Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 049, 24 pages      arXiv:1409.8177
Contribution to the Special Issue on New Directions in Lie Theory

A Combinatorial Formula for Certain Elements of Upper Cluster Algebras

Kyungyong Lee ab, Li Li c and 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

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

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.

Key words: cluster algebra; upper cluster algebra; Dyck path.

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