Denominator vectors and compatibility degrees in cluster algebras of finite type

Ceballos, Cesar; Pilaud, Vincent

doi:10.1090/S0002-9947-2014-06239-9

Denominator vectors and compatibility degrees in cluster algebras of finite type
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Trans. Amer. Math. Soc. 367 (2015), 1421-1439 Request permission

Abstract:

We present two simple descriptions of the denominator vectors of the cluster variables of a cluster algebra of finite type, with respect to any initial cluster seed: one in terms of the compatibility degrees between almost positive roots defined by S. Fomin and A. Zelevinsky, and the other in terms of the root function of a certain subword complex. These descriptions only rely on linear algebra. They provide two simple proofs of the known fact that the $d$-vector of any non-initial cluster variable with respect to any initial cluster seed has non-negative entries and is different from zero.

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