We generalize the almost positive roots model for cluster algebras from finite type to a uniform finite/affine type model. We define a subset Φc of the root system and a compatibility degree on Φc , given by a formula that is new even in finite type. The clusters (maximal pairwise compatible sets of roots) define a complete fan Fanc (Φ). Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of Fanc (Φ) induced by real roots to the g-vector fan of the associated cluster algebra. We show that Φc is the set of denominator vectors of the associated acyclic cluster algebra and conjecture that the compatibility degree also describes denominator vectors for non-acyclic initial seeds. We extend results on exchangeability of roots to the affine case.

An affine almost positive roots model / Reading, N.; Stella, S.. - In: JOURNAL OF COMBINATORIAL ALGEBRA. - ISSN 2415-6302. - 4:1(2020), pp. 1-59. [10.4171/JCA/37]

An affine almost positive roots model

Stella S.
2020

Abstract

We generalize the almost positive roots model for cluster algebras from finite type to a uniform finite/affine type model. We define a subset Φc of the root system and a compatibility degree on Φc , given by a formula that is new even in finite type. The clusters (maximal pairwise compatible sets of roots) define a complete fan Fanc (Φ). Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of Fanc (Φ) induced by real roots to the g-vector fan of the associated cluster algebra. We show that Φc is the set of denominator vectors of the associated acyclic cluster algebra and conjecture that the compatibility degree also describes denominator vectors for non-acyclic initial seeds. We extend results on exchangeability of roots to the affine case.
2020
affine cluster algebras
01 Pubblicazione su rivista::01a Articolo in rivista
An affine almost positive roots model / Reading, N.; Stella, S.. - In: JOURNAL OF COMBINATORIAL ALGEBRA. - ISSN 2415-6302. - 4:1(2020), pp. 1-59. [10.4171/JCA/37]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1411058
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