Some elementary properties of Laurent phenomenon algebras

TL;DR

This paper explores fundamental properties of Laurent phenomenon algebras, including seed uniqueness, mutation connectivity, invariance of upper bounds, and bases of lower bounds, extending results to related cluster algebras.

Contribution

It establishes seed uniqueness, mutation connectivity, invariance of the upper bound, and bases of the lower bound in Laurent phenomenon algebras, also applicable to cluster algebras.

Findings

Each seed is uniquely determined by its cluster.

Seeds with n-1 common variables are connected by a mutation.

Upper bound invariance under seed mutations under certain conditions.

Abstract

Let $\Sigma$ be Laurent phenomenon (LP) seed of rank $n$, $\mathcal{A}(\Sigma)$, $\mathcal{U}(\Sigma)$ and $\mathcal{L}(\Sigma)$ be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively. We prove that each seed of $\mathcal{A}(\Sigma)$ is uniquely defined by its cluster, and any two seeds of $\mathcal{A}(\Sigma)$ with $n-1$ common cluster variables are connected with each other by one step of mutation. The method in this paper also works for (totally sign-skew-symmetric) cluster algebras. Moreover, we show that $\mathcal{U}(\Sigma)$ is invariant under seed mutations when each exchange polynomials coincides with its exchange Laurent polynomials of $\Sigma$. Besides, we obtain the standard monomial bases of $\mathcal{L}(\Sigma)$. We also prove that $\mathcal{U}(\Sigma)$ coincides with $\mathcal{L}(\Sigma)$ under certain conditions.