arXiv subjects

Jian-Rong Li

Publications and source records attributed to Jian-Rong Li.

At least 37 records · Page 2Linked to original sources

arXiv2025

Compatibility of Drinfeld presentations for split affine Kac-Moody quantum symmetric pairs

Let $(\mathbf{U}, \mathbf{U}^\imath)$ be a split affine quantum symmetric pair of type $\mathsf{B}_n^{(1)}, \mathsf{C}_n^{(1)}$ or $\mathsf{D}_n^{(1)}$. We prove factorization and coproduct formulae for the Drinfeld-Cartan operators $Θ_i(z)$ in the Lu-Wang Drinfeld-type presentation, generalizing the type $\mathsf{A}_n^{(1)}$ result from [Prz23]. As an application, we show that a boundary analogue of the $q$-character map, defined via the spectra of these operators, is compatible with the usual $q$-character map. As an auxiliary result, we also produce explicit reduced expressions for the fundamental weights in the extended affine Weyl groups of classical types.

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arXiv2025

Cluster structures on spinor helicity and momentum twistor varieties

We study the homogeneous coordinate rings of partial flag varieties and Grassmannians in their Plücker embeddings and exhibit an embedding of the former into the latter. Both rings are cluster algebras and the embedding respects the cluster algebra structures in the sense that there exists a seed for the Grassmannian that restricts to a seed for the partial flag variety (\textit{i.e.} it is obtained by freezing and deleting some cluster variables). The motivation for this project stems from the application of cluster algebras in scattering amplitudes: spinor helicity and momentum twistor varieties describe massless scattering without assuming dual conformal symmetry. Both may be obtained from Grassmanninas which model the dual conformal case. They are instances of partial flag varieties and their cluster structures reveal information for the scattering amplitudes. As an application of our main result we exhibit the relation between these cluster algebras.

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arXiv2024

Verlinde rings and cluster algebras arising from quantum affine algebras

We formulate a positivity conjecture relating the Verlinde ring associated with an untwisted affine Lie algebra at a positive integer level and a subcategory of finite-dimensional representations over the corresponding quantum affine algebra with a cluster algebra structure. Specifically, we consider a ring homomorphism from the Grothendieck ring of this representation category to the Verlinde ring and conjecture that every object in the category has a positive image under this map. We prove this conjecture in certain cases where the underlying simple Lie algebra is simply-laced with level 2 or of type $A_1$ at an arbitrary level. The proof employs the close connection between this category and cluster algebras of finite cluster type. As further evidence for the conjecture, we show that for any level, all objects have positive quantum dimensions under the assumption that some Kirillov-Reshetikhin modules have positive quantum dimensions.

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arXiv2024

Hecke and Artin monoids and their homomorphisms

The aim of the present work is to systematically study homomorphisms of Hecke and Artin monoids and thus to develop their comprehensive theory. Our original motivation was the striking observation that parabolic projections of Hecke monoids respect all parabolic elements. We found other classes of homomorphisms of Hecke monoids with the same property and discovered that many of them lift to homomorphisms of covering Artin monoids with a similar property. It turned out that they belong to a much larger class (in fact, a category) of homomorphisms of Artin monoids, most of which appear to be new.

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arXiv2024

Braid group actions on grassmannians and extended crystals of type $A$

Let $σ_i$ be the braid actions on infinite Grassmannian cluster algebras induced from Fraser's braid group actions. Let $\mathsf{T}_i$ be the braid group actions on (quantum) Grothendieck rings of Hernandez-Leclerc category ${\mathscr C}_\mathfrak{g}^0$ of affine type $A_n^{(1)}$, and $\mathsf{R}_i$ the braid group actions on the corresponding extended crystals. In the paper, we prove that the actions $σ_i$ coincide with the braid group actions $\mathsf{T}_i$ and $\mathsf{R}_i$.

