Spring Term 2020
In Spring Term 2020, the LAC will be hosted by City, University of London. Each talk will take place at 17:0018:00 in the room ELG 04 (unless otherwise stated). Tea and coffee will be available before seminars from about 16:15 in the Mathematics Common Room E 215.
23 January  Brita Nucinkis (Royal Holloway, University of London)  An irrational slope Thompson’s group 
30 January  Justin Lynd (University of Louisiana)  The BensonSolomon fusion systems 
6 February  Cheryl Praeger (The University of Western Australia)  Diagonal structures and primitive permutation groups 
13 February  Dave Benson (University of Aberdeen)  Some exotic tensor categories in prime characteristic 
20 February  Jay Taylor (University of Southern California)  Unitriangularity of Decomposition Matrices of Unipotent Blocks 
12 March  John Murray (National University of Ireland, Maynooth)  Brauer characters and normal subgroups 
Abstracts:
Brita Nucinkis
Title: 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 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.
Justin Lynd
Title: The BensonSolomon 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 psubgroup
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 plocal 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 oneparameter 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 BensonSolomon
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 2modular “simple modules” would the principal 2block of Sol(q)
have if it were a group? In various combinations, this is joint work
with E. Henke, A. Libman, and J. Semeraro.
Cheryl Praeger
Title: 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’.
Dave Benson
Title: Some exotic tensor categories in prime characteristic
Abstract:
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/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.
Jay Taylor
Title: 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.
John Murray
Title: 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 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
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 realvalued nonorthogonal irreducible
2Brauer 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 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.
Previous seminars
Autumn 2019
In Autumn Term 2019, the LAC was hosted by Imperial College. Each talk will be 4:455:45, and will take place in room Huxley 139 (unless otherwise stated). Tea and coffee available from about 4:00 in the Mathematics and Computing Common Room, Huxley 549.
10th October  Alastair Litterick (Essex)  Rigidity and representation varieties 
17th October  Lewis Topley (Birmingham/Kent)  Yangians and representations of the general linear Lie algebra in positive characteristic 
24th October  Nick Gill (South Wales)  Some interesting statistics concerning finite primitive permutation groups 
31st October  Michele Zordan (Imperial)  Zeta functions of groups, model theory and rationality 
7th November  Anitha Thillaisundaram (Lincoln)  Maximal subgroups of GrigorchukGuptaSidki (GGS)groups 
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) 
Alastair Litterick
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 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.
Lewis Topley
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 Walgebra. The main result of this talk is a joint work with Simon Goodwin, in which we show that these restricted finite Walgebras can be described explicitly as certain subquotients of a Yangian.
Nick Gill
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 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.
Anitha Thillaisundaram
Abstract: 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.
Francois Thilmany
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 will be hosted in Queen Mary, University of London in room W316 of the Queens’ Building (with one exception). Each talk will be 4:455:45.
17th January  John R. Parker (Durham)  Constructing fundamental polyhedra for groups generated by complex reflections 
24th January  Eleonore Faber (Leeds)  Reflections, rotations, and singularities via the McKay correspondence 
7th February  Vladimir Dotsenko (Trinity College Dublin)  Three guises of toric varieties of Loday’s associahedra and related algebraic structures 
14th February  Zeinab Toghani (QMUL)  Tropical Differential Algebra 
21st February  TBA  TBA 
28th February  Felipe Rincón (QMUL)  CSM cycles of matroids 
Felipe Rincón (Queen Mary).
CSM cycles of matroids.
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.
Zeinab Toghani (Queen Mary).
Tropical Differential Algebra.
Let 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.
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.
Vladimir Dotsenko (Trinity College, Dublin).
Three guises of toric varieties of Loday’s associahedra and related
algebraic structures.
Associahedra 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.
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
socalled 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 CoxeterDynkin diagrams.
In this talk I will first review the history of this fascinating result, and then give an outlook on recent joint work with RagnarOlaf
Buchweitz and Colin Ingalls about a McKay correspondence for finite reflection groups in GL(n,C).
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 (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.
