BiRep Seminar Archive

The talk is based on the paper "Exceptional Components of Wild Hereditary Algebras" by O. Kerner.
For a wild hereditary algebra the regular components of the AR-quiver are of type ZA_∞. Kerner defines such a component to be regular by the existence and vanishing of maps from an indecomposable module in the component to iterated τ-shifts of that module. In my talk I will give an example of Kerner's construction of exceptional components which arise from embeddings of the module category of a tame algebra. In this example, the exceptional components resemble the tubes in the tame case. Furthermore, I will give a new example of an exceptional component arising from an embedding of the module category of a representation-finite algebra.

Schur algebras are finite-dimensional (quasihereditary) algebras obtained from an algebraic group G by truncating its representation theory at a finite set of dominant weights. This enables one to study the representation theory of G through finite-dimensional algebras. One may replace G by a Frobenius kernel G_r of G or its thickening G_r T by a maximal torus T in G. This leads to "infinitesimal" Schur algebras. There are also q-analogues, obtained by replacing G with an appropriate quantum group. I will give a brief introduction to this area, focusing mainly on Type A.

For an algebraically closed field, Reineke gave a monoid structure on closed irreducible varieties of the representation space. There is a natural submonoid, the composition monoid, which is given by considering representations having a composition series with fixed composition factors in fixed order. We desribe the relation between the composition monoid and the composition algebra at q=0 for Dynkin and extended Dynkin quivers.

For a field k of characteristic p we classify all simple finite dimensional representations of the algebraic group Aut(k[t]/t^p) which is essentially due to Deligne. This enables us to compute the representation ring of this group. The motivation for this is to obtain a universal description of a setting arising from the Frobenius morphism for smooth varieties in K-Theory.

In this talk we consider O(3)-invariants of several matrices over a field of characteristic different from 2. We compute an upper bound on degrees of elements of a minimal generating system and estimate its deviation. This work is based on our previous results describing relations between generators of matrix O(n)-invariants.

Let A be a finite dimensional algebra and D^b(A) the bounded derived module category. Given two partial tilting objects in D^b(A), I will show that under some homological conditions one can obtain a tilting object. These conditions arise naturally in the recollement structure on D^b(A). I will present some examples as well as the construction. This is a joint work with L. Angeleri Hügel and S. König.

Auslander and Ringel-Tachikawa have shown that for an artinian ring R of finite representation type, every R-module is the sum of finitely generated indecomposable R-modules. In this talk, we will adapt this result to finite representation type full subcategories of the module category of an artinian ring which are closed under subobjects and extensions and contain all projective modules.

In particular, the results in this talk hold for subspace representations of a poset, in case this subcategory is of finite type.

There is an "almost" associative multiplication structure for indecomposable objects in a 2-period triangulated category. As an application, we give a new proof of a theorem of Peng and Xiao which provides a way of constructing Lie algebras from two periodic triangulated categories.

The Artin groups of type A are the braid groups; for any Coxeter group, there is an associated Artin group, which is called finite type if the Coxeter group is finite. There is a standard presentation for Artin group analogous to the standard presentation for Coxeter groups. A useful property of the standard presentation for Artin groups of finite type is that there is an associated Garside structure. This gives, for example, an algorithm for computing a normal form for elements of the Artin group.

Bessis introduced a "dual" presentation for finite type Artin groups (extending work of Birman-Ko-Lee in type A) which also has this Garside property, and which is, in some respects, computationally preferable. The proofs of Bessis's results make use of type-by-type arguments and computer checks for the exceptional types. I will explain an alternative approach to this Garside structure (in crystallographic cases only) using the representation theory of Dynkin quivers in which the proofs are carried out in a uniform way. Time permitting, I will also discuss conjectural applications to non-finite-type Artin groups.

Given an embedding of a subgroup $A$ in a finite abelian $p$-group $B$, the LR-sequence for the embedding $(B,A)$ is a sequence of partitions which records the isomorphism types of the elementary embeddings $(B/p^iA,p^{i-1}A/p^iA)$. It turns out that there is no hope for a combinatorial description of the $p^3$-bounded embeddings $(B/p^iA,p^{i-3}A/p^iA)$.

In the talk I will discuss prototypes which were introduced by T. Klein as refinements of LR-sequences. Prototypes are used as tool to give structure to the computation of Hall polynomials, but they also record exactly the isomorphism types of the $p^2$-bounded embeddings $(B/p^iA,p^{i-2}A/p^iA)$.

Let g_e be the centraliser of a nilpotent element e in a finite dimensional simple Lie algebra g of rank l. We consider the algebra S(g_e)^{g_e} of symmetric invariants of g_e and, more generally, the coadjoint representation of g_e. If g is of type A or C, then S(g_e)^{g_e} is always a graded polynomial algebra in l variables. One can explicitly construct the generator. Unfortunately, for the minimal nilpotent element in type E_8 the algebra S(g_e)^{g_e} is not free. This provides also a counterexample to the conjecture of Joseph on the semi-invariants of (bi)parabolics.

For coprime dimension vectors certain torus fixed points of the Kronecker moduli space are indecomposable tree modules. Using their stability and the reflection functor we show that for arbitrary roots there exist indecomposable tree modules of the Kronecker quiver as submodules of these torus fixed points.

In the modular representation theory of finite groups, methods from relative homological algebra appear in interplay of Auslander-Reiten components and Green correspondence. In this talk, I will employ such techniques to discuss AR-components of a finite group G that contain an indecomposable module which is projective relative to some p-point of the group algebra kG. Our aim is to understand the behavior of the Jordan types of modules belonging to components of this type.