BIREP — Representations of finite dimensional algebras at Bielefeld

# Seminar on Representation Theory (SoSe 2015)

( = Seminar zur Darstellungstheorie (eKVV 241038))
Wednesday 10-12 h in V5-227 (2 SWS)
Organizer: Prof. Dr. Henning Krause, Dr. Julia Sauter
Content: Selected topics from representation theory of finite dimensional algebras
This is a seminar with varying topics, suggestions are always welcome. In particular, phd students are encouraged to formulate their research needs and interests and we will try to incorporate this in the seminar. Usually we speak English and in exceptional cases German.
At the moment, we plan the following topics (with the person in charge of planning the talks):
• Counting in root posets (Julia Sauter)
8th of April - 29th of April
We follow Ringel's lecture notes, see [R] which is based on a series of talks from the ICRA in 2014. He gave four talks called
• numbers
• hyperplanes
• tilting
• lattices
Additional material for his talks can be found here. We start with his third talk recalling the bijections between directed antichains, thick subcategories and support tilting modules (in the module category of a hereditary artin algebra) due to Ingalls-Thomas and [ONFR]. We follow the appendix of [ONFR].
Then we continue with his talk on lattices. We then take a step backwards and return to root systems and have a closer look at Ringel's Dynkin functions. We also might have a short introduction into species to understand the root systems of type B,C etc. better. At last, we come to hyperplane arrangements (the one defined by the reflections associated to the Weyl group is our standard example) and the associated rings of derivations, here the standard book is Orlik-Terao [OT].
• Kac's theorem (Andrew Hubery)
is a generalization of the well-known theorem of Gabriel. We follow lecture notes of Crawley-Boevey.
• String algebras
String algebras are an interesting class of tame artin algebras. Their indecomposable modules have a description in terms of so-called strings and bands. The main method to produce these modules is by functorial filtration (see the talk on the 8th of July).
• Many related concepts... (each speaker on its own)
We investigate many structures to 'filter' or 'decompose' categories (of modules or their derived categories). Our focus here is to understand the implications or relations between them. A(n incomplete, unordered) list is the following:
torsion theories - t-structures - tilting - exceptional collections - mutations of hearts - recollements - semi-orthogonal decompositions - simple minded systems - quasi-hereditary structures - standardly stratified algebras - cellular algebras - Koszulness
• Orbit closures as quotients (Julia Sauter)
Let Q be a Dynkin quiver. Following Cerulli Irelli, Feigin, Reineke's article [CFR] we construct (Dynkin quiver) orbit closures as affine quotient schemes for certain graded quiver varieties. There are still many questions open like: Can this be used to find a conceptional proof of normality of these orbit closures? Also, there are examples of other representation finite algebras where the same construction seem to work but is not covered by the previous construction (e.g. for the truncated polynomial ring, Kraft and Procesi in [KP] did use precisely this construction).
Furthermore, we have some research talks:
• On relative derived categories speaker: Rasool Hafezi (Isfahan), date: May 13
Abstract: In this talk, I will introduce relative derived category and then discuss about its properties and its connection with ordinary derived category. If time permits, I will explain a triangle equivalence between a sub-triangulated category of homotopy category of Gorenstein projective modules and a localization of homotopy category of acyclic complex of projective modules.
• Tropical curve counting and canonical bases speaker: Travis Mandel (Aarhus, Denmark), date: May 27 in V3-201
Abstract: Gross, Hacking, Keel, and Kontsevich recently constructed certain canonical bases for cluster algebras. The construction is combinatoric, but the bases are conjecturally controlled by the Gromov-Witten theory of the mirror cluster variety. I will discuss a new construction of these bases in terms of certain tropical curve counts which one expects to correspond to the predicted holomorphic curve counts. I will also discuss a refinement of the tropical counts which produces quantized versions of the canonical bases.

# Schedule of talks

Please be aware that the schedule might change.
• Apr 08: Ingalls-Thomas bijections, [ONFR], [R] (Apolonia Gottwald)
• Apr 15: Lattices,[R] (Florian Gellert)
• Apr 22: Root systems and Dynkin functions, [R] (Philipp Lampe)
• Apr 29: Hyperplanes, [OT],[R] (Mina Aquilino)
• May 06: no seminar
• May 13: On relative derived categories, (Rasool Hafezi)
• May 20: Quasi-hereditary structures and recollements, (Rebecca Reischuk)
• May 27: Tropical curve counting and canonical bases, (Travis Mandel) in V3-201
• June 03: Recollements and Tilting objects, (Fajar Yuliawan)
• June 10: Torsion theories and t-structures, (Chao Zhang)
• June 17: Kac's Theorem - part 1, (Oegmundur Eiriksson)
• June 24: Kac's Theorem - part 2, (Andrew Hubery)
• July 01: Orbit closures as quotients, [CFR], [KP] (Julia Sauter)
• July 08: Functorial filtrations, (William Crawley-Boevey)
• July 15: