Combinatorics of $\mathcal{X}$variables in finite type cluster algebras
Abstract
We compute the number of $\mathcal{X}$variables (also called coefficients) of a cluster algebra of finite type when the underlying semifield is the universal semifield. For classical types, these numbers arise from a bijection between coefficients and quadrilaterals (with a choice of diagonal) appearing in triangulations of certain marked surfaces. We conjecture that similar results hold for cluster algebras from arbitrary marked surfaces, and obtain corollaries regarding the structure of finite type cluster algebras of geometric type.
 Publication:

arXiv eprints
 Pub Date:
 March 2018
 arXiv:
 arXiv:1803.02492
 Bibcode:
 2018arXiv180302492S
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Rings and Algebras
 EPrint:
 22 pages, 6 figures. Version 2: Updated introduction to include Corollary 1.2, modified the definition of "quadrilateral", added minor clarifications elsewhere