Denominator vectors and compatibility degrees in cluster algebras of finite type
Combinatorics
2015-05-27 v4 Rings and Algebras
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 denominator vector of any non-initial cluster variable with respect to any initial cluster seed has non-negative entries and is different from zero.
Keywords
Cite
@article{arxiv.1302.1052,
title = {Denominator vectors and compatibility degrees in cluster algebras of finite type},
author = {Cesar Ceballos and Vincent Pilaud},
journal= {arXiv preprint arXiv:1302.1052},
year = {2015}
}
Comments
19 pages, 2 figures. v4-section 7 shortened (examples are found in v3). This is the final published version