Below is a preliminary timetable. Scroll down for a detailed programme including the titles and abstracts we have received so far.

Registration will be from 08:30 on Wednesday 24th august, in the foyer for the Langdale Building, where all lectures will be held.

Titles & Abstracts

Wednesday

9:00 - Markus Linckelmann (City, University of London): Block Theory over Local Rings (part 1)

10:00 - Alexander Zimmermann (University of Picardie): Clifford theorem for orbit categories

Abstract: Orbit categories first appeared in Gabriel and Riedtmann's work on representation finite algebras. They were used in the sequel in particular in the context of Hall algebras and quantum groups, and more recently in categorifications of cluster combinatorics. We prove a Clifford like theorem on orbit categories. The necessary hypotheses are motivated by Auslander-Kleiner's approach to Green correspondence for pairs of adjoint functors.

Morning Coffee (11:00 - 11:30)

11:30 - Mark Wildon (Royal Holloway, University of London): An introduction to modular plethysms

Abstract: The plethysm product on symmetric functions corresponds to composition of polynomial representations of general linear groups. In recent work with de Boeck and Paget we used representation theory to prove new results on plethysms and give unified proofs of several older results. In my talk I will give an introduction to this area. I will finish with a recent result obtained with McDowell that give a version of Hermite reciprocity valid over an arbitrary commutative ring, and explain some of the plethystic identities that it categorifies. No prior knowledge of plethysms will be assumed.

Lunch Break (12:30 - 14:00)

14:00 - Florian Eisele (University of Manchester): Rigidity and Lifting to Local Rings (part 1)

Afternoon Tea (15:00 - 15:30)

15:30 - John MacQuarrie (Federal University of Minas Gerais): Blocks and representations of finite and profinite groups

Abstract: I'll give an elevator pitch for several aspects of block and representation theory involving inverse limits (that is,"pseudocompact" things).  I'll discuss a generalization from 2018 of Weiss' famous theorem for detecting permutation modules for finite p-groups, which applies to infinitely generated pseudocompact lattices over a complete DVR in mixed characteristic (joint work with Peter Symonds and Pavel Zalesskii).  I'll also talk about more recent work (separate projects with Ricardo Franquiz Flores and Peter Symonds) where we study the local-global approach to the block theory of profinite groups: I'll mention very clean analogues of Brauer's First Main Theorem and of the characterization of blocks with cyclic defect group.  I will try to emphasise how remarkably well behaved the block theory of profinite groups seems to be, and my belief that a huge amount of  block theory should pass unscathed from finite to profinite groups.  I'll mention, when they come up, some points where DVRs are better, and others where fields are better.  WARNING: I don't know what an elevator pitch is and probably the lift will go up and down several times.

Thursday

9:00 - Markus Linckelmann (City, University of London): Block Theory over Local Rings (part 2)

10:00 - Caroline Lassueur (TU Kaiserslautern): On the trivial source character tables of finite groups

Abstract: The aim of this talk is to review results obtained towards the calculation of trivial source character tables of ”small” finite groups, and the creation of a database of such tables.

Morning Coffee (11:00 - 11:30)

11:30 - Bernhard Böhmler (TU Kaiserslautern): On the computation of trivial source character tables using computer algebra

Abstract: Trivial source modules, also known as p-permutation modules, arise naturally in the representation theory of finite groups. They are the building blocks for p-permutation equivalences, splendid Morita equivalences and splendid derived equivalences. In order to do calculations with trivial source modules the ordinary characters of their lifts from positive characteristic p to characteristic zero are of particular interest. The “trivial source character tables” or “species tables” collect information about the character values of trivial source modules with all possible vertices, as well as those of their Brauer constructions.

Let G be a finite group and k be a large enough field of characteristic p dividing |G|. The Brauer correspondence establishes a bijection between the isomorphism classes of indecomposable kG-modules with trivial source and vertex P and the isomorphism classes of non-zero projective indecomposable k[N_G(P)/P]-modules. Therefore, the projective indecomposable modules are of particular interest when focusing on computer algebra.

In this talk, we explain how to explicitly calculate matrix representations of projective indecomposable modules using the open source computer algebra system GAP. Moreover, we comment on how these results can be applied to the computation of the whole trivial source character table of kG.

12:00 - Jialin Wang (Nanyang Technological University): The rank varieties of some simple modules for symmetric groups

Abstract: For a field F of characteristic p>0, Carlson defined the rank varieties of modules for elementary abelian p-groups over F.

For a module M of any finite group G over F, one can also define the rank variety of M restricted to some elementary abelian p-subgroup of G. In particular, such rank varieties detect the projectivity and the corresponding dimensions are closely related to the complexities. In this talk, I will examine some specific simple modules for symmetric groups and determine the corresponding rank varieties and complexities.

Lunch Break (12:30 - 14:00)

14:00 - Michael Livesey (University of Manchester): New Developments on Donovan's Conjecture (part 1)

Afternoon Tea (15:00 - 15:30)

15:30 - Leo Margolis (ICMAT): Modular Isomorphism Problem - progress, solution and open challenges

Abstract: One instance where the benefit of working over local rings instead of fields is evident are isomorphism problems of group rings: while it is known since the late 1980’s that when G is a finite p-group an isomorphism of group rings RG ≅ RH implies that G and H are isomorphic for R the ring of p-adic integers, the question whether this is also true when R a field of characteristic p, known as the Modular Isomorphism Problem, has been open for a long time and only rather weak positive results are available.

I will point out why it seems more difficult to work with fields and report on progress achieved with various colleagues during the last years, including a negative solution of the problem for 2-groups, as well as mention questions remaining open. This will include joint work with Diego García-Lucas, Tobias Moede, Ángel del Río, Taro Sakurai and Mima Stanojkovski.

Friday

9:00 - Florian Eisele (University of Manchester): Rigidity and Lifting to Local Rings (part 2)

10:00 - Cedric Bonnafé (CNRS, University of Montpellier): From finite to p-adic reductive groups

Abstract: The knowledge of representations of finite reductive groups (in transverse characteristic) helps to understand the level 0 representations of a p-adic reductive group. We will explain recent works of Dat, Helm, Lanard, Li, Shotton in this direction.

Morning Coffee (11:00 - 11:30)

11:30 - Cesare Ardito (University of Manchester): Classifying blocks

Abstract: I will give an overview of the current methods used to classify blocks up to Morita equivalence, with a few examples (some work in progress).

12:00 - John McHugh (UCSC): On the image of the trivial source ring in the ring of virtual characters of a finite group

Abstract: We examine the cokernel of the canonical homomorphism from the trivial source ring of a finite group to the ring of p-rational complex characters. We use Boltje and Coşkun's theory of fibered biset functors to determine the structure of the cokernel. An essential tool in the determination of this structure is Bouc's theory of rational p-biset functors.

Lunch Break (12:30 - 14:00)

14:00 - Michael Livesey (University of Manchester): New Developments on Donovan's Conjecture (part 2)

Afternoon Tea (15:00 - 15:30)

15:30 - Robert Boltje (UCSC): The trivial source ring, coherent character tuples, and orthogonal units

Abstract: Direct summands of permutation modules over complete discrete valuation rings seem to play a special role in the representation theory of finite groups. We show that their representation ring, the trivial source ring, is isomorphic to a ring of coherent tuples of characters. For this description it is essential to work over a complete discrete valuation ring versus working over its residue field. As an application we describe the group of orthogonal units of the trivial source ring, a group that is related to groups of auto-equivalences of blocks of group algebras. This is joint work with Rob Carman.