Conference: Perspectives in Representation Theory

Schedule of Talks

Titles and Abstracts

Bangming Deng (Beijing Normal University, China)
Hall algebras of cyclic quivers and affine quantum Schur algebras
In this talk we first define surjective algebra homomorphisms from double Hall algebras of cyclic quivers to affine quantum Schur algebras and then discuss the structure of affine quantum Schur algebras. This is based on a joint work with J. Du and Q. Fu.

Peter Fiebig (University of Freiburg, Germany)
Moment graphs in representation theory and topology
We give an introduction to the theory of moment graphs and its applications towards multiplicity conjectures in representation theory and modular smoothness of Schubert varieties.

Stéphane Gaussent (University of Nancy, France)
Onesketeton galleries and Hall polynomials
I will report on a joint work with Peter Littelmann. To any finite Weyl group, one can associate the algebra of symmetric polynomials. It has two natural bases: the Schur polynomials and the monomial polynomials. The Hall-Littlewood ones form another basis that interpolates between those two. Using a geometrical interpretation of them, we obtain a combinatorial formula for the coefficients appearing in the expansion of Hall-Littlewood polynomials in terms of monomial ones. This formula is in the same spirit as the one that Schwer proved. Except that we use oneskeleton galleries instead of galleries of alcoves, therefore it should be thought of as a compression of Schwer's one.

Meinolf Geck (University of Aberdeen, UK)
Some problems in the representation theory of Hecke algebras
Hecke algebras arise naturally in the representation theory of finite groups of Lie type, as endomorphism algebras of certain induced representations. We will discuss some important open questions in this area, most notably James' Conjecture and Lusztig's Conjectures on Hecke algebras with unequal parameters.

Anne Henke (University of Oxford, UK)
Brauer algebras and their Schur algebras
Schur-Weyl duality relates the representation theory of two algebras. In 1937, Brauer asked the following question: which algebra has to replace the group algebra of the symmetric groups in the set-up of Schur-Weyl duality if one replaces the general linear group by its orthogonal or symplectic subgroup. As an answer he defined an algebra which is a special case of what nowadays is called Brauer algebra. In the situation of the symmetric groups $\Sigma_r$ and general linear groups $GL_n$, Schur algebras are defined as the ring of those endomorphisms of the $r$-fold tensor space of an $n$-dimensional vector space that commute with the symmetric group action. Alternatively, a Morita equivalent version of the Schur algebra can be defined as the endomorphism ring of permutation modules for symmetric groups. In the situation of the Brauer algebras and orthogonal/symplectic groups, the two definitions indicated above lead to different Schur algebras. The talks will discuss Schur algebras corresponding to Brauer algebras, in particular the Schur algebras defined via permutation modules.

Michail Kapovich (University of California, Davis, USA)
Lie theory for exotic finite dihedral groups
I will describe our work with Arkady Berenstein on developing a "Lie Theory'' where the role of Weyl groups is played by arbitrary dihedral groups. In particular, I will talk about geometry of associated spherical and Euclidean buildings, decomposition of tensor products and a candidate construction for the actual infinite-dimensional Lie algebra.

Radha Kessar (University of Aberdeen, UK)
Numerical equalities and categorical equivalences in block theory
I will discuss some approaches to and results on the structure of blocks of modular group algebras.

Bernard Leclerc (University of Caen, France)
Cluster algebras and representation theory
Cluster algebras have been introduced by Fomin and Zelevinsky in 2001, motivated by various combinatorial problems arising notably in Lie theory. After a quick introduction to cluster algebras, I will review a series of joint works with C. Geiss and J. Schröer about cluster algebra structures on coordinate rings of unipotent groups and flag varieties. In the last lecture I will discuss a recent joint paper with D. Hernandez which shows that certain tensor categories of representations of quantum affine algebras also have interesting cluster structures.

Jacques Thévenaz (EPFL Lausanne, Switzerland)
Endo-trivial modules for finite groups
1. Endo-trivial modules for finite groups. Endo-trivial modules for a finite group G are representations of G in characteristic p which play an important role in modular representation theory and block theory. In the case of a p-group, they have been classified in 2004. This talk will give an introduction to the subject. 2. Endo-trivial modules: present and future. After the classification of all endo-trivial modules for p-groups, several results have been obtained recently for other groups. However, a classification in general seems to be hard. This talk will give an overview on some of the questions involved in the subject.

Will Turner (University of Aberdeen, UK)
Highest weight categories which are Calabi-Yau-0
Category theory is a notoriously pointless subject. If we wish to make it more pointed, we must place strong restrictions on the theorised categories. Two strong categorical restrictions are the highest weight restriction and the Calabi-Yau-0 restriction. The highest weight restriction shows up in the representation theory of algebraic groups, whilst the Calabi- Yau-0 restriction shows up in the representation theory of finite groups. I will discuss recent work on categories in which both of these restrictions are assumed to hold. There are connections to representation theory, the theory of tilings, and algebraic geometry.

