# Denominator vectors and compatibility degrees in cluster algebras of finite type

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, and provide 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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Conference papers

https://hal.inria.fr/hal-01229727
Contributor : Alain Monteil <>
Submitted on : Tuesday, November 17, 2015 - 10:20:39 AM
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dmAS0108.pdf
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• HAL Id : hal-01229727, version 1

### Citation

Cesar Ceballos, Vincent Pilaud. Denominator vectors and compatibility degrees in cluster algebras of finite type. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.85-96. ⟨hal-01229727⟩

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