Unfolding of acyclic sign-skew-symmetric cluster algebras and applications to positivity and $F$-polynomials

Min Huang; Fang Li

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arXiv · 1609.05981

Unfolding of acyclic sign-skew-symmetric cluster algebras and applications to positivity and $F$-polynomials

Min Huang·Fang Li

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Abstract

In this paper, we build the unfolding approach from acyclic sign-skew-symmetric matrices of finite rank to skew-symmetric matrices of infinite rank, which can be regard as an improvement of that in the skew-symmetrizable case. Using this approach, we give a positive answer to the problem by Berenstein, Fomin and Zelevinsky in \cite{fz3} which asks whether an acyclic sign-skew-symmetric matrix is always totally sign-skew-symmetric. As applications, the positivity for cluster algebras in acyclic sign-skew-symmetric case is given; further, the $F$-polynomials of cluster algebras are proved to have constant term 1 in acyclic sign-skew-symmetric case.

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BibTeX RIS

Min Huang, Fang Li. 2016-09-20. Unfolding of acyclic sign-skew-symmetric cluster algebras and applications to positivity and $F$-polynomials. https://arxiv.org/abs/1609.05981

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