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

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.

arXiv comment48 pages, 1 figure

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Cited by in corpus (3)

• Categorification of sign-skew-symmetric cluster algebras and some conjectures on g-vectors
• Study on cluster algebras via abstract pattern and two conjectures on d-vectors and g-vector
• Positivity for quantum cluster algebras from unpunctured orbifolds