Proof of a positivity conjecture of M. Kontsevich on non-commutative cluster variables

Abstract

We prove a conjecture of Kontsevich, which asserts that the iterations of the noncommutative rational map $F_r:(x,y)-->(xy{x}^{-1},(1+y^r)x^{-1})$ are given by noncommutative Laurent polynomials with nonnegative integer coefficients.

arXiv comment13 pages; This paper supersedes the first author's preprint "A step towards the cluster positivity conjecture" (arXiv:1103.2726). The main improvement is that we give expressions in terms of subpaths of certain lattice paths

References in corpus (1)

• Noncommutative integrability, paths and quasi-determinants

Cited by in corpus (2)

• Greedy bases in rank 2 quantum cluster algebras
• The existence of greedy bases in rank 2 quantum cluster algebras