arXiv subjects

Jian-Rong Li

Publications and source records attributed to Jian-Rong Li.

45 records · Page 3Linked to original sources

arXiv2018

Cluster algebras and snake modules

Snake modules introduced by Mukhin and Young form a family of modules of quantum affine algebras. The aim of this paper is to prove that the Hernandez-Leclerc conjecture about monoidal categorifications of cluster algebras is true for prime snake modules of types $A_{n}$ and $B_{n}$. We prove that prime snake modules are real. We introduce $S$-systems consisting of equations satisfied by the $q$-characters of prime snake modules of types $A_{n}$ and $B_{n}$. Moreover, we show that every equation in the $S$-system of type $A_n$ (respectively, $B_n$) corresponds to a mutation in the cluster algebra $\mathscr{A}$ (respectively, $\mathscr{A}'$) constructed by Hernandez and Leclerc and every prime snake module of type $A_n$ (respectively, $B_n$) corresponds to some cluster variable in $\mathscr{A}$ (respectively, $\mathscr{A}'$). In particular, this proves that the Hernandez-Leclerc conjecture is true for all prime snake modules of types $A_{n}$ and $B_{n}$.

math.QA↗

arXiv2018

On cacti and crystals

In the present work we study actions of various groups generated by involutions on the category $\mathscr O^{int}_q(\mathfrak g)$ of integrable highest weight $U_q(\mathfrak g)$-modules and their crystal bases for any symmetrizable Kac-Moody algebra $\mathfrak g$. The most notable of them are the cactus group and (yet conjectural) Weyl group action on any highest weight integrable module and its lower and upper crystal bases. Surprisingly, some generators of cactus groups are anti-involutions of the Gelfand-Kirillov model for $\mathscr O^{int}_q(\mathfrak g)$ closely related to the remarkable quantum twists discovered by Kimura and Oya.

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arXiv2017

Three-term recurrence relations of minimal affinizations of type $G_2$

Minimal affinizations form a class of modules of quantum affine algebras introduced by Chari. We introduce a system of equations satisfied by the $q$-characters of minimal affinizations of type $G_2$ which we call the M-system of type $G_2$. The M-system of type $G_2$ contains all minimal affinizations of type $G_2$ and only contains minimal affinizations. The equations in the M-system of type $G_2$ are three-term recurrence relations. The M-system of type $G_2$ is much simpler than the extended T-system of type $G_2$ obtained by Mukhin and the second author. We also interpret the three-term recurrence relations in the M-system of type $G_2$ as exchange relations in a cluster algebra constructed by Hernandez and Leclerc.

math.QA↗

arXiv2015

On the extended T-system of type $C_3$

We continue the study of extended T-systems of quantum affine algebras. We find a sub-system of the extended T-system of the quantum affine algebra $U_q \hat{\mathfrak{g}}$ of type $C_3$. The sub-system consisting of four systems which are denoted by I, II, III, and IV. Each of the systems I, II, III, IV is closed. The systems I-IV can be used to compute minimal affinizations with weights of the form $λ_1 ω_1 + λ_2 ω_2 + λ_3 ω_3$, where at least one of $λ_1$, $λ_2$, $λ_3$ are zero. Using the systems I-IV, we compute the characters of the restrictions of the minimal affinizations in the systems to $ U_q \mathfrak{g} $ and obtain some conjectural decomposition formulas for the restrictions of some minimal affinizations.

math.QA↗

arXiv2015

Graded limits of minimal affinizations over the quantum loop algebra of type $G_2$

The aim of this paper is to study the graded limits of minimal affinizations over the quantum loop algebra of type $G_2$. We show that the graded limits are isomorphic to multiple generalizations of Demazure modules, and obtain defining relations of them. As an application, we obtain a polyhedral multiplicity formula for the decomposition of minimal affinizations of type $G_2$ as a $U_q(\mathfrak{g})$-module, by showing the corresponding formula for the graded limits.

math.QA↗

arXiv2015

M-systems and Cluster algebras

The aim of this paper is two-fold: (1) introduce four systems of equations called M-systems and dual M-systems of types $A_{n}$ and $B_{n}$ respectively; (2) make a connection between M-systems (dual M-systems) and cluster algebras and prove that the Hernandez-Leclerc conjecture is true for minimal affinizations of types $A_n$ and $B_n$.

math.QA↗

arXiv2015

On the minimal affinizations over the quantum affine algebras of type $C_n$

In this paper, we study the minimal affinizations over the quantum affine algebras of type $C_n$ by using the theory of cluster algebras. We show that the $q$-characters of a large family of minimal affinizations of type $C_n$ satisfy some systems of equations. These equations correspond to mutation equations of some cluster algebras. Furthermore, we show that the minimal affinizations in these equations correspond to cluster variables in these cluster algebras.

math.QA↗

arXiv2015

On the minimal affinizations of type $F_4$

In this paper, we apply the theory of cluster algebras to study minimal affinizations for the quantum affine algebra of type $F_4$. We show that the $q$-characters of a large family of minimal affinizations of type $F_4$ satisfy a system of equations. Moreover, a minimal affinization in this system corresponds to some cluster variable in some cluster algebra $\mathscr{A}$. For the other minimal affinizations of type $F_4$ which are not in this system, we give some conjectural equations which contains these minimal affinizations. Furthermore, we introduce the concept of dominant monomial graphs to study the equations satisfied by $q$-characters of modules of quantum affine algebras.

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arXiv2013

Extended $T$-System of Type $G_2$

We prove a family of 3-term relations in the Grothendieck ring of the category of finite-dimensional modules over the affine quantum algebra of type $G_2$ extending the celebrated $T$-system relations of type $G_2$. We show that these relations can be used to compute classes of certain irreducible modules, including classes of all minimal affinizations of type $G_2$. We use this result to obtain explicit formulas for dimensions of all participating modules.

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