arXiv subjects

Jian-Rong Li

Publications and source records attributed to Jian-Rong Li.

At least 19 recordsLinked to original sources

arXiv2026

Artin monoids, their homomorphisms and twins

Motivated by the twin homomorphism problem for Coxeter groups and the corresponding Hecke monoids, we find a large class of its solutions originating from standard homomorphisms of Artin monoids and their compositions. These homomorphisms are expected to be injective when they are optimal and injective on generators, which generalizes the homogeneous homomorphisms and the famous Tits conjecture settled by Crisp and Paris. We classify disjoint standard homomorphisms and conjecture the complete classification when the domain is of rank two.

math.QA↗

arXiv2026

Monoidal categorification of generalized cluster algebras and conjectures of Fraser and Gleitz

Hernandez and Leclerc introduced the notion of monoidal categorification of cluster algebras. We define similarly the notion of monoidal categorifications of generalized cluster algebras: an abelian monoidal category $\mathcal M$ is said to be a monoidal categorification of a generalized cluster algebra $\mathcal A$ if the Grothendieck ring of $\mathcal M$ is isomorphic to the upper generalized cluster algebra $\mathcal A^{\mathrm{up}}$, and if cluster monomials (resp. cluster variables) of $\mathcal A$ correspond to classes of real simple (resp. real prime simple) objects of $\mathcal M$. Let $\varepsilon$ be a root of unity such that $\varepsilon^{2\ell}=1$ for some $\ell\in\mathbb{Z}_{\geq 2}$. Denote by $\mathcal{C}_{\varepsilon}$ the category of finite-dimensional modules of the restricted quantum loop algebra $U_\varepsilon^{\res}(L\mathfrak{sl}_k)$ at root $\varepsilon$ of unity, and let $\mathcal{C}_{\varepsilon, \xi}$ be a full subcategory of $\mathcal{C}_{\varepsilon}$ determined by a bipartition $\xi: I \to \{0,1\}$ of the Dynkin diagram. For $k=3$, Gleitz conjectured that the Grothendieck ring of $\mathcal C_{\varepsilon,\xi}$ is isomorphic to a generalized cluster algebra of rank $2\ell-2$, and that generalized cluster monomials correspond to classes of simple modules. This conjecture is a special case of a more general conjecture of Fraser. In this paper, we prove the first part of Gleitz's conjecture. More precisely, for $k=3$ and arbitrary $\ell\ge2$, we prove that the Grothendieck ring of $\mathcal C_{\varepsilon,\xi}$ is isomorphic to a generalized cluster algebra of rank $2\ell-2$. We also classify the real Kirillov--Reshetikhin modules of $U^{\mathrm{res}}_\varepsilon(L\mathfrak{sl}_3)$ and obtain mutation sequences for the real Kirillov--Reshetikhin modules from the initial seed of the generalized cluster algebra.

math.RT↗

arXiv2026

Representations of shifted twisted quantum affine algebras

In this paper, we introduce and study shifted twisted quantum affine algebras which provide a twisted counterpart of the theory of shifted quantum affine algebras. The shifted twisted quantum affine algebra $\U_q^{\mu_+,\mu_-}(\hgs)$ is obtained from the Drinfeld current presentation of twisted quantum loop algebras by shifting the Cartan--Drinfeld currents $\phi_i^\pm(z)$ according to a coweight pair $(\mu_+,\mu_-)$. We prove that it admits a triangular decomposition and that, up to isomorphism, they depend only on the total shift $\mu=\mu_+ + \mu_-$. For each shift $\mu$, we define a category $\mathcal O_\mu$ of representations of $\U_q^\mu(\hgs) = \U_q^{0,\mu}(\hgs)$ and prove a rationality theorem for the Cartan currents: on every weight space, the two currents $\phi_i^+(z)$ and $\phi_i^-(z)$ are expansions of the same rational operator-valued function, whose degree is prescribed by $\alpha_i(\mu)$. As a consequence, we classify the simple objects of $\mathcal O_\mu$ by rational $\ell$-weights of the corresponding degrees. We then construct a deformed Drinfeld coproduct and use it to define a fusion product on the direct sum $\mathcal{O}^{sh}$ of the categories $\mathcal O_\mu$. This fusion product is compatible with $q$-characters. We also classify finite-dimensional simple modules in $\mathcal{O}^{sh}$ in terms of dominant rational $\ell$-weights, with a separate treatment of type $A_{2n}^{(2)}$. Finally, we construct restriction representations relating representations of twisted quantum affine Borel algebras to representations of shifted twisted quantum affine algebras, and establish a $q$-characters formula for simple finite-dimensional representations of shifted twisted quantum affine algebras in terms of the $q$-characters of the corresponding simple representations of the twisted quantum affine Borel algebra $\U_q(\bs)$.

