Autumn 2022
In Autumn Term 2022, the LAC is hosted at Queen Mary University of London on Thursdays at 4pm in MB503, organised by Ivan Tomasic.
29 September  Constanze Roitzheim (Kent) 
Homotopy theory of finite total orders, trees and chicken feetAbstract: 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) 
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20 October  Robert Laugwitz (Nottingham) 
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27 October  Dugald Macpherson (Leeds) 
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03 November  Brita Nucinkis (Royal Holloway) 
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10 November  YK? 
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17 November  CH? 
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24 November  Thomas Oliver (Teesside) 
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1 December  Karin Baur (Leeds) 
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15 December  Frank Neumann (Leicester) 
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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 groupsAbstract: 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 ChuangKessar. 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 nonprojective part of tensor powers of modular representations of finite groupsAbstract: The core of a finitedimensional modular representation $M$ of a finite group is its largest nonprojective 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 Omegaalgebraic. 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 KtheoryAbstract: 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 Ktheory. 

17 March  Damiano Rossi (City University of London) 
Dade’s Conjecture and the Character Triple ConjectureAbstract: Dade’s Conjecture plays an important role in modular representation theory of finite groups as it implies most of the so called LocalGlobal 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 BrouéMalleMichel and CabanesEnguehard we show how to understand the distribution of characters into blocks for groups of Lie type by providing a description of the socalled BrauerLusztig blocks in terms of generalized HarishChandra series. Moreover, we show how to obtain a parametrization of generalized HarishChandra 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 ProblemAbstract: 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 socalled 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 socalled 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 pgroup 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 invariantsAbstract:Let B be a block algebra of a finite group with respect to an algebraically closed field of positive characteristic. 

7 April  Serge Bouc (University of Picardie, Amiens) 
Functorial equivalence of blocksAbstract: 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 modulesAbstract: Let G be a finite group and k a field of positive characteristic p dividing G. Endotrivial kGmodules are finitely generated kGmodules M such that the set of klinear transformations End_k(M) forms a permutation kGmodule. The stable isomorphism classes of endotrivial kGmodules 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 elementAbstract: 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 Girreducible 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 Girreducible. 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 groupsAbstract: 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 primesAbstract: If p and q are different primes and G is a finite group, it is not generally reasonable to expect meaningful interactions between the prepresentation theory of G and its qrepresentation theory. However there are some exceptions, specially when we deal with principal blocks. For instance, if B_p (G) denotes the principal pblock 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 GoldschmidtSims conjectureAbstract: A triple of finite groups (H,M,K), usually written H> M<K, is called a primitive amalgam if M is a subgroup of both H and K, and each of the following holds: Primitive amalgams arise naturally in many different contexts across pure mathematics, from Tutte’s study of vertextransitive groups of automorphisms of finite, connected, trivalent graphs; to Thompson’s classification of simple Ngroups; to Sims’ study of point stabilizers in primitive permutation groups, and beyond. In this talk, we will discuss some recent progress on the central conjecture from the theory of primitive amalgams: the Goldschmidt–Sims conjecture. Joint work with L. Pyber. 

16 December 3 pm  Giles Gardam (WWU Muenster) 
The Kaplansky conjecturesAbstract: Three conjectures on group rings of torsionfree groups are commonly attributed to Kaplansky, namely the unit, zero divisor and idempotent conjectures. For example, the zero divisor conjecture predicts that if K is a field and G is a torsionfree group, then the group ring K[G] has no zero divisors. I will discuss these conjectures and their relationship to other conjectures and properties of groups. I will then explain how modern solvers for Boolean satisfiability can be applied to them, producing the first counterexample to the unit conjecture. 

