Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Sebastian Herpel
Ruhr Universität Bochum
On the smoothness of centralizers in reductive groups
Let G be a connected reductive algebraic group over an algebraically closed field. The question whether
the scheme-theoretic centralizer of a closed subgroup of G is smooth, or equivalently whether the
dimensions of the global and infinitesimal centralizers coincide, occurs naturally in many contexts. We
introduce a condition for the characteristic of the ground field that is slightly weaker than the notion
of “very good” characteristic. We go on to show that this condition is necessary and sufficient for the
smoothness of all centralizers of closed subgroup schemes. Reductive groups defined in such “pretty
good” characteristic are closely related to so called standard groups, for instance to groups satisfying the
standard hypotheses of Jantzen.
Gerhard Hiß
RWTH Aachen
Hecke algebras in representation theory
Hecke algebras i.e., endomorphism rings of suitable modules of group algebras, constitute an essential
tool in the representation theory of finite groups. In groups of Lie types, Hecke algebras also arise as
deformations of group algebras of Weyl groups. In this area, applications of Hecke algebras are rather
varied: in non-defining characteristics Hecke algebras serve to classify irreducible representations, in
defining characteristics a famous conjecture of Alperin can be proved with the help of Hecke algebras.
A possible generalisation of this approach to arbitrary groups is investigated in a recent joint paper with
Steffen König and Natalie Naehrig. A further area of applications of Hecke algebras are the condensation
methods in computational representation theory. My talk will give a survey, largely free of technical details,
on the various applications of Hecke algebras.
Bernhard Keller
Université Paris Diderot
Quiver mutation and quantum dilogarithm identities
Quiver mutation is an elementary operation on quivers which appeared in physics in Seiberg duality in
the 1990s and in mathematics in Fomin-Zelevinsky’s definition of cluster algebras in 2002. In this talk,
I will show how, by comparing sequences of quiver mutations, one can construct identities between
products of quantum dilogarithm series. These identities generalize the classical pentagon identity of
Faddeev-Kashaev-Volkov and the identities obtained recently by Reineke. Morally, the new identities
follow from Kontsevich-Soibelman’s theory of refined Donaldson-Thomas invariants. They can be proved
rigorously using the theory linking cluster algebras to quiver representations.
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