math.QA↗

arXiv2026

The Diagrammar of Quantum Magnusian

The logarithm of the time-evolution operator has been termed Magnusian, on account of the fact that its expansion describes the Magnus series. The diagrammatic expansion and computation of the classical Magnusian have been completely established in terms of tree graphs and their Hopf algebra. Recent works initiated extensions into quantum field theory, revealing general structures of loop expansions while finding intriguing relations between different diagrams. In this work, we advance the loop expansion further by providing an efficient diagrammatic algorithm to calculate the weight factor of each graph in the quantum Magnusian, known as the Murua coefficient. This is achieved by incorporating two complementary perspectives on the Magnusian at the same time: the color basis and the black-and-white basis. We extract the Murua coefficients from the Magnus series by utilizing these two bases while implementing an exponentiated Wick contraction. In turn, we identify the loop-level extension of Murua's recursive formula. Eventually, we establish a set of edge-contraction rules which facilitate a direct recursive computation of the Murua coefficients at the purely diagrammatic level, without referencing or directly manipulating the underlying Magnus expansion. This shows that the matrix elements of the quantum Magnusian can be computed from graph manipulations alone.

hep-th↗

arXiv2026

Hecke monoids, their homomorphisms and parabolicity

We study homomorphisms of Hecke monoids, notably parabolic homomorphisms, which map parabolic elements to parabolic elements, and injective ones. The importance of the first class stems from the fact that parabolic elements form a rather mysterious submonoid of the Hecke monoid, and we found a plethora of parabolic homomorphisms. Concerning injective ones, as a first step towards their classification, we classified all locally injective connected homomorphisms between Hecke monoids of classical types and expect all of them to be injective. As a surprising byproduct of our study of parabolic and injective homomorphisms we described, to some extent, all homomorphisms between Hecke monoids.

math.RT↗

arXiv2026

GKLO representations for shifted quantum affine symmetric pairs

In this note, we introduce shifted quantum affine symmetric pairs of split simply-laced type, and construct their GKLO representations, following similar recent developments in the case of shifted twisted Yangians. A full proof that our formulas yield a representation is given.

math.QA↗

arXiv2026

Monomial bialgebras

Starting from a single solution of QYBE (or CYBE) we produce an infinite family of solutions of QYBE (or CYBE) parametrized by transitive arrays and, in particular, by signed permutations. We are especially interested in cases when such solutions yield quasi-triangular structures on direct powers of Lie bialgebras and tensor powers of Hopf algebras. We obtain infinite families of such structures as well and study the corresponding Poisson-Lie structures and co-quasi-triangular algebras.

