for Brauer-Severi varieties.
This is joint work with Daniel Bergh (Copenhagen).

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