16 December 4 pm  Katrin Tent (WWU Muenster) 
Simple sharply 2transitive groupAbstract: Until recently all known sharply 2transitive groups were of the form K x K^* for some (near)field K. I will explain recent constructions of sharply 2transitive groups without abelian normal subgroups, how to build simple sharply 2transitive groups and discuss other conditions one might ask of such permutation groups. 
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 behaviourAbstract: 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 (ClermontFerrand) 
The problem of decomposition numbers of finite classical groupsAbstract: 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 nondefining characteristic. The combinatorics of higherlevel 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 E_{8}(q)Abstract: The last two talks I gave at various places were on the maximal subgroups of ^{2}E_{6}(q) and the maximal subgroups of E_{7}(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 E_{8}(q), and the progress so far made. If time permits, some indications of the new difficulties that present themselves with E_{8}, rather than smaller groups, will be discussed. 

25 March  Lucia Morotti (Hannover) 
Decomposition matrices for spin representations of symmetric groupsAbstract: When studying decomposition matrices for spin representations of symmetric groups a problem, which does not arise for the nonspin 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 bimodulesAbstract: 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 bihooksAbstract: 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 handson 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 TheoryAbstract: 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 selfmap, the considerations take on a topological dynamics flavour, and we discuss some connections with symbolic dynamics. Disclaimer: this theory is unrelated to PicardVessiot style Galois theory of linear difference equations. 

22 April  Carolina Vallejo (Madrid) 
Character tables and generation of Sylow 2subgroupsAbstract: 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. 

13 May  Louise Sutton (Manchester) 
Tilting modules for SL_{2}Abstract: The family of tilting modules plays a crucial role in the representation theory of the special linear group SL_{n} 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 SL_{2}, where we study them as objects inside a monoidal category governed by wellknown TemperleyLieb 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 JonesWenzl 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 JonesWenzl 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 groupsAbstract: (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” selfequivalence 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 algebraAbstract: 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. 
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 groupAbstract: 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 treepair 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 BensonSolomon fusion systemsAbstract: The fusion system of a finite group G at a prime p is a category whose objects are the subgroups of a fixed Sylow psubgroup S, and where the morphisms are the conjugation homomorphisms induced by the elements of G. The notion of a saturated fusion system is 

6 February  Cheryl Praeger (The University of Western Australia) 
Diagonal structures and primitive permutation groupsAbstract: 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 characteristicThis 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/2graded) 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 BlocksAbstract: 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 subgroupsAbstract: 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 2Brauer characters. We begin by showing that if \theta is an irreducible 2Brauer character of N, then G has a realvalued irreducible 2Brauer character over \theta if and only if \theta is Gconjugate to its complex conjugate. Now suppose that \theta is realvalued. Then it is a remarkable fact that \theta has a unique real extension to its stabilizer in G. So G has a unique realvalued irreducible 2Brauer character \mu such that \theta occurs with odd multiplicity in the restriction to N of \mu. Next let \phi be a realvalued irreducible 2Brauer 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 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 2block of N. We show that set of 2blocks of G which lie over b and which are weakly regular with respect to N contains a unique real 2block. 
Autumn 2019
In Autumn Term 2019, the LAC was hosted by Imperial College.
10th October  Alastair Litterick (Essex) 
Rigidity and representation varietiesAbstract: 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 Gaction. 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 Gorbits 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


24th October  Nick Gill (South Wales) 
Some interesting statistics concerning finite primitive permutation groupsAbstract: 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 pointwisestabilizer 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 GrigorchukGuptaSidki (GGS)groupsAbstract: The GGSgroups 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 GGSgroups do not have maximal subgroups of infinite index. In this talk, I will consider the remaining nontorsion GGSgroups. 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 socalled 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 reflectionsA complex reflection is a complex linear map given by a matrix A for which (AI) 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. 

24th January  Eleonore Faber (Leeds) 
Reflections, rotations, and singularities via the McKay correspondenceThe 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 socalled binary polyhedral groups. This investigation is at the origin of singularity theory. 

