arXiv subjects

Bing Duan

Publications and source records attributed to Bing Duan.

16 recordsLinked to original sources

arXiv2025

Classification of real modules in monoidal categorifications of cluster algebras

In this paper, we propose a conjectural formula for the highest $\ell$-weight monomial of an arbitrary real module over a simply-laced quantum affine algebra. We verify the conjecture under a multiplicative reachability condition, answering the Hernandez--Leclerc classification problem in monoidal categorifications of cluster algebras under this condition. Moreover, we introduce the notion of cluster modules, generalizing Kirillov--Reshetikhin modules and Hernandez--Leclerc modules as special cases. We prove that cluster modules are reachable real modules, and obtain a system of equations governing $q$-characters of the prime cluster modules, providing a natural generalization of both the classical T-system relations for Kirillov--Reshetikhin modules and the exchange relations for Hernandez--Leclerc modules.

math.RT↗

arXiv2025

Maximal green sequences for $\mathcal{Q}^N$ quivers

We introduce $\mathcal{Q}^N$ quivers and construct maximal green sequences for these quivers. We prove that any finite connected full subquiver of the quivers defined by Hernandez and Leclerc, arising in monoidal categorifications of cluster algebras, is a special case of $\mathcal{Q}^N$ quivers. Moreover, we prove that the trees of oriented cycles introduced by Garver and Musiker are special cases of $\mathcal{Q}^N$ quivers. This result resolves an open problem proposed by Garver and Musiker, providing a construction of maximal green sequences for quivers that are trees of oriented cycles. Furthermore, we prove that quivers that are mutation equivalent to an orientation of a type AD Dynkin diagram can also be recognized as special cases of $\mathcal{Q}^N$ quivers.

math.AC↗

arXiv2023

A Split-and-Privatize Framework for Large Language Model Fine-Tuning

Fine-tuning is a prominent technique to adapt a pre-trained language model to downstream scenarios. In parameter-efficient fine-tuning, only a small subset of modules are trained over the downstream datasets, while leaving the rest of the pre-trained model frozen to save computation resources. In recent years, a popular productization form arises as Model-as-a-Service (MaaS), in which vendors provide abundant pre-trained language models, server resources and core functions, and customers can fine-tune, deploy and invoke their customized model by accessing the one-stop MaaS with their own private dataset. In this paper, we identify the model and data privacy leakage risks in MaaS fine-tuning, and propose a Split-and-Privatize (SAP) framework, which manage to mitigate the privacy issues by adapting the existing split learning architecture. The proposed SAP framework is sufficiently investigated by experiments, and the results indicate that it can enhance the empirical privacy by 62% at the cost of 1% model performance degradation on the Stanford Sentiment Treebank dataset.

cs.CL↗

arXiv2023

Generalized Hernandez-Leclerc modules and cluster algebras

We introduce generalized Hernandez-Leclerc modules over $U_q(\widehat{\mathfrak{sl}_{n+1}})$ as a generalization of Hernandez-Leclerc modules of type A, and prove that they are real and prime via monoidal categorifications of cluster algebras.

math.QA↗

arXiv2023

Real simple modules over simply-laced quantum affine algebras and categorifications of cluster algebras

Let $\mathscr{C}$ be the category of finite-dimensional modules over a simply-laced quantum affine algebra $U_q(\widehat{\mathfrak{g}})$. For any height function $ξ$ and $\ell\in \mathbb{Z}_{\geq 1}$, we introduce certain subcategories $\mathscr{C}^{\leq ξ}_\ell$ of $\mathscr{C}$, and prove that the quantum Grothendieck ring $K_t(\mathscr{C}^{\leq ξ}_\ell)$ of $\mathscr{C}^{\leq ξ}_\ell$ admits a quantum cluster algebra structure. Using $F$-polynomials and monoidal categorifications of cluster algebras, we classify all real simple modules in $\mathscr{C}^{\leq ξ}_1$ in terms of their highest $\ell$-weight monomials, among them the families of type $D$ and type $E$ are new. For any $\ell$, inspired by Hernandez and Leclerc's work, we propose two conjectures for the study of real simple modules, and prove them for the subcategories $\mathscr{C}^{\leq ξ}_\ell$ whose Grothendieck rings are cluster algebras of finite type.

math.QA↗

arXiv2023

A combinatorial model for $q$-characters of fundamental modules of type $D_{n}$

In this paper, we introduce a combinatorial path model of representation of the quantum affine algebra of type $D_n$, inspired by Mukhin and Young's combinatorial path models of representations of the quantum affine algebras of types $A_n$ and $B_n$. In particular, we give a combinatorial formula for $q$-characters of fundamental modules of type $D_{n}$ by assigning each path to a monomial or binomial. By counting our paths, a new expression on dimensions of fundamental modules of type $D_n$ is obtained.

math.QA↗

arXiv2020

Hernandez-Leclerc modules and snake graphs

In 2010, Hernandez and Leclerc studied connections between representations of quantum affine algebras and cluster algebras. In 2019, Brito and Chari defined a family of modules over quantum affine algebras, called Hernandez-Leclerc modules. We characterize the highest $\ell$-weight monomials of Hernandez-Leclerc modules. We give a non-recursive formula for $q$-characters of Hernandez-Leclerc modules using snake graphs, which involves an explicit formula for $F$-polynomials. We also give a new recursive formula for $q$-characters of Hernandez-Leclerc modules.

