We present an initial-seed-mutation formula for d-vectors of clus- ter variables in a cluster algebra. We also give two rephrasings of this recursion: one as a duality formula for d-vectors in the style of the g-vectors/c-vectors dualities of Nakanishi and Zelevinsky, and one as a formula expressing the highest powers in the Laurent expansion of a cluster variable in terms of the d-vectors of any cluster containing it. We prove that the initial-seed-mutation recursion holds in a varied collection of cluster algebras, but not in general. We conjecture further that the formula holds for source-sink moves on the initial seed in an arbitrary cluster algebra, and we prove this conjecture in the case of surfaces.

Initial-seed recursions and dualities for d-vectors / Reading, N.; Stella, S.. - In: PACIFIC JOURNAL OF MATHEMATICS. - ISSN 0030-8730. - 293:1(2018), pp. 180-206. [10.2140/pjm.2018.293.179]

Initial-seed recursions and dualities for d-vectors

Stella S.
2018

Abstract

We present an initial-seed-mutation formula for d-vectors of clus- ter variables in a cluster algebra. We also give two rephrasings of this recursion: one as a duality formula for d-vectors in the style of the g-vectors/c-vectors dualities of Nakanishi and Zelevinsky, and one as a formula expressing the highest powers in the Laurent expansion of a cluster variable in terms of the d-vectors of any cluster containing it. We prove that the initial-seed-mutation recursion holds in a varied collection of cluster algebras, but not in general. We conjecture further that the formula holds for source-sink moves on the initial seed in an arbitrary cluster algebra, and we prove this conjecture in the case of surfaces.
2018
cluster algebras; d-vectors
01 Pubblicazione su rivista::01a Articolo in rivista
Initial-seed recursions and dualities for d-vectors / Reading, N.; Stella, S.. - In: PACIFIC JOURNAL OF MATHEMATICS. - ISSN 0030-8730. - 293:1(2018), pp. 180-206. [10.2140/pjm.2018.293.179]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1411063
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