Laurent phenomenon and simple modules of quiver Hecke algebras

In this paper we study consequences of the results of Kang et al. [Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), 349{426] on a monoidal categorification of the unipotent quantum coordinate ring Aq(n(w)) together with the Laurent phenomenon of cluster algebras. We show that if a simple module S in the category C-w strongly commutes with all the cluster variables in a cluster [C], then [S] is a cluster monomial in [C]. If S strongly commutes with cluster variables except for exactly one cluster variable [M-k], then [S] is either a cluster monomial in [C] or a cluster monomial in mu(k)([C]). We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Qin [Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. 166 (2017), 2337{2442]) of the localization (A) over tilde (q) (n(w)) of e A(q) (n(w)) at the frozen variables. A characterization on the commutativity of a simple module S with cluster variables in a cluster [C] is given in terms of the denominator vector of [S] with respect to the cluster [C].