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arXiv2024

Tropical Geometry, Quantum Affine Algebras, and Scattering Amplitudes

The goal of this paper is to make a connection between tropical geometry, representations of quantum affine algebras, and scattering amplitudes in physics. The connection allows us to study important and difficult questions in these areas: (1) We give a systematic construction of prime modules (including prime non-real modules) of quantum affine algebras using tropical geometry. We also introduce new objects which generalize positive tropical Grassmannians. (2) We propose a generalization of Grassmannian string integrals in physics, in which the integrand is no longer a finite, but rather an infinite product indexed by prime modules of a quantum affine algebra. We give a general formula of $u$-variables using prime tableaux (corresponding to prime modules of quantum affine algebras of type $A$) and Auslander-Reiten quivers of Grassmannian cluster categories. (3) We study limit $g$-vectors of cluster algebras. This is another way to obtain prime non-real modules of quantum affine algebras systematically. Using limit $g$-vectors, we construct new examples of non-real modules of quantum affine algebras.

math.QA↗

arXiv2023

Quasi-homomorphisms of quantum cluster algebras

In this paper, we study quasi-homomorphisms of quantum cluster algebras, which are quantum analogy of quasi-homomorphisms of cluster algebras introduced by Fraser. For a quantum Grassmannian cluster algebra $\mathbb{C}_q[{\rm Gr}(k,n)]$, we show that there is an associated braid group and each generator $σ_i$ of the braid group preserves the quasi-commutative relations of quantum Plücker coordinates and exchange relations of the quantum Grassmannian cluster algebra. We conjecture that $σ_i$ also preserves $r$-term ($r \ge 4$) quantum Plücker relations of $\mathbb{C}_q[{\rm Gr}(k,n)]$ and other relations which cannot be derived from quantum quantum Plücker relations (if any). Up to this conjecture, we show that $σ_i$ is a quasi-automorphism of $\mathbb{C}_q[{\rm Gr}(k,n)]$ and the braid group acts on $\mathbb{C}_q[{\rm Gr}(k,n)]$.

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arXiv2023

The amplituhedron crossing and winding numbers

In \cite{arkani2018unwinding}, Arkani-Hamed, Thomas and Trnka formulated two conjectural descriptions of the tree amplituhedron $\ampli$ depending on the parity of $m$. When $m$ is even, the description involves the winding number and when $m$ is odd the description involves the crossing number. In this paper, we prove that if a point of the amplituhedron is in the image of the positive Grassmannian by the amplituhedron map, then it satisfies the winding or crossing descriptions depending on the parity of $m$. When $m=2$, we also prove the other direction: a point satisfying the winding description is inside the amplituhedron.

math.CO↗

arXiv2023

Real roots in the root system $\mathsf{T}_{2,p,q}$

Motivated by the recent advances in the categorification of the cluster structure on the coordinate rings of Grassmannians of $k$-subspaces in $n$-space, we investigate a particular construction of root systems of type $\mathsf{T}_{2,p,q}$, including the type $\mathsf{E}_n$. This construction generalizes Manin's ``hyperbolic construction'' of $\mathsf{E}_8$ and reveals a lot of otherwise hidden regularities in this family of root systems.

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arXiv2023

Rigid Indecomposable Modules in Grassmannian Cluster Categories

The coordinate ring of the Grassmannian variety of $k$-dimensional subspaces in $\mathbb{C}^n$ has a cluster algebra structure with Plücker relations giving rise to exchange relations. In this paper, we study indecomposable modules of the corresponding Grassmannian cluster categories ${\rm CM}(B_{k,n})$. Jensen, King, and Su have associated a Kac-Moody root system $J_{k,n}$ to ${\rm CM}(B_{k,n})$ and shown that in the finite types, rigid indecomposable modules correspond to roots. In general, the link between the category ${\rm CM}(B_{k,n})$ and the root system $J_{k,n}$ remains mysterious and it is an open question whether indecomposables always give roots. In this paper, we provide evidence for this association in the infinite types: we show that every indecomposable rank 2 module corresponds to a root of the associated root system. We also show that indecomposable rank 3 modules in ${\rm CM}(B_{3,n})$ all give rise to roots of $J_{3,n}$. For the rank 3 modules in ${\rm CM}(B_{3,n})$ corresponding to real roots, we show that their underlying profiles are cyclic permutations of a certain canonical one. We also characterize the rank 3 modules in ${\rm CM}(B_{3,n})$ corresponding to imaginary roots. By proving that there are exactly 225 profiles of rigid indecomposable rank 3 modules in ${\rm CM}(B_{3,9})$ we confirm the link between the Grassmannian cluster category and the associated root system in this case. We conjecture that the profile of any rigid indecomposable module in ${\rm CM}(B_{k,n})$ corresponding to a real root is a cyclic permutation of a canonical profile.