An application of this algorithm is that it gives fundamental domains for all known
(commensurability classes) of nonarithmetic lattices in PU(2,1).
Autumn 2018
In Autumn Term 2018, the LAC was hosted in Birkbeck, University of London in room 745 of the main Malet Street building. Directions available on request. We will convene for tea and coffee in the same room at 3:15. Each talk will be 3:454:45.
Oct 4th  Richard Webb (Cambridge) CANCELLED  The conjugacy problem in mapping class groups 
Oct 11th  Sibylle Schroll (Leicester)  On the geometric model for the bounded derived category of gentle algebras 
Oct 18th  Katerina Hristova (Warwick)  Frobenius Reciprocity for Topological Groups 
Oct 25th  Noah Arbesfeld (Imperial)  Virasoro algebras and the YangBaxter equation 
Nov 1st  Maura Paterson (Birkbeck)  ReciprocallyWeighted External Difference Families and the Bimodal Property 
Nov 8th  Gerald Williams (Essex)  Generalized graph groups with balanced presentations 
Nov 15th  Brendan Masterson (Middlesex)  On the table of marks of a direct product of finite groups 
Nov 22nd  Radha Kessar (City)  Weight conjectures for fusion systems 
Nov 29th  Derek Holt (Warwick)  Polynomial time computation in matrix groups over finite fields 
Dec 6th  Haralampos Geranios (York)  New families of decomposable Specht modules 
Haralampos Geranios (York) New families of decomposable Specht modules
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.
Derek Holt (Warwick) Polynomial time computation in matrix groups over finite fields
The new results described in this talk were proved jointly work with Charles LeedhamGreen 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.
Radha Kessar (City) Weight conjectures for fusion systems
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.
Brendan Masterson (Middlesex) On the table of marks of a direct product of finite groups
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.
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.
Gerald Williams (Essex) Generalized graph groups with balanced presentations
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.
Maura Paterson (Birkbeck) ReciprocallyWeighted External Difference Families and the Bimodal Property
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.
Noah Arbesfeld (Imperial) Virasoro algebras and the YangBaxter equation
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.
Katerina Hristova (Warwick) Frobenius Reciprocity for Topological Groups
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.
Sibylle Schroll (Leicester) On the geometric model for the bounded derived category of gentle algebras
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.
Richard Webb (Cambridge) The conjugacy problem in mapping class groups
I will discuss the conjugacy problem in mapping class groups of surfaces, specifically the result that it can be solved in polynomial time, or even quadratic time, in the word length of the input. This includes the braid groups and outer automorphism groups of surface groups. Joint work with Mark Bell.
Summer 2018
In Summer Term 2018, the following LAC talks took place at City, room ELG04. The talks will begin at 14.00, with tea and coffee served afterwards.
19th June  Sigiswald Barbier (Gent)  A minimal representation of the orthosymplectic Lie superalgebra 
2nd July  Michael Batanin (Macquarie University)  Deformation complex of a tensor category is an E_3algebra. 
Sigiswald Barbier (Gent)
A minimal representation of the orthosymplectic Lie superalgebra
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,q2n) using Jordan superalgebras. This representation also has an annihilator ideal equal to a Josephlike ideal. I will also mention the obstacles which prevent a straightforward generalisation to other Lie superalgebras.
——————
Michael Batanin (Macquarie University)
Deformation complex of a tensor category is an E_3algebra.
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.
DavydovYetter 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 2cubes we get little 3cubes 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 (room ELG08), organized by Joseph Chuang and Jorge Vitória. The talks begin at 17.00, with tea being served in the common room at 16.30.
