Cartan matrices are of fundamental importance in representation theory. Foralgebras defined by quivers (i.e. directed graphs) with relations thecomputation of the entries of the Cartan matrix amounts to counting nonzeropaths in the quivers, leading naturally to a combinatorial setting. In thispaper we study a refined version, so-called q-Cartan matrices, where eachnonzero path is weighted by a power of an indeterminant q according to itslength. Specializing q=1 gives the classical Cartan matrix. Our main motivationare derived module categories and their invariants: the invariant factors, andhence the determinant, of the Cartan matrix are preserved by derivedequivalences. The paper deals with the important class of (skewed-) gentlealgebras which occur naturally in representation theory, especially in thecontext of derived categories. These algebras are defined in purelycombinatorial terms. We determine normal forms for the Cartan matrices of(skewed-) gentle algebras. In particular, we give explicit combinatorialformulae for the invariant factors and thus also for the determinant of theCartan matrices of skewed-gentle algebras. As an application of our mainresults we show how one can use our formulae for the notoriously difficultproblem of distinguishing derived equivalence classes.
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