In this talk, I will explain our work on the monoidal categorification of the quantum coordinate ring $A_q(n(w))$ of the unipotent subgroup associated with a symmetric Kac-Moody algebra $g$ and an element $w$ of the Weyl group. This is a joint work with Seok-Jin Kang, Masaki Kashiwara, and Se-jin Oh.

The notion of monodical categorification of cluster algebras was introduced by Hernandez and Leclerc: an abelian monodical category $C$ is called a monodical categorification of a cluster algebra $A$ if the Grothendieck ring of $C$ is isomorphic to $A$ and the cluster monomials of $A$ belong to the classes of real simple objects of $C$.

The existence of a monodical categorification of a cluster algebra $A$ implies several nice properites of $A$ in a natural way, for example, the positivity of the coefficients of the expansion of cluster monomials with respect to an arbitrary cluster.

Our main result is that a subcategory $C_w$ of category of finite-dimensional graded modules over the symmetric quiver Hecke algebra is a monodical categorification of the (quantum) cluster algebra $A_q(n(w))$.

Combining the results of Khovanov-Lauda, Rouquier and Varagnolo-Vasserot, we conclude that the cluster monomials of $A_q(n(w))$ belongs to the upper global basis (dual canonical basis). It answers the conjecture by Kimura and Geiss-Leclerc-Schr{\"o}er, which can be also regarded as a sharpened version of a question asked by Fomin-Zelevinsky.