Unfolding of Acyclic Sign-skew-symmetric Cluster Algebras and Applications to Positivity and F -polynomials

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 [6] which asks whether an acyclic signskew-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.