Laurent phenomenon and simple modules of quiver Hecke algebras

TL;DR

This paper explores the relationship between simple modules and cluster variables in monoidal categorifications of quantum coordinate rings, providing new proofs and characterizations related to Laurent phenomenon and basis structures.

Contribution

It offers new insights into the commutativity of simple modules with cluster variables and proves the upper global basis as a common triangular basis.

Findings

Simple modules that strongly commute with all cluster variables are cluster monomials.

Characterization of module commutativity via denominator vectors.

New proof that the upper global basis is a common triangular basis.

Abstract

We study consequences of a monoidal categorification of the unipotent quantum coordinate ring $A_q(\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category $\mathcal C_w$ strongly commutes with all the cluster variables in a cluster $[ \mathscr C]$, then $[S]$ is a cluster monomial in $[ \mathscr C ]$. If $S$ strongly commutes with cluster variables except exactly one cluster variable $[M_k]$, then $[S]$ is either a cluster monomial in $[\mathscr C ]$ or a cluster monomial in $\mu_k([ \mathscr C ])$. We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Fan Qin) of the localization $\widetilde A_q(\mathfrak{n}(w))$ of $A_q(\mathfrak{n}(w))$ at the frozen variables. A characterization on the commutativity of a simple module $S$ with cluster variables in a cluster $[ \mathscr C]$ is given in terms of the denominator vector of $[S]$ with respect to the cluster $[ \mathscr C]$.