Carlos André (University of Lisbon, Portugal)
Irreducible characters of groups associated with finite involutive algebras
An algebra group is a group of the form G = 1+J where J = J(A) is the Jacobson radical of a finite-dimensional associative algebra A (with identity). A theorem of Z. Halasi asserts that, in the case where A is defined over a finite field F, every irreducible complex representation of G is induced by a linear representation of a subgroup of the form H = 1+J(B) for some subalgebra B of A. If we assume that F has odd characteristic p and (A,s) is an algebra with involution, then s naturally defines a group automorphism of G = 1+J, and we may consider the fixed point subgroup G(s). In this situation, we show that every irreducible complex representation of G(s) is induced by a linear representation of a subgroup of the form H(s) where H = 1+J(B) for some s-invariant subalgebra B of A. As a particular case, the result holds for the Sylow p-subgroups of the classical groups of Lie type.

Hannes Bierwirth (University of Málaga, Spain)
On graded Lie algebras
I study the Lie structure of graded associative algebras. We derive, as consequences, examples of algebras of quotients of graded Lie algebras and show that the Lie algebra of graded derivations of a graded associative algebra is graded strongly nondegenerate when the center does not contain nozero associative ideals.

Evgeny Feigin (Lebedev Physics Institute, Moscow, Russia)
Multi-component KP construction and the WDVV equation
The celebrated construction of the semi-infinite wedge space due to M.Jimbo and T.Miwa allows to study certain system of differential equations (called the KP hierarchy) in terms of the representation theory of the Lie algebra of infinite matrices. It turned out that the natural multi-component generalization of the semi-infinite wedge space can be used in order to construct solutions of the WDVV (associativity) equation. In the talk we will briefly recall main points of the notions and constructions above. We will also make a link between the representation theory of certain loop groups (acting on the wedge space)and the formalism of Frobenius structures due to A.Givental.

Burkhard Külshammer (University of Jena, Germany)
The depth of subalgebras and subgroups
The depth is a numerical invariant which can be attached to a Frobenius extension of rings. In the talk, we will explore this notion in the context of finite-dimensional algebras, in particular group algebras. This will be a report on joint work with R. Boltje, S. Burciu, S. Danz and L. Kadison.

Artem Lopatin (University of Bielefeld, Germany)
A minimal generating system for semi-invariants of quivers for dimension $(2,\ldots,2)$
We work over an infinite field $K$ of arbitrary characteristic. Given a quiver $Q$, we denote by $I (Q,\underline{n})$ ($SI(Q,\underline{n})$, respectively) the algebra of invariants (semi- invariants, respectively) of representations of $Q$ for dimension vector $\underline{n}$. The generators for $SI(Q,\underline{n})$ are known. In this talk we will explicitly describe a minimal (by inclusion) generation system for $SI(Q,\underline {n})$ for $\underline{n}=(2,\ldots,2)$. Note that the known generating system for $I (Q,\underline{n})$ is ``simpler'' than that for $SI (Q,\underline{n})$. Moreover, the ideal of relations between generators for invariants is known in contrast to semi-invariants. Nevertheless, a minimal generating system for $I(Q,(2,\ldots,2))$ is not known and it is unclear how it can be explicitly described.

Dmitry Malinin (University of Vlora, Albania)
Finite arithmetic groups and Galois operation
abstract

Jürgen Müller (RWTH Aachen, Germany)
Verifying the abelian defect group conjecture for sporadic simple groups
This talk will emphasize the interplay between theoretical and computational techniques. This is a joint work with S. Koshitani and F. Noeske.

Felix Noeske (RWTH Aachen, Germany)
TMatching Simple Modules of Condensed Algebras
Let A be a finite dimensional algebra over a finite field F. Condensing an A-module V with two different idempotents e and e' leads to the problem that to compare the composition series of Ve and Ve', we need to match the composition factors of both modules. In other words, given a composition factor S of Ve, we have to find a composition factor S' of Ve' such that there exists a composition factor T of V with Te = S and Te' = S', or prove that no such S' exists. In this talk, we present a computationally tractable solution to this problem, and illustrate how its application was a crucial step in the recent proof of the 3-modular character table of the sporadic simple Harada Norton group.

Armin Shalile (University of Oxford, UK)
Modular Characters for Brauer algebras
We define modular characters for Brauer algebras which share many of the features of Brauer characters defined for groups. Since notions such as conjugacy classes and orders of elements are not a priori meaningful for Brauer algebras, we show which structure replaces the conjugacy classes and determine eigenvalues associated to these.