MIT Combinatorics Seminar

On Cartan Determinants For Special Quivers With Relations

Christine Bessenrodt (University of Hannover)

Wednesday, June 28, 2006   4:15 PM;   Building 4, room 270


An important topic in the representation theory of algebras is the study of derived equivalences between their derived module categories; only few invariants for these equivalences are known. In joint work with T. Holm we have investigated special algebras which have come up several times in this context and which are defined in purely combinatorial terms by a quiver (i.e., a finite directed graph) and homogeneous relations; we also allow weights on the arrows of the quiver.

In this situation the weighted Cartan matrix collects the weighted counts of the non-zero paths in the quiver. For our special quivers, we have obtained an explicit formula for the weighted Cartan determinant in terms of the combinatorics of the quivers. This gives new and easy ways to compute combinatorial invariants for the special algebras mentioned above.