Laurent phenomenon and simple modules of quiver Hecke algebras

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 $[ \mathscr C]$.