Monoidal categorification of cluster algebras

We prove that the quantum cluster algebra structure of a unipotent quantum coordinate ring A_q(\mathfrak{n}(w)), associated with a symmetric Kac–Moody algebra and its Weyl group element w, admits a monoidal categorification via the representations of symmetric Khovanov–Lauda–Rouquier algebras. In order to achieve this goal, we give a formulation of monoidal categorifications of quantum cluster algebras and provide a criterion for a monoidal category of finite-dimensional graded R-modules to become a monoidal categorification, where R is a symmetric Khovanov–Lauda–Rouquier algebra. Roughly speaking, this criterion asserts that a quantum monoidal seed can be mutated successively in all the directions, once the first-step mutations are possible. Then, we show the existence of a quantum monoidal seed of A_q(\mathfrak{n}(w)) which admits the first-step mutations in all the directions. As a consequence, we prove the conjecture that any cluster monomial is a member of the upper global basis up to a power of q^{1/2}. In the course of our investigation, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.