# Research Seminar in Representation Theory

11 March 2013, Room A-317

13:00-14:30, Kotaro Kawatani
Introduction to stability conditions I

Abstract: Bridgeland constructed the theory stability conditions on triangulated categories. I will explain algebro-geometric backgrounds of stability conditions and Bridgeland's conjecture on the derived categories of K3 surfaces.

14:45-16:15, Kotaro Kawatani
Introduction to stability conditions II

Abstract: I will give the precise definition of stability conditions and some examples. I also explain some technical issues of the construction of stability conditions on projective manifold. If I have a time, I will state my results for the space of stability conditions on K3 surfaces.

### Previous talks

20 February 2013, Room A-317

13:00-14:30, Martin Kalck
Spherical subcategories

Abstract: Let X be an object with two dimensional graded endomorphism algebra in a Hom-finite algebraic triangulated category T with Serre functor. Then there is a unique maximal subcategory U of T, such that X is a spherical object in U. We discuss examples from representation theory and geometry. This is joint work with Andreas Hochenegger and David Ploog.

13 February 2013, Room A-317

13:00-14:30, Torkil Utvik Stai
The n-periodic derived category

Abstract: Our main objective is to show that if A is an iterated tilted algebra, then the orbit category of D^b(Mod A) modulo shift by n is triangulated in such a way that the canonical functor from D^b(Mod A) becomes a triangle functor.

We shall start, however, by explaining how certain features of an additive category C carry over to the category of n-periodic complexes over C. Then, letting C be the module category of A equipped with a non-canonical exact structure, we will eventually obtain the "n-periodic derived category", which is triangulated and turns out to be equivalent to the above orbit category.

4 February 2013, Room A-317

13:00-14:30, Masahide Konishi
A classification of cyclotomic KLR-algebras of type $A_{n}^{(1)}$.

Abstract: A cyclotomic KLR-algebra is defined from three data; quiver $\Gamma$, and two weights $\alpha$ and $\Lambda$ for vertices of $\Gamma$. Each data gives relations, generators (especially KLR-idempotents) and an ideal respectively. In this talk, we fix $\Gamma$ as type $A_{n}^{(1)}$. In the most "essential" case, we can check that all the KLR-idempotents are primitive(*). Then naturally a question arises; in which case the statement (*) follows? Our goal is giving the answer to the question.

30 January 2013, Room A-317

13:00-14:30, Ryoichi Kase
The pre-projective part of the tilting quiver over a path algebra with some conditions

Abstract: D. Happel and L. Unger defined a partial order on a set of isomorphism classes of basic tilting modules. A tilting quiver is the Hasse-quiver of this poset. In this talk we will give a combinatorial characterization of the pre-projective part of the tilting quiver over a path algebra satisfying some conditions and discuss its structure.

23 January 2013, Room A-317

13:00-14:30, Dong Yang
The singularity category of a radical square zero algebra

Abstract: In this talk, I will speak about the singularity category of a radical square zero algebra, in particular, its relationship to the associated Leavitt path algebra and to a certain trivial extension.

9 January 2013, Room A-317

13:00-14:30, Kenneth Chan
Noncommutative Invariant theory for AS-regular algebras

Abstract: The study of the finite symmetries of k[u,v] is a classical subject with connections to representation theory, algebraic geometry and more recently, noncommutative algebra. The noncommutative incarnation of this theory replaces the k[u,v] with a "quantum polynomial ring" and explores quantum analogues of classical results. These quantum polynomial rings, called Artin Schelter (AS) regular algebras, tend to have small automorphism groups, so we consider Hopf algebra actions in order to reveal their hidden symmetries.

Our main result is the classification of all finite dimensional Hopf algebras acting (satisfying some natural hypotheses) on AS regular algebras of global dimension 2. In this talk, I will proceed from the basic definitions of Hopf algebra actions on rings to explain our classification result.

19 December 2012, Room A-317

14:00-15:30, Boris Lerner
Orders on surfaces

12 December 2012, Room A-317

10:30-12:00, Hyohe Miyachi
On a generic lower bound for diagonal Cartan invariants of finite groups of Lie type in non-defining characteristic

Abstract: By applying Auslander-Reiten theory to finite group algebras and a series of works due to Webb, Erdmann, Okuyama, Benson-Parker, we know the tree classes of Auslander-Reiten components. Especially in the odd prime wild type case, the type is A_{\infty}.

Kawata, Michler and Uno [KMU] tried to determine the positions of simples. In most of the cases, they are at the edge of AR quivers.

Their main idea is to use Kawata's theorem which makes a link between the Cartan invariant and the position of simples.

Motivated those works, we find a lower bound for those Cartan invariants for non isolated (very wide classes) wild blocks of finite groups of Lie type in non-defining characteristic. Along the approach of [KMU], we will know that generically the simples are at the edge.

It turns out that if a (so called) defect group is too small such as p=3 and the elementary abelian group of order 9, I can't find general argument. Actually, in such cases, KMO found a counter example.

21 November 2012, Room I-409

13:00-14:30, Kota Yamaura
When are trivial extensions Iwanaga-Gorenstein algebras?

2 July 2012, Room A-328

15:00-16:30, Erik Darpö
Tensoring modular representations of cyclic two-groups

25 June 2012, Room A-328

15:00-16:30, Laurent Demonet
Categorification of cluster algebra structures of coordinate rings of Grassmanian varieties through representations of preprojective algebras

23 April 2012, Room A-328

15:00-16:30, Hiroyuki Minamoto
n-regular modules from a view point of noncommutative algebraic geometry

2 April 2012, Room A-328

The connection between support tau-tilting modules, silting complexes and cluster-tilting objects

Abstract: In my talk, I shall give a bijection between support tau-tilting modules and two-term silting complexes. Iyama and Reiten show that support tau-tilting modules and cluster-tilting objects are one-to-one correspondence. I shall directly give a bijection between two-term silting complexes and cluster-tilting objects.

26 March 2012, Room A-328

15:00-16:30, Gustavo Jasso
2-representation-finite quasi-canonical algebras

19 March 2012, Room A-428

15:00-16:30, Laurent Demonet
Skew group algebras and bimodule Calabi-Yau algebras

5 March 2012, Room A-328

15:00-16:30, Laurent Demonet
Group actions of categories

27 February 2012, Room A-328

15:00-16:30, Martin Herschend
Higher dimensional Auslander-Reiten theory and quivers with potential II

Abstract: I will continue my previous talk, explaining about the use of QPs in higher Auslander-Reiten theory.

20 February 2012, Room A-328

13:00-14:30, Martin Herschend
Higher dimensional Auslander-Reiten theory and quivers with potential

Abstract: In my talk I will introduce basic notions of higher dimensional Auslander-Reiten theory such as n-representation finite and n-representation infinite algebras and explain how these can be characterized using (n+1)-preprojective algebras. Since 3-preprojective algebras are given by quivers with potential (QPs) this leads to a study of certain graded QPs. I will discuss various properties of QPs that appear in this way and give many examples.