Vol. 229, No. 1, 2007

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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
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
Authors
Christine Bessenrodt
Institut für Algebra, Zahlentheorie und Diskrete Mathematik
Fakultät für Mathematik und Physik
Leibniz Universität Hannover
Welfengarten 1
D-30167 Hannover
Germany
http://www-ifm.math.uni-hannover.de/~bessen/
Thorsten Holm
Department of Pure Mathematics
University of Leeds
Leeds LS2 9JT
England
Institut für Algebra und Geometrie
Otto-von-Guericke-Universität Magdeburg
Postfach 4120
D-39016 Magdeburg
Germany
http://www.maths.leeds.ac.uk/~tholm