q-Cartan matrices and combinatorial invariants of derived categories for skewed-gentle algebras

Abstract

Cartan matrices are of fundamental importance in representation theory. For algebras defined by quivers (i.e. directed graphs) with relations the computation of the entries of the Cartan matrix amounts to counting nonzero paths in the quivers, leading naturally to a combinatorial setting. In this paper we study a refined version, so-called q-Cartan matrices, where each nonzero path is weighted by a power of an indeterminant q according to its length. Specializing q=1 gives the classical Cartan matrix. Our main motivation are derived module categories and their invariants: the invariant factors, and hence the determinant, of the Cartan matrix are preserved by derived equivalences. The paper deals with the important class of (skewed-) gentle algebras which occur naturally in representation theory, especially in the context of derived categories. These algebras are defined in purely combinatorial terms. We determine normal forms for the Cartan matrices of (skewed-) gentle algebras. In particular, we give explicit combinatorial formulae for the invariant factors and thus also for the determinant of the Cartan matrices of skewed-gentle algebras. As an application of our main results we show how one can use our formulae for the notoriously difficult problem of distinguishing derived equivalence classes.