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EXPLICIT CONSTRUCTION OF COMPANION BASES

Part of: Algebraic combinatorics Arithmetic rings and other special rings Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  21 July 2015

MARK JAMES PARSONS
MARK JAMES PARSONS*
Affiliation:
Institut für Mathematik und Wissenschaftliches Rechnen, Universität Graz Heinrichstraße 36, A-8010 Graz, Austria e-mail: [email protected]
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Abstract

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A companion basis for a quiver Γ mutation equivalent to a simply-laced Dynkin quiver is a subset of the associated root system which is a $\mathbb{Z}$-basis for the integral root lattice with the property that the non-zero inner products of pairs of its elements correspond to the edges in the underlying graph of Γ. It is known in type A (and conjectured for all simply-laced Dynkin cases) that any companion basis can be used to compute the dimension vectors of the finitely generated indecomposable modules over the associated cluster-tilted algebra. Here, we present a procedure for explicitly constructing a companion basis for any quiver of mutation type A or D.

MSC classification

Primary: 13F60: Cluster algebras
Secondary: 05E10: Combinatorial aspects of representation theory 17B22: Root systems
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 2 , May 2016 , pp. 357 - 384
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

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