Log-concavity and unimodality of cluster monomials in fan coordinates of type A_n

Abstract

We prove that every cluster monomial of type A_n is log-concave and unimodal with respect to its Laurent expansion in a fan initial cluster. We derive a coefficient formula for laminar interval products arising from compatible diagonals in the polygon model. This gives partial affirmative answers to the log-concavity conjecture of Chen-Huang-Sun and the unimodality conjecture of Chen. Our result for cluster A-varieties also complements Shen's study on the supports of products of canonical basis elements for cluster X-varieties of type A_n.

arXiv comment25 pages, 4 figures. Clarified the proofs, improved the exposition, and corrected minor typos. Any comments are welcome!