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EXPLICIT CONSTRUCTION OF COMPANION BASES
Part of:
Algebraic combinatorics
Lie algebras and Lie superalgebras
Arithmetic rings and other special rings
Published online by Cambridge University Press: 21 July 2015
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
- Type
- Research Article
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- Copyright
- Copyright © Glasgow Mathematical Journal Trust 2015
References
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