The unipotent quantum coordinate ring $A_q(n(w))$ is isomorphic to the Grothendieck ring of a monoidal category $C_w$ consisting of some finite dimensional graded modules over a quiver Hecke algebra. Moreover $A_q(n(w))$ has a quantum cluster algebra structure, and it is shown that the cluster monomials are classes of real simple modules in $C_w$. In this talk, I will present some interesting consequences of this "monoidal categorification" of $A_q(n(w))$ with the Laurent phenomenon of the cluster algebras. This is a joint work with Masaki Kashiwara.

The venue is Maskawa Building for Education and Research Room 507.