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

Kyungyong Lee; Ralf Schiffler

doi:10.1112/S0010437X12000425

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

Quantum Algebra 2019-02-20 v1 Algebraic Geometry Combinatorics

Authors: Kyungyong Lee , Ralf Schiffler

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Abstract

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

Keywords

non-commutative cluster variables positivity conjecture Kontsevich non-commutative rational map Laurent polynomials nonnegative integer coefficients

Cite

@article{arxiv.1109.5130,
  title  = {Proof of a positivity conjecture of M. Kontsevich on non-commutative cluster variables},
  author = {Kyungyong Lee and Ralf Schiffler},
  journal= {arXiv preprint arXiv:1109.5130},
  year   = {2019}
}

Comments

13 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

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

Abstract

Keywords

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