Laurent phenomenon and simple modules of quiver Hecke algebras

Abstract

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$A_{q}(\mathfrak{n}(w))$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 for exactly one cluster variable$[M_{k}]$, then$[S]$is either a cluster monomial in$[\mathscr{C}]$or a cluster monomial in$\unicode[STIX]{x1D707}_{k}([\mathscr{C}])$. We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Qin [Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math.166(2017), 2337–2442]) 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 $[\mathscr{C}]$.