math.QA↗

arXiv2026

Tropical symmetries of cluster algebras

We study tropicalisations of quasi-automorphisms of cluster algebras and show that their induced action on the g-vectors can be realized by tropicalising their action on the homogeneous $\hat{y}$ (or $\mathcal{X}$) variables of a chosen initial cluster. This perspective allows us to interpret the action on g-vectors as a change of coordinates in the tropical setting. Focusing on Grassmannian cluster algebras, we analyse tropicalisations of quasi-automorphisms in detail. We derive tropical analogues of the braid group action and the twist map on both g-vectors and tableaux. We introduce the notions of unstable and stable fixed points for quasi-automorphisms, which prove useful for constructing cluster monomials and non-real modules, respectively. As an application, we demonstrate that the counts of prime non-real tableaux with a fixed number of columns in $\mathrm{SSYT}(3, [9])$ and $\mathrm{SSYT}(4, [8])$, arising from the braid group action on stable fixed points, are governed by Euler's totient function. Furthermore, we apply our findings to scattering amplitudes in physics, providing a novel interpretation of the square root associated with the four-mass box integral via stable fixed points of quasi-automorphisms of the Grassmannian cluster algebra $\CC[\Gr(4,8)]$.

math.RT↗

arXiv2026

Compatibility of Drinfeld presentations and $q$-characters for affine Kac-Moody quantum symmetric pairs: quasi-split case

Let $(\mathbf{U}, \mathbf{U}^\imath)$ be a quasi-split affine quantum symmetric pair of type $\mathsf{AIII}$. This case is of particular interest thanks to the existence of geometric realizations and Schur--Weyl dualities. We establish factorization and coproduct formulae for the Drinfeld--Cartan series $\boldsymbol\Theta_i(z)$ in the Lu--Pan--Wang--Zhang `new Drinfeld'-style presentation, generalizing the split type results from [Prz23, LP25a]. As an application, we construct a boundary analogue of the $q$-character map, and show that it is compatible with Frenkel and Reshetikhin's original $q$-character homomorphism.

math.QA↗

arXiv2025

Langlands branching rule for type B snake modules

We prove that each snake module of the quantum Kac-Moody algebra of type $B_n^{(1)}$ admits a Langlands dual representation, as conjectured by Frenkel and Hernandez (Lett. Math. Phys. (2011) 96:217-261). Furthermore, we establish an explicit formula, called the Langlands branching rule, which gives the multiplicities in the decomposition of the character of a snake module of the quantum Kac-Moody algebra of type $B_n^{(1)}$ into a sum of characters of irreducible representations of its Langlands dual algebra.

math.RT↗

arXiv2025

From dual canonical bases to positroidal subdivisions

The Grassmannian cluster algebra $\mathbb{C}[\text{Gr}(k, n)]$ admits a distinguished basis known as the dual canonical basis, whose elements correspond to rectangular semi-standard Young tableaux with $k$ rows and with entries in $[n]$. We establish that each such tableau induces a positroidal subdivision of the hypersimplex $\Delta(k,n)$ via a map introduced by Speyer and Williams. For $\text{Gr}(2,n)$, we prove that non-frozen prime tableaux correspond precisely to the coarsest positroidal subdivisions of $\Delta(2,n)$. Furthermore, we present computational evidence extending these results to $k>2$. In the process, we formulate a conjectural formula for the number of split positroidal subdivisions of $\Delta(k,n)$ for any $k \ge 2$ and explore the deep connections between the polyhedral combinatorics of $\Delta(k,n)$ and the dual canonical basis of $\mathbb{C}[\text{Gr}(k, n)]$.

math.CO↗

arXiv2025

Boundary $q$-characters of evaluation modules for split quantum affine symmetric pairs

We study evaluation modules for quantum symmetric pair coideal subalgebras of affine type $\mathsf{AI}$. By computing the action of the generators in Lu and Wang's Drinfeld-type presentation on Gelfand-Tsetlin bases, we determine the spectrum of a large commutative subalgebra arising from the Lu-Wang presentation. This leads to an explicit formula for boundary analogues of $q$-characters in the setting of quantum affine symmetric pairs. We interpret this formula combinatorially in terms of semistandard Young tableaux. Our results imply that boundary $q$-characters share familiar features with ordinary $q$-characters - such as a version of the highest weight property - yet they also display new phenomena, including an extra symmetry. In particular, we provide the first examples of boundary $q$-characters for quantum affine symmetric pairs that do not arise from restriction of ordinary $q$-characters, thereby revealing genuinely new structures in this new setting.