math.QA↗

arXiv2019

Combinatorial model for m-cluster categories in type E

We revisit the geometric description of cluster categories in type E in terms of colored diagonals in a polygon and generalize it to the case of m-cluster categories. As an application, we relate colored diagonals in a polygon to semi-standard Young tableaux, in type E_6,E_7,E_8. This provides a new compatibility description of semi--standard Young tableaux in Grassmannian cluster algebras in type E_6, E_8 and in a sub-cluster algebra of type E_7.

math.CO↗

arXiv2019

Quiver mutations and Boolean reflection monoids

In 2010, Everitt and Fountain introduced the concept of reflection monoids. The Boolean reflection monoids form a family of reflection monoids (symmetric inverse semigroups are Boolean reflection monoids of type $A$). In this paper, we give a family of presentations of Boolean reflection monoids and show how these presentations are compatible with quiver mutations of orientations of Dynkin diagrams with frozen vertices. Our results recover the presentations of Boolean reflection monoids given by Everitt and Fountain and the presentations of symmetric inverse semigroups given by Popova respectively. Surprisingly, inner by diagram automorphisms of irreducible Weyl groups and Boolean reflection monoids can be constructed by sequences of mutations preserving the same underlying diagrams. Besides, we show that semigroup algebras of Boolean reflection monoids are cellular algebras.

math.RA↗

arXiv2019

Quantum affine algebras and Grassmannians

We study the relation between quantum affine algebras of type A and Grassmannian cluster algebras. Hernandez and Leclerc described an isomorphism from the Grothendieck ring of a certain subcategory $\mathcal{C}_{\ell}$ of $U_q(\hat{\mathfrak{sl}_n})$-modules to a quotient of the Grassmannian cluster algebra in which certain frozen variables are set to 1. We explain how this induces an isomorphism between the monoid of dominant monomials, used to parameterize simple modules, and a quotient of the monoid of rectangular semistandard Young tableaux. Via the isomorphism, we define an element ch(T) in a Grassmannian cluster algebra for every rectangular tableau T. By results of Kashiwara, Kim, Oh, and Park, and also of Qin, every Grassmannian cluster monomial is of the form ch(T) for some T. Using formulas of Arakawa-Suzuki, we give an explicit expression for ch(T), and also give explicit q-character formulas for finite-dimensional $U_q(\hat{\mathfrak{sl}_n})$-modules. We give a tableau-theoretic rule for performing mutations in Grassmannian cluster algebras. We suggest how our formulas might be used to study reality and primeness of modules, and compatibility of cluster variables.

math.RT↗

arXiv2019

A geometric $q$-character formula for snake modules

Let $\mathscr{C}$ be the category of finite dimensional modules over the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ of a simple complex Lie algebra ${\mathfrak{g}}$. Let $\mathscr{C}^-$ be the subcategory introduced by Hernandez and Leclerc. We prove the geometric $q$-character formula conjectured by Hernandez and Leclerc in types $\mathbb{A}$ and $\mathbb{B}$ for a class of simple modules called snake modules introduced by Mukhin and Young. Moreover, we give a combinatorial formula for the $F$-polynomial of the generic kernel associated to the snake module. As an application, we show that snake modules correspond to cluster monomials with square free denominators and we show that snake modules are real modules. We also show that the cluster algebras of the category $\mathscr{C}_1$ are factorial for Dynkin types $\mathbb{A,D,E}$.

math.QA↗

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↗

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 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.

math.QA↗

arXiv2001

Independent Eigenstates of Angular Momentum in a Quantum N-body System

The global rotational degrees of freedom in the Schrödinger equation for an $N$-body system are completely separated from the internal ones. After removing the motion of center of mass, we find a complete set of $(2\ell+1)$ independent base functions with the angular momentum $\ell$. These are homogeneous polynomials in the components of the coordinate vectors and the solutions of the Laplace equation, where the Euler angles do not appear explicitly. Any function with given angular momentum and given parity in the system can be expanded with respect to the base functions, where the coefficients are the functions of the internal variables. With the right choice of the base functions and the internal variables, we explicitly establish the equations for those functions. Only (3N-6) internal variables are involved both in the functions and in the equations. The permutation symmetry of the wave functions for identical particles is discussed.

physics.atom-ph↗

arXiv2000

Generalized Radial Equations in a Quantum N-Body Problem

We demonstrate how to separate the rotational degrees of freedom in a quantum N-body problem completely from the internal ones. It is shown that any common eigenfunction of the total orbital angular momentum ($\ell$) and the parity in the system can be expanded with respect to $(2\ell+1)$ base-functions, where the coefficients are the functions of the internal variables. We establish explicitly the equations for those functions, called the generalized radial equations, which are $(2\ell+1)$ coupled partial differential equations containing only $(3N-6)$ internal variables.

physics.atom-ph↗