Linear independence of global monomials on positive spaces

Abstract

In this paper, we prove that global monomials on positive spaces are linearly independent, extending the basic fact that Laurent monomials in a Laurent polynomial algebra are linearly independent to a much more general setting. We also establish a global monomial avoidance phenomenon for positive spaces. Our approach is based on the study of Newton polytopes of Laurent expansions. These general results apply to positive spaces arising from cluster algebras (including the totally sign-skew-symmetric case), $Y$-patterns, and Laurent phenomenon algebras whose clusters are related by subtraction-free birational transformations. In particular, we obtain the proper Laurent monomial property and the linear independence of cluster monomials for all cluster algebras and Laurent phenomenon algebras under consideration. Notably, the proper Laurent monomial property follows from the global monomial avoidance phenomenon for positive spaces.