math.RT↗

arXiv2025

A path description for $\varepsilon$-characters of representations of type $A$ restricted quantum loop algebras at roots of unity

Fix $\varepsilon^{2\ell}=1$ with $\ell \geq 2$. In this paper, we show that all finite-dimensional simple modules of any restricted quantum loop algebra $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$ in a certain category can be transformed into snake modules. We obtain an effective and concrete path description for $\varepsilon$-characters of any simple module with highest $l$-weight of degree two and any Kirillov-Reshetikhin module of $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$. As an application of our path description, we obtain a necessary and sufficient condition for the tensor product of two fundamental representations of $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$ to be irreducible. Additionally, we obtain a necessary condition for the tensor product of two or more fundamental representations of $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$ to be irreducible.

math.QA↗

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.

math.RT↗

arXiv2024

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

Let $\sigma_i$ 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 $\sigma_i$ coincide with the braid group actions $\mathsf{T}_i$ and $\mathsf{R}_i$.

math.RT↗

arXiv2024

Auslander-Reiten combinatorics and $q$-characters of representations of affine quantum groups

For each simple Lie algebra $\mathfrak{g}$ of simply-laced type, Hernandez and Leclerc introduced a certain category $\mathcal{C}_{\mathbb{Z}}$ of finite-dimensional representations of the quantum affine algebra of $\mathfrak{g}$, as well as certain subcategories $\mathcal{C}_{\mathbb{Z}}^{\leq \xi}$ depending on a choice of height function adapted to an orientation of the Dynkin graph of $\mathfrak{g}$. In our previous work we constructed an algebra homomorphism $\widetilde{D}_{\xi}$ whose domain contains the image of the Grothendieck ring of $\mathcal{C}_{\mathbb{Z}}^{\leq \xi}$ under the truncated $q$-character morphism $\widetilde{\chi}_q$ corresponding to $\xi$. We exhibited a close relationship between the composition of $\widetilde{D}_{\xi}$ with $\widetilde{\chi}_q$ and the morphism $\overline{D}$ recently introduced by Baumann, Kamnitzer and Knutson in their study of the equivariant homology of Mirkovi\'c-Vilonen cycles. In this paper, we extend $\widetilde{D}_{\xi}$ in order to investigate its composition with Frenkel-Reshetikhin's original $q$-character morphism. Our main result consists in proving that the $q$-characters of all standard modules in $\mathcal{C}_{\mathbb{Z}}$ lie in the kernel of $\widetilde{D}_{\xi}$. This provides a large family of new non-trivial rational identities suggesting possible geometric interpretations.

math.RT↗

arXiv2024

Dual conformal invariant kinematics and folding of Grassmannian cluster algebras

Grassmannian manifolds $\Gr(4,n)$ are closely related to the kinematic space of $n$-particle scattering processes in $D=4$, and their combinatorial and geometric structures have played an important role in the study of conformal invariant theories and scattering amplitudes. He, Li, and Yang \cite{HLY26} observed that restricting $D=4$ kinematics to a $D=3$ subspace can be interpreted as a folding of the Grassmannian cluster algebra $\CC[\Gr(4,n)]$ for $n\leq 8$. In this paper, we derive general expressions for the $D=3$ kinematic constraints in terms of Pl\"ucker coordinates of $\Gr(4,n)$ directly from the three-dimensional kinematic condition. We then construct a family of foldable seeds for $\CC[\Gr(4,n)]$, obtained explicitly from the standard initial seed by mutation, whose folding conditions reproduce these kinematic constraints. This establishes the connection between $D=3$ kinematics and folding of Grassmannian cluster algebras for general $n$.

math-ph↗

arXiv2024

Cluster structures on spinor helicity and momentum twistor varieties

We study the homogeneous coordinate rings of partial flag varieties and Grassmannians in their Pl\"ucker 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.

math.AG↗