18th January  Simon Peacock (Bristol)  Representation dimension and separable equivalences 
25th January  Ivan Tomašić (Queen Mary)  Cohomology of difference algebraic groups 
1st February  Joseph Karmazyn (Sheffield)  Equivalences of singularity categories via noncommutative algebras 
8th February  Eugenio Giannelli (Cambridge)  Restriction of characters to Sylow psubgroups 
15th February  Ivo Dell’Ambrogio (Lille)  A categorification of the representation theory of finite groups 
1st March  Greg Stevenson (Glasgow) (CANCELLED)  (Some of) What I don’t know about the Kronecker quiver 
22nd March  David Pauksztello (Lancaster)  Silting theory and stability spaces 
29th March  Wajid Mannan (Queen Mary)  Nonstandard syzygies over quaternion groups 
5th April  Nadia Mazza (Lancaster)  On a prop group of upper triangular matrices 
12th April  Sira Gratz (Glasgow)  Homotopy invariants of singularity categories 
Abstracts
Simon Peacock (Bristol)
Representation dimension and separable equivalences
The representation dimension of an algebra is a finite integer that is supposed to indicate how complicated an algebra’s module category is. This dimension was first introduce by Auslander in 1971 and is, in general, notoriously hard to compute. This measure is related to the representation type of an algebra: an algebra has finite representation type if and only if it’s representation dimension is less than 3.
Separable equivalence is an equivalence relation on finite dimensional algebras. Over a field of a characteristic p, a group algebra is separably equivalent to the group algebra of its Sylow psubgroup. We use this relationship between a group and its Sylows to put an upper bound on the representation dimension of a group algebra for any finite group with a elementaryabelian Sylow subgroup.
——————
Ivan Tomašić (Queen Mary)
Cohomology of difference algebraic groups
Difference algebra studies algebraic structures equipped with an endomorphism/difference operator, and difference algebraic varieties are defined by systems of difference polynomial equations over difference rings and fields. In this talk, we will:
— argue that twisted groups of Lie type are best viewed as difference algebraic groups;
— develop the cohomology theory of difference algebraic groups;
— compute the cohomology in a number of interesting cases, and discuss its applications.
——————
Joseph Karmazyn (Sheffield)
Equivalences of singularity categories via noncommutative algebras
Singularity categories are triangulated categories occurring as invariants associated to singular algebras. For hypersurface singularities these categories can be realised via matrix factorisations, and in this case Knorrer periodicity constructs equivalences between the singularity categories of many different hypersurfaces.
I will discuss these ideas, and talk about how equivalences of singularity categories in the nonhypersurface (and nonGorenstein) setting can be constructed by considering quasihereditary noncommutative resolutions produced from certain geometric situations. In addition, Ringel duality has a very explicit description and interpretation for these quasihereditary algebras.
——————
Eugenio Giannelli (Cambridge)
Restriction of characters to Sylow psubgroups
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.
I will present some recent results on the topic.
——————
Ivo Dell’Ambrogio (Lille)
A categorification of the representation theory of finite groups
Dress’s theory of Mackey functors is a successful axiomatization of the representation theory of finite groups, capturing the formal aspects of such classical invariants as the character ring or group (co)homology. But, typically, each such invariant is only a partial shadow (consisting of abelian groups and homomorphisms) of a richer structure (consisting of additive, abelian or triangulated categories and suitable functors between them).
In joint work with Paul Balmer, we develop a theory of “Mackey 2functors” in order to study this higher structure, thus explaining certain phenomena which, though invisible to classical Mackey functors, occur throughout equivariant mathematics.
In this talk I will provide examples of Mackey 2functors, such as derived and stable module categories in representation theory or equivariant stable homotopy categories in topology, I will motivate our axioms and explain the first results of the theory.
——————–
Greg Stevenson (Glasgow) (CANCELLED)
(Some of) What I don’t know about the Kronecker quiver
I’ll discuss some open questions concerning representations of the Kronecker quiver. By now it’s fair to say we understand the finite dimensional representations fairly well, so we’ll concentrate on what can be said about understanding the representations of infinite dimension. There are a number of challenges still remaining and, as it happens, we know a lot less about the Kronecker quiver (in some senses) than we do about polynomial rings for instance. I’ll make precise what it might mean to ‘understand’ the infinite dimensional representations, give an overview of what I do know, discuss what I hope is tractable, and try to indicate why better understanding the Kronecker quiver is a good warmup for a number of important problems in representation theory and geometry.
———————
David Pauksztello (Lancaster)
Silting theory and stability spaces
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.
