## Autumn 2022

In Autumn Term 2022, the LAC is hosted at Queen Mary University of London on Thursdays at 4pm in MB-503, organised by Ivan Tomasic.

 29 September Constanze Roitzheim (Kent) Homotopy theory of finite total orders, trees and chicken feet Abstract: A transfer system is a graph on a lattice satisfying certain restriction and composition properties. They were first studied on the lattice of subgroups of a finite group in order to examine equivariant homotopy commutativity, which then unlocked a wealth of links to combinatorial methods. On a finite total order [n], transfer systems can be used to classify different homotopy theories on [n]. The talk will involve plenty of examples and not assume any background knowledge. 13 October Maud De Visscher (City) Title Abstract: 20 October Robert Laugwitz (Nottingham) Title Abstract: 27 October Dugald Macpherson (Leeds) Title Abstract: 03 November Brita Nucinkis (Royal Holloway) Title Abstract: 10 November YK? Title Abstract: 17 November CH? Title Abstract: 24 November Thomas Oliver (Teesside) Title Abstract: 1 December Karin Baur (Leeds) Title Abstract: 15 December Frank Neumann (Leicester) Title Abstract:

## Spring 2022

In Spring Term 2022, the LAC was hosted at City University of London.

 10 February Michael Livesey (University of Manchester) RoCK blocks for double covers of symmetric groups Abstract: In this talk I will give an overview of RoCK blocks and their significance in attacking Broue’s abelian defect group conjecture. I will then go on to talk about RoCK blocks for double covers of symmetric groups and the unexpected difficulties one encounters when compared to the symmetric group situation studied by Chuang-Kessar. The main tool used in overcoming these difficulties is that of Quiver Hecke superalgebras, mirroring the work of A. Evseev who used KLR algebras in the symmetric group setting. (This is ongoing joint work with A. Kleshchev.) 24 February Aparna Upadhyay (University of Arizona) The non-projective part of tensor powers of modular representations of finite groups Abstract: The core of a finite-dimensional modular representation $M$ of a finite group is its largest non-projective summand. Dave Benson and Peter Symonds introduced a new invariant for $M$. This invariant is a result of studying the asymptotics of the core of tensor powers of $M$. In this talk, we will see some interesting properties of this invariant. We obtain a closed formula for computing this invariant for signed permutation modules of the symmetric groups. We prove that the sequence of dimensions of the cores of $M^{\otimes n}$ have algebraic Hilbert series when $M$ is Omega-algebraic. When $M$ is $\Omega ^+$-algebraic then these dimension sequences are eventually linearly recursive. This partially answers a conjecture by Benson and Symonds. 3 March Özgür Bayindir (City University of London) Adjoining roots to ring spectra and algebraic K-theory Abstract: In this work, we develop a new method to adjoin roots to ring spectra and show that this process results in interesting splittings in algebraic K-theory. In the first part of my talk, I will provide motivation for algebraic K-theory and highly structured ring spectra. After this, I will discuss trace methods, a program that provides computational tools for algebraic K-theory, and introduce our results. This is joint work with Tasos Moulinos and Christian Ausoni. 17 March Damiano Rossi (City University of London) Dade’s Conjecture and the Character Triple Conjecture Abstract: Dade’s Conjecture plays an important role in modular representation theory of finite groups as it implies most of the so called Local-Global conjectures. In 2017 Spaeth introduced a strengthening of Dade’s Conjecture, known as the Character Triple Conjecture, and showed that if this new statement holds for quasisimple groups, then Dade’s Conjecture holds for every finite groups. In this talk we consider this problem for quasisimple groups of Lie type in nondefining characteristic. First, by extending results of Broué-Malle-Michel and Cabanes-Enguehard we show how to understand the distribution of characters into blocks for groups of Lie type by providing a description of the so-called Brauer-Lusztig blocks in terms of generalized Harish-Chandra series. Moreover, we show how to obtain a parametrization of generalized Harish-Chandra theory which is compatible with Clifford thery and with the action of automorphisms. Finally, we apply these results and show how this parametrization can be used to prove the Character Triple Conjecture for quasisimple groups of Lie type. This reduces the verification of the Character Triple Conjecture to certain questions on the extendibility of characters of Levi subgroups that are being studied by other authors. 24 March Leo Margolis (VU Brussel) The Modular Isomorphism Problem Abstract: Say we are given only the ring structure of a group ring RG of a finite group G over a commutative ring R. Can we then find the isomorphism type of G as a group? This so-called Isomorphism Problem has obvious negative answers, considering e.g. abelian groups over the complex numbers, but more specific formulations have led to many deep results and beautiful mathematics. The last classical open formulation was the so-called Modular Isomorphism Problem: Does the isomorphism type of kG as a ring determine the isomorphism type of G as a group, if G is a p-group and k a field of characteristic p? After giving an overview of some history of general isomorphism problems and the state of knowledge on the modular formulation, I will present recent results on the problem which include a counterexample to the Modular Isomorphism Problem. 31 March Benjamin Sambale (University of Hannover) Counting characters by local Cartan invariants Abstract:Let B be a block algebra of a finite group with respect to an algebraically closed field of positive characteristic. We show that the number of irreducible ordinary characters of B can be bounded by the Cartan matrix of (a Brauer correspondent of) B. An inductive method of Usami-Puig, often allows the construction of Cartan matrices on a local level. An computer implementation of their method leads to new evidence for Brauer’s k(B)-Conjecture. 7 April Serge Bouc (University of Picardie, Amiens) Functorial equivalence of blocks Abstract: After recalling the basic definitions of category theory and block theory of finite groups, I will introduce the notion of functorial equivalence of blocks, developed in a recent joint work with Deniz Yilmaz. For a commutative ring $R$, and a field $k$ of characteristic $p>0$, we introduce the category of diagonal $p$-permutation functors over $R$. To a pair $(G,b)$ of a finite group $G$ and a block idempotent $b$ of $kG$, we associate a diagonal $p$-permutation functor $F_{G,b}$, and we say that two such pairs $(G,b)$ and $(H,c)$ are functorially equivalent over $R$ if the functors $F_{G,b}$ and $F_{H,c}$ are isomorphic. We show that the category of diagonal $p$-permutation functors over an algebraically closed field of characteristic 0 is semisimple. We obtain a full description of the simple functors, and explicit formulas for their multiplicities as summands of $F_{G,b}$. It follows that functorial equivalence preserves the defect groups of blocks and their number of simple modules. This also leads to characterizations of nilpotent blocks, and to a finiteness theorem in the spirit of Donovan’s finiteness conjecture. 14 April Nadia Mazza (University of Lancaster) On endotrivial modules Abstract: Let G be a finite group and k a field of positive characteristic p dividing |G|. Endotrivial kG-modules are finitely generated kG-modules M such that the set of k-linear transformations End_k(M) forms a permutation kG-module. The stable isomorphism classes of endotrivial kG-modules form a finitely generated abelian group. In this talk, we will review the background and we will present the main results. We will end with a selection of examples on endotrivial modules for Very Important Groups.

