Laurent phenomenon and simple modules of quiver Hecke algebras

• Published in: Compositio mathematica 2019-12, Vol.155 (12), p.2263-2295, Article 2263
• Main Authors: Kashiwara, Masaki, Kim, Myungho
• Format: Article
• Language: English
• Subjects: Algebra; Clusters; Commutativity; Mathematical analysis; Modules; Variables
• ISSN: 0010-437X; 1570-5846
• DOI: 10.1112/s0010437x19007565

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  $[\mathscr{C}]$ .