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.