Denominators in cluster algebras of affine type
Aslak Bakke Buan, Bethany Marsh
TL;DR
This paper provides representation-theoretic formulas for the denominators of cluster variables in affine type cluster algebras, linking algebraic expressions to homomorphism dimensions in cluster categories.
Contribution
It introduces new formulas for denominators of cluster variables in affine types using homomorphism spaces, applicable to any initial cluster.
Findings
Formulas relate denominators to homomorphism dimensions
Applicable to any initial cluster in affine type
Enhances understanding of cluster variable structure
Abstract
The Fomin-Zelevinsky Laurent phenomenon states that every cluster variable in a cluster algebra can be expressed as a Laurent polynomial in the variables lying in an arbitrary initial cluster. We give representation-theoretic formulas for the denominators of cluster variables in cluster algebras of affine type. The formulas are in terms of the dimensions of spaces of homomorphisms in the corresponding cluster category, and hold for any choice of initial cluster.
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