2018/11/06 by Kashiwara, Masaki, Kim, Myungho
#13F60 #16G #17B37 #81R50 #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1811.02237
We study consequences of a monoidal categorification of the unipotent quantum coordinate ring Aq(\mathfrakn(w)) together with the Laurent phenomenon of cluster algebras. We show that if a simple module S in the category \mathcal Cw 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 [Mk], then [S] is either a cluster monomial in [\mathscr C ] or a cluster monomial in μk([ \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 Aq(\mathfrakn(w)) of Aq(\mathfrakn(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].