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تسجيل مستخدم جديدMonoidal categorification of cluster algebras (merged version)
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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.
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s_{i_{-1}}s_{i_0}s_{i_1} cdots$ of simple reflections, we introduce subcategories $mathcal{C}_{mathfrak{g}}^{[a,b]}$ of $mathcal{C}_{mathfrak{g}}^0$ for all $a le b in mathbb{Z}sqcup{ pm infty }$. Associated with a certain chain $mathfrak{C}$ of intervals in $[a,b]$, we construct a real simple commuting family $M(mathfrak{C})$ in $mathcal{C}_{mathfrak{g}}^{[a,b]}$, which consists of Kirillov-Reshetikhin modules. The category $mathcal{C}_{mathfrak{g}}^{[a,b]}$ provides a monoidal categorification of the cluster algebra $K(mathcal{C}_{mathfrak{g}}^{[a,b]})$, whose set of initial cluster variables is $[M(mathfrak{C})]$. In particular, this result gives an affirmative answer to the monoidal categorification conjecture on $mathcal{C}_{mathfrak{g}}^-$ by Hernandez-Leclerc since it is $mathcal{C}_{mathfrak{g}}^{[-infty,0]}$, and is also applicable to $mathcal{C}_{mathfrak{g}}^0$ since it is $mathcal{C}_{mathfrak{g}}^{[-infty,infty]}$.
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