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Denominator vectors and compatibility degrees in cluster algebras of finite type. (English) Zbl 1350.13020
Summary: 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.

##### MSC:
 13F60 Cluster algebras 20F55 Reflection and Coxeter groups (group-theoretic aspects) 05E15 Combinatorial aspects of groups and algebras (MSC2010) 05E45 Combinatorial aspects of simplicial complexes
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