BIREP — Representations of finite dimensional algebras at Bielefeld

• BIREP
• Faculty of Mathematics
• Bielefeld University

 

Seminar on Representation Theory (WS 2014/2015)

( = Seminar zur Darstellungstheorie (eKVV 241030))
Thursday 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 (and beyond)
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):

• Gröbner basis methods for functor categories (Philipp Lampe),
  16th of October - 21th of November,
  Program (pdf)
  The goal of this part of the seminar will be to understand Steven Sam’s proof of the artinian conjecture in generic representation theory. This conjecture is due to Lionel Schwartz (c.f. Kuhn [K1, 3.12]), motivated from algebraic topology and was open for twenty years. Recently, Sam gave a prove in a private communication [S]. Subsequently, together with Putnam and Snowden, respectively, he gave two written proofs [PS, SS]. Both proofs are similar and generalize ideas from the theory of Gröbner bases. We follow Sam and Snowden’s exposition.
• Classification of Lie superalgebras (Julia Sauter)
  Simple Lie superalgebras have been classified by Kac (1975). The main difference to the usual Lie theory is that the definition of the Weyl group for Lie superalgebras is not straight forward, in fact there are several different competing definitions of it. We follow Serganovas approach of the Weyl groupoid and review her more recent notion of a generalised root system which recovers (partly) the classification result of Kac. Possible other topics related only to type A Lie superalgebras are Schur superalgebras, Super Schur Weyl duality, polynomial superfunctors.

Schedule of talks

Please be aware that the schedule might change.

• Oct 16: Introduction to Gröbner basis theory, (Tobias Roßmann)
• Oct 23: Gröbner categories, (Philipp Lampe)
• Oct 30: Quasi Gröbner categories, (Philipp Lampe)
• Nov 6: Higman's lemma and the conclusion to the proof (Li Zhi-Wei)
• Nov 13: (no seminar)
• Nov 21 at 13:15h in X-E0-202: FI-modules, (Kai-Uwe Bux)
• Nov 21 at 15:15h in X-E0-202: Topological origins of the Artinian Conjecture, (Henning Krause)
• Nov 27: Submodule-closed subcategories in type A, (Apolonia Gottwald)
• Dec 4: Mutation Sequences and tilting of hearts, (Florian Gellert)
• Dec 11: Introduction to Lie superalgebras, (Estanislao Herscovich)
• Dec 18: Classification of Lie superalgebras using generalised root systems, (Julia Sauter)
• Jan 8: Tannakian Categories, (Greg Stevenson)
• Jan 15: Generalised Rickard idempotents in tensor triangulated categories, following [BF], (Fajar Yuliawan)
• Jan 22: The Monoidal structure on strict polynomial functors, (Mina Aquilino)
• Jan 29: --- no seminar --
• Feb 5: From submodule categories to the preprojective algebra, following [RZ], (Oegmundur Eiriksson)

Literature for Gröbner basis methods for functor categories

• [A] M. Auslander: Functors and morphisms determined by objects. Representation Theory of Algebras, Lecture Notes in Pure Appl. Math., vol. 37, Marcel Dekker, New York (1978), 1–244.
• [AS] M. Auslander, S. Smalø: Preprojective modules over Artin algebras. Journal of Algebra 66 (1980), 61–122.
• [B] B. Buchberger: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Dissertation, Innsbruck (1965).
• [CLO] D. Cox, J. Little, D. O’Shea: Ideals, Varieties, and Algorithms . Springer, New York (1992).
• [D] A. Djament: La conjecture artinienne, d’apr`es Steven Sam . Private communication.
• [H] G. Higman: Ordering by divisibility in abstract algebras . Proc. London Math. Soc. (3) 2 (1952), 326–336.
• [K] J.B. Kruskal: The theory of well-quasi-ordering: A frequently discovered concept. Journal of Combinatorial Theory A 13 (1972), 297–305.
• [K1] N. Kuhn: Generic representations of the finite general linear groups and the Steenrod algebra: II . K-Theory 8 (1994), 395–428.
• [K2] N. Kuhn: The generic representation theory of finite fields: a survey of basic structure. Infinite Length Modules, Proc. Bielefeld, 1998, Trends in Mathematics, Birkhäuser, Basel (2000), 193–212.
• [N] C. Nash-Williams: On well-quasi ordering finite trees. Proc. Cambridge Phil. Soc. 59 (1963), 833–835.
• [PS] A. Putnam, S. Sam: Representation stability and finite linear groups . arXiv:1408.3694.
• [S] S. Sam: Noetherianity of polynomial functors over finite fields . Private communication.
• [SS] S. Sam, A. Snowden: Gröbner methods for representations of combinatorial categories . arXiv:1409.1670.
• [S2] B. Sturmfels: What is a Gröbner basis? Notices of the AMS 52 (2005), 1199–1200.

Literature on Lie superalgebras

• L. Frappat, A. Sciarrino, P. Sorba: Dictionary on Lie Superalgebras. arXiv:hep-th/9607161
• Serganova, Vera: On generalizations of root systems. Comm. Algebra 24 (1996), no. 13

Further literature

• [BF] P.Balmer, G. Favi: Generalized tensor idempotents and the telescope conjecture
• Ringel, Zhang: From submodule categories to preprojective algebras