Laurent phenomenon and simple modules of quiver Hecke algebras

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 $[\mathcal{C}]$, then $[S]$ is a cluster monomial in $[\mathcal{C}]$. If $S$ strongly commutes with cluster variables except exactly one cluster variable $[M_k]$, then $[S]$ is either a cluster monomial in $[\mathcal{C}]$ or a cluster monomial in ${\mu }_k([\mathcal{C}])$. 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 ${\tilde{A}}_q(\mathfrak{n}(w))$ 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 $[\mathcal{C}]$ is given in terms of the denominator vector of $[S]$ with respect to the cluster $[\mathcal{C}]$.

38 pages, v.2: small change, one reference added

References in corpus (1)

• 2-Kac-Moody algebras

Cited by in corpus (1)

• Newton-Okounkov polytopes of Schubert varieties arising from cluster structures