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arXiv2022

Dual canonical bases for unipotent groups and base affine spaces

Denote by $N \subset SL_k$ the subgroup of unipotent upper triangular matrices. In this paper, we show that the dual canonical basis of $\mathbb{C}[N]$ (and base affine spaces) can be parameterized by semi-standard Young tableaux. Moreover, we give an explicit formula for every element in the the dual canonical basis using the data of the corresponding semistandard Young tableau. We apply our results to study cluster variables in $\mathbb{C}[N]$.

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arXiv2022

Equivariant multiplicities via representations of quantum affine algebras

For any simply-laced type simple Lie algebra $\mathfrak{g}$ and any height function $ξ$ adapted to an orientation $Q$ of the Dynkin diagram of $\mathfrak{g}$, Hernandez-Leclerc introduced a certain category $\mathcal{C}^{\leq ξ}$ of representations of the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$, as well as a subcategory $\mathcal{C}_Q$ of $\mathcal{C}^{\leq ξ}$ whose complexified Grothendieck ring is isomorphic to the coordinate ring $\mathbb{C}[\mathbf{N}]$ of a maximal unipotent subgroup. In this paper, we define an algebraic morphism $\widetilde{D}_ξ$ on a torus $\mathcal{Y}^{\leq ξ}$ containing the image of $K_0(\mathcal{C}^{\leq ξ})$ under the truncated $q$-character morphism. We prove that the restriction of $\widetilde{D}_ξ$ to $K_0(\mathcal{C}_Q)$ coincides with the morphism $\overline{D}$ recently introduced by Baumann-Kamnitzer-Knutson in their study of equivariant multiplicities of Mirković-Vilonen cycles. This is achieved using the T-systems satisfied by the characters of Kirillov-Reshetikhin modules in $\mathcal{C}_Q$, as well as certain results by Brundan-Kleshchev-McNamara on the representation theory of quiver Hecke algebras. This alternative description of $\overline{D}$ allows us to prove a conjecture by the first author on the distinguished values of $\overline{D}$ on the flag minors of $\mathbb{C}[\mathbf{N}]$. We also provide applications of our results from the perspective of Kang-Kashiwara-Kim-Oh's generalized Schur-Weyl duality. Finally, we define a cluster algebra $\overline{\mathcal{A}}_Q$ as a subquotient of $K_0(\mathcal{C}^{\leq ξ})$ naturally containing $\mathbb{C}[\mathbf{N}]$, and suggest the existence of an analogue of the Mirković-Vilonen basis in $\overline{\mathcal{A}}_Q$ on which the values of $\widetilde{D}_ξ$ may be interpreted as certain equivariant multiplicities.

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arXiv2021

Construction of Rank $2$ Indecomposable Modules in Grassmannian Cluster Categories

The category ${\rm CM}(B_{k,n}) $ of Cohen-Macaulay modules over a quotient $B_{k,n}$ of a preprojective algebra provides a categorification of the cluster algebra structure on the coordinate ring of the Grassmannian variety of $k$-dimensional subspaces in $\mathbb C^n$, \cite{JKS16}. Among the indecomposable modules in this category are the rank $1$ modules which are in bijection with $k$-subsets of $\{1,2,\dots,n\}$, and their explicit construction has been given by Jensen, King and Su. These are the building blocks of the category as any module in ${\rm CM}(B_{k,n}) $ can be filtered by them. In this paper we give an explicit construction of rank 2 modules. With this, we give all indecomposable rank 2 modules in the cases when $k=3$ and $k=4$. In particular, we cover the tame cases and go beyond them. We also characterise the modules among them which are uniquely determined by their filtrations. For $k\ge 4$, we exhibit infinite families of non-isomorphic rank 2 modules having the same filtration.

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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↗

arXiv2020

Primitive orthogonal idempotents of Brandt semigroup algebras

A complete set of primitive orthogonal idempotents plays an important role in the representation theory of an associative algebra. In this paper, we construct a complete set of primitive orthogonal idempotents for any finite Brandt semigroup algebra.

math.RA↗

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.

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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.

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