7th February  Vladimir Dotsenko (Trinity College Dublin) 
Three guises of toric varieties of Loday’s associahedra and related algebraic structuresAssociahedra are remarkable CWcomplexes 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 DeligneMumford 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 AlgebraLet I be an ideal of the ring of Laurent polynomials with coefficients in a realvalued 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. 

28th February  Felipe Rincón (QMUL) 
CSM cycles of matroidsAbstract: I will introduce ChernSchwartzMacPherson 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 algebrasAbstract: 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 PierreGuy Plamondon and Sebastian Opper. Our model is based on the representation theory of gentle algebras. By work of HaidenKatzarkovKontsevich and LekiliPolishchuk this gives a model of the partially wrapped Fukaya category of surfaces with stops. 

Oct 18th  Katerina Hristova (Warwick) 
Frobenius Reciprocity for Topological GroupsAbstract: 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 YangBaxter equationAbstract: Generalizing work of Maulik and Okounkov, we explain how to use certain intertwiners of highestweight modules for Virasoro algebras to produce solutions to the YangBaxter equation. The proof uses the geometry of the Hilbert scheme of points on a surface. 

Nov 1st  Maura Paterson (Birkbeck) 
ReciprocallyWeighted External Difference Families and the Bimodal PropertyAbstract: Let G be a finite abelian group of order n. An (n,k,λ) mExternal 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 reciprocallyweight 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 presentationsAbstract: 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 trianglefree 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 groupsAbstract: The table of marks of a finite group G characterises the actions of G on the transitive Gsets, 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. 

Nov 22nd  Radha Kessar (City) 
Weight conjectures for fusion systemsAbstract: I will present joint work with Markus Linckelmann, Justin Lynd, and Jason Semeraro connecting localglobal 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 fieldsAbstract: The new results described in this talk were proved jointly work with Charles LeedhamGreen and Eamonn O’Brien. 

Dec 6th  Haralampos Geranios (York) 
New families of decomposable Specht modulesAbstract: 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 wellknown 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 superalgebraAbstract: 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). 

2nd July  Michael Batanin (Macquarie University) 
Deformation complex of a tensor category is an E_3algebra.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 2cubes operad. 
Spring 2018
In Spring Term 2018, the LAC was held at City University.
18th January  Simon Peacock (Bristol) 
Representation dimension and separable equivalencesAbstract: 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. 

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


1st February  Joseph Karmazyn (Sheffield) 
Equivalences of singularity categories via noncommutative algebrasAbstract: 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. 

8th February  Eugenio Giannelli (Cambridge) 
Restriction of characters to Sylow psubgroupsAbstract: 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 psubgroups. 

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

22nd March  David Pauksztello (Lancaster) 
Silting theory and stability spacesAbstract: 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 socalled siltingdiscrete 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) 
Nonstandard syzygies over quaternion groupsAbstract: 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 prop group of upper triangular matricesAbstract: In this talk, we will discuss a prop group G whose finite quotient groups give your “favourite” Sylow psubgroups 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 padic variant of G and Hausdorff dimensions of some closed subgroups. 

12th April  Sira Gratz (Glasgow) 
Homotopy invariants of singularity categoriesAbstract: 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 selfinjective 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^1homotopy 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) 
HarishChandra Induction and Lusztig’s Jordan Decomposition of Characters  
22nd June  Arik Wilbert (Bonn) 
Twoblock Springer fibers and Springer representations in type DAbstract: We explain how to construct an explicit topological model for every twoblock Springer fiber of type D. These socalled topological Springer fibers are homeomorphic to their corresponding algebrogeometric Springer fiber. They are defined combinatorially using cup diagrams which appear in the context of finding closed formulas for parabolic KazhdanLusztig 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 modulesAbstract: 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 Zgrading 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 semiorthogonal decompositionsAbstract: Motivated by the local flavor of several wellknown semiorthogonal 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, blowups and root constructions. Our technique simplifies the proof of these decompositions and establishes them in greater generality. We also discuss semiorthogonal decompositions for BrauerSeveri varieties. 