Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Simon Riche
Université Blaise Pascal, Clermont Ferrand II
Constructible sheaves of affine Grassmannians and geometry of the dual group
Works of many others (including Kazhdan-Lusztig, Ginzburg, Mirkovic-Vilonen, Bezrukavnikov) show that
there exist fruitful connections between the algebraic geometry of some varieties attached to a reductive
group and the constructible geometry (or topology) of some varieties attached to the loop group of the
dual reductive group (in the sense of Langlands). I will describe new results in this direction obtained in
joint works with Pramod Achar and Victor Ginzburg.
Sarah Scherotzke
Université Paris Diderot - Paris 7
Linear recurrence relations for cluster variables of affine quivers
Recently, Assem Reutenauer and Smith have introduced families of sequences associated to the vertices
of an acyclic quiver Q. These sequences consist of cluster variables. They proved that if the sequences
associated with Q satisfy linear recurrence relations, then Q is necessarily affine or Dynkin. Conversely,
they conjectured that the sequences associated with a quiver of Dynkin or affine type always satisfy linear
recurrence relations. In my talk I will present a proof of the Assem-Reutenauer-Smith conjecture using the
representation-theoretic approach to cluster algebras. More precisely, our main tool is the categorification
of acyclic cluster algebras via cluster categories. This is joint work with Bernhard Keller.
Leonard Scott
University of Virginia
New graded methods in representation theory
This talk describes recent work with Brian Parshall. Given a finite dimensional algebra A, there is a familar
graded algebra grA obtained by summing quotients of successive powers of the radical of A. Forming grA
might be a first step in, say, trying to find a Koszul algebra related to A. However, it is notoriously difficult to
prove Koszulity of grA or other nice properities, such as quasi-heredity (even when A is quasi-hereditary).
We succeed in proving both Koszulity and quasi-heredity of grA in some familiar cases involving algebras
arising from quantum and algebraic groups. Known Koszulity results of Andersen-Jantzen-Soergel for the
small quantum group and characteristic p > 0 restricted enveloping algebra are a starting point, though
some methods are quite general. The results apply in type A to q-Schur algebras at a root of unity (char.
0) and Schur algebras (char. p), with restrictions.
Literatur
H. Andersen, J. Jantzen, W. Soergel, Representations of quantum groups at a pth root of unity and of
semisimple groups in characteristic p, Astérique 220 (1994).
B. Parshall and L. Scott, New graded methods in representation theory. arXiv:0910.0633.
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