———————
Wajid Mannan (Queen Mary)
Nonstandard syzygies over quaternion groups
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.
————————
Nadia Mazza (Lancaster)
On a prop group of upper triangular matrices
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.
————————
Sira Gratz (Glasgow)
Homotopy invariants of singularity categories
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 on Thursdays throughout the term, and will begin at 5pm unless otherwise stated. The room will be Huxley 130 unless otherwise stated. Organizer: John Britnell
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 2  no colloquium  
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 Double  3.30pm Charlotte Kestner (Imperial)  Strongly Minimal Semigroups  
5.00pm 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 (ELG04 City University)  Jay Taylor (Arizona)  HarishChandra Induction and Lusztig’s Jordan Decomposition of Characters 
22nd June (ELG04 City University)  Arik Wilbert (Bonn)  Twoblock Springer fibers and Springer representations in type D 
29th June (ELG08 City University)  Benjamin Briggs (Bonn)  The characteristic action of Hochschild cohomology, and Koszul duality 
4th July (ELG08 City University)  Andrew Mathas (Sydney)  Jantzen filtrations and graded Specht modules 
20th July (ELG08 City University)  Olaf Schnürer (Bonn)  Geometric applications of conservative descent for semiorthogonal decompositions 
Abstracts:
Title: Geometric applications of conservative descent for semiorthogonal decompositions
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.
This is joint work with Daniel Bergh (Copenhagen).
Title: Jantzen filtrations and graded Specht modules
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.
Title: Twoblock Springer fibers and Springer representations in type D
Abstract: 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.
Spring 2017
In Winter/Spring 2017 the LAC was hosted at Queen Mary, University of London, organized by John Bray. The seminars will usually begin at 4:45pm (the traditional LAC start time). They all take place in the Fogg Lecture Theatre, Fogg Building (SBCS).
19 January (5pm)  Alex Fink (Queen Mary)  Characteristic polynomials from reciprocal planes in two ways 
26 January  Ben Fairbairn (Birkbeck)  A Baby, Some Bathwater & What I Did on my Holidays 
2 February  Behrang Noohi (Queen Mary)  Explicit HRStilting 
9 February  Rieuwert J. Blok (Bowling Green State University, Ohio, visiting Birmingham (UK))  CANCELLED, owing to illness. 
16 February  Chimere S. Anabanti (Birkbeck)  Three questions of Bertram on locally maximal sumfree sets. 
23 February  Rieuwert J. Blok (Bowling Green State University, Ohio, visiting Birmingham (UK))  3spherical Curtis–Tits groups 
2 March  Michael Wibmer (University of Pennsylvania)  Differential Embedding Problems over Complex Function Fields 
16 March  Susama Agarwala (US Naval Academy)  Mixed Tate Motives from Graphs 
23 March  Ben Smith (QMUL)  An Algebraic Approach to Generalised Frobenius Numbers 
30 March  Alla Detinko (St Andrews)  TBA 
In Autumn 2016 the LAC was hosted at City University, organized by Chris BowmanScargill. The seminars began at 5pm. Room details will be added to the list of seminars below shortly.
6 October ELG08  Michael Bate (York)  Geometric Invariant Theory without Etale Slices 
13 October EM01  Lewis Topley (Bristol)  Modular finite Walgebras and their applications 
20 October EM01  Jan Grabowski (Lancaster)  Recovering automorphisms of quantum spaces 
27 October EM01  Robert Marsh (Leeds)  Dimer models and cluster categories of Grassmannians 
3 November EM01  Jorge Vitoria (City)  Silting modules and ring epimorphisms 
10 November EM01  Florian Eisele (City)  Tame blocks 
17 November EM01  Neil Saunders (City) 
On the Exotic Springer Correspondence

24 November EM01  Kevin McGerty (Oxford)  Kirwan surjectivity for quiver varieties 
1 December ELG08  Mark Wildon (Royal Holloway)  Plethysms: permutations, weights and Schur functions 
8 December ELG08  Tim Burness (Bristol)  Generating simple groups and their subgroups 