This term’s seminars

Autumn 2018

In Autumn Term 2018, the LAC will be 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 3:45-4:45.

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 Yang-Baxter equation
Nov 1st Maura Paterson (Birkbeck) Reciprocally-Weighted 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) TBA

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 Leedham-Green and Eamonn O’Brien.

Over the past 30 years, an algorithm CompositionTree has been developed for enabling practical computation in large matrix groups over finite fields. The principal aim is to find a composition series and membership test for an input group GGL(d,q). This has been implemented in Magma and performs well in practice.

In 2009, Babai, Beals and Seress published a polynomial time algorithm (assuming oracles for integer factorization and discrete logs) with the same aims for odd q. But this is not suitable for implementation.

We have been asked whether it is feasible to show that CompositionTree can be easily adapted to run in polynomial time, and we can now prove that this is possible with a few provisos.

The main new idea is that we can modify our black box algorithms for constructive recogniton of the finite nonabelian simple groups so that, if the input group is not simple, then a nontrivial element in a proper normal subgroup is output.


Radha Kessar (City) Weight conjectures for fusion systems

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.


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 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.


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 triangle-free then the corresponding group cannot be trivial or finite of rank greater than 2. This is joint work with Johannes Cuno.


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

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.


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

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.


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 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.


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 will take 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_3-algebra.

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,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.

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

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.

Previous seminars

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 p-subgroups
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) Non-standard syzygies over quaternion groups
5th April Nadia Mazza (Lancaster) On a pro-p 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 p-subgroup. We use this relationship between a group and its Sylows to put an upper bound on the representation dimension of a group algebra for any finite group with a elementary-abelian Sylow subgroup.

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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.

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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 non-hypersurface (and non-Gorenstein) setting can be constructed by considering quasi-hereditary noncommutative resolutions produced from certain geometric situations. In addition, Ringel duality has a very explicit description and interpretation for these quasi-hereditary algebras.

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

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

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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 2-functors” in order to study this higher structure, thus explaining certain phenomena which, though invisible to classical Mackey functors, occur throughout equivariant mathematics.
In this talk I will provide examples of Mackey 2-functors, such as derived and stable module categories in representation theory or equivariant stable homotopy categories in topology, I will motivate our axioms and explain the first results of the theory.

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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.

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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 so-called silting-discrete algebras. This talk will be a discussion of joint work with Nathan Broomhead, David Ploog, Manuel Saorin and Alexandra Zvonareva.

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Wajid Mannan (Queen Mary)
Non-standard 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.

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

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

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

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

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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 self-injective algebras. Using work of Tabuada, we’ll show how to deduce from this knowledge the homotopy invariants of the ungraded stable categories for such algebras. As another illustration of these ideas we’ll show that cluster categories of Dynkin type A_n, for even n, are “A^1-homotopy phantoms”. All this is based on joint work with Greg Stevenson.

Autumn 2017

The Autumn 2017 seminars will be 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) Harish-Chandra Induction and Lusztig’s Jordan Decomposition of Characters
22nd June (ELG04 City University) Arik Wilbert (Bonn) Two-block 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 semi-orthogonal decompositions

Abstracts:

Title: Geometric applications of conservative descent for semi-orthogonal decompositions

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

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 Z-grading because they are isomorphic to cyclotomic KLR algebras. I will explain how to give an easy proof of the Jantzen sum formula for the Specht modules of the cyclotomic Hecke algebras of type A using the KLR grading. I will discuss some consequences and applications of this approach.

Title: Two-block Springer fibers and Springer representations in type D

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

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 HRS-tilting
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 sum-free sets.
23 February Rieuwert J. Blok (Bowling Green State University, Ohio, visiting Birmingham (UK)) 3-spherical 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 Bowman-Scargill. The seminars began at 5pm. Room details will be added to the list of seminars below shortly.

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6 October ELG08 Michael Bate (York) Geometric Invariant Theory without Etale Slices
13 October EM01 Lewis Topley (Bristol) Modular finite W-algebras 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
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