## Autumn 2021

In Autumn Term 2021, the LAC is hosted Imperial College London

 18 November Ben Martin (University of Aberdeen) Subgroups of reductive groups containing a regular unipotent element Abstract: Let G be a linear algebraic group over an algebraically closed field k. A major strand of algebraic group theory is to study the subgroup structure of G: can we describe the subgroups H of G (up to conjugacy) and understand how they fit together? The problem becomes more tractable if we put extra hypotheses on H. For instance, we have a good understanding of the set of connected reductive subgroups H when G is simple. Suppose G is connected and reductive. A subgroup H of G is said to be G-irreducible if it is not contained in any proper parabolic subgroup of G. Recently we proved the following result: if H is a connected reductive subgroup of G that contains a regular unipotent element of G then H is G-irreducible. This result was proved by Testerman and Zalesski and later extended by Malle and Testerman. Our proof is short and carries over nicely to the case when H or G is nonconnected. We have also proved analogous results for Lie algebras and finite groups of Lie type. I will discuss these results and sketch the ideas behind the proofs. This is joint work with Michael Bate and Gerhard Röhrle. 25 November Adam Thomas (University of Warwick) The classification of extremely primitive groups Abstract: Let G be a finite primitive permutation group acting on a set X with nontrivial point stabiliser G_x. We say that G is extremely primitive if G_x acts primitively on every orbit in X \ {x}. These groups arise naturally in several different contexts and their study can be traced back to work of Manning in the 1920s. After surveying previous results, we will discuss joint work with Tim Burness towards completing this classification dealing with the almost simple groups with socle an exceptional group of Lie type. We will describe the various techniques used in the proof and, discuss the results we proved on bases for primitive actions of exceptional groups. 09 December 3 pm Noelia Rizo Carrion (University of the Basque Country) Some questions on principal blocks for different primes Abstract: If p and q are different primes and G is a finite group, it is not generally reasonable to expect meaningful interactions between the p-representation theory of G and its q-representation theory. However there are some exceptions, specially when we deal with principal blocks. For instance, if B_p (G) denotes the principal p-block of G, Bessenrodt, Navarro and Olsson proved that Irr(B_p(G))=Irr(B_q(G)) if and only if p and q do not divide |G|. Another nice interaction between B_p(G) and B_q(G) is studied in works by Malle and Navarro and by Liu, Willems, Wang, and Zhang, who prove a version of Brauer’s Height Zero conjecture for principal blocks and two primes. In this talk we give an overview of this kind of results and we propose some problems involving principal blocks for different primes. 09 December 4 pm Gareth Tracey (University of Birmingham) Primitive amalgams the Goldschmidt-Sims conjecture Abstract: A triple of finite groups (H,M,K), usually written H> M

## Spring 2021

In Spring Term 2021, the LAC was hosted online by Queen Mary University of London, organised by Matt Fayers.

 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 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 Maud de Visscher (City) Combinatorial representation theory of the partition algebra Abstract: The partition algebra is closely connected to the symmetric group. In fact, it can be defined as the centraliser algebra of the diagonal action of the symmetric group on the tensor product of the natural permutation module. In this talk I will explain how one can generalise much of the combinatorics used to study the symmetric group to the partition algebra. I will also discuss how this can help us shed new light on the mysterious Kronecker coefficients which appear in the representation theory of the symmetric group. This is based on joint work with C. Bowman, J. Enyang and R. Orellana and recent results by S. Creedon.

## 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 G ≤ GL(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.