Kyungyong Lee, Li Li, Matthew R. Mills, “A Combinatorial Formula for Certain Elements of Upper Cluster Algebras”, SIGMA, 2015, Volume 11,049, 24 pp.

This article is cited in 4 papers

A Combinatorial Formula for Certain Elements of Upper Cluster Algebras

Kyungyong Lee^ab, Li Li^c, Matthew R. Mills^a

^a Department of Mathematics, Wayne State University, Detroit, MI 48202, USA
^b Korea Institute for Advanced Study, Seoul, Republic of Korea 130-722
^c Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USA

Abstract: We develop an elementary formula for certain non-trivial elements of upper cluster algebras. These elements have positive coefficients. We show that when the cluster algebra is acyclic these elements form a basis. Using this formula, we show that each non-acyclic skew-symmetric cluster algebra of rank 3 is properly contained in its upper cluster algebra.

Keywords: cluster algebra; upper cluster algebra; Dyck path.

MSC: 13F60

Received: September 30, 2014; in final form June 22, 2015; Published online June 26, 2015

Language: English

DOI: 10.3842/SIGMA.2015.049