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

Christine Bessenrodt and Thorsten Holm

Vol. 229 (2007), No. 1, 25–47

DOI: 10.2140/pjm.2007.229.25

Abstract

Cartan matrices are of fundamental importance in representation theory. For algebras defined by quivers with monomial relations, the computation of the entries of the Cartan matrix amounts to counting nonzero paths in the quivers, leading naturally to a combinatorial setting. For derived module categories, the invariant factors, and hence the determinant, of the Cartan matrix are preserved by derived equivalences.

In the generalization called q-Cartan matrices (the classical Cartan matrix corresponding to q = 1), each nonzero path is weighted by a power of an indeterminate q according to its length. We study q-Cartan matrices for gentle and skewed-gentle algebras, which occur naturally in representation theory, especially in the context of derived categories. We determine normal forms for these matrices in the skewed-gentle case, giving explicit combinatorial formulae for the invariant factors and the determinant. As an application, we show how to use our formulae for the difficult problem of distinguishing derived equivalence classes.

Keywords

Cartan matrices, derived categories, skewed-gentle algebras

Mathematical Subject Classification 2000

Primary: 16G10, 18E30, 05E99, 05C38, 05C50

Milestones

Received: 1 May 2005

Revised: 22 September 2005

Accepted: 9 December 2005

Published: 1 January 2007