arXiv subjects

Euiyong Park

Publications and source records attributed to Euiyong Park.

At least 19 recordsLinked to original sources

arXiv2026

Monoidal seeds of the categories $\mathcal{C}_{wv}$ over quiver Hecke algebras

In this paper, when the quiver Hecke algebra R is symmetric, we present a new construction of quantum monoidal seeds $ \mathscr{S}_{w,v}$ for $\mathcal{C}_{wv}$ using the reflection functors $\mathcal{F}_i$ and the newly introduced operators $\mathcal{K}_i$. The monoidal seed $ \mathscr{S}_{w,v}$ is obtained as a subseed of the monoidal seed of $\mathcal{C}_w$ constructed by applying $\mathcal{K}_i$ and $\mathcal{F}_i$ along the special KF sequence determined by a reduced expression of $w$ and $v$. We further prove that the monoidal seed $\mathscr{S}_{w,v}$ coincides with the set of all prime factors of the determinantial modules $M(w_{\le k } Λ_{i_k}, v_{\le k} Λ_{i_k} )$. We prove that the Grothendieck ring $K(\mathcal{C}_{wv}) $ lies between the cluster algebra and the upper cluster algebra.

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arXiv2026

Unipotent quantum coordinate ring and cominuscule prefundamental representations

We continue the study of realization of the prefundamental modules $L_{r,a}^{\pm}$, introduced by Hernandez and Jimbo, in terms of unipotent quantum coordinate rings as in [J-Kwon-Park, Int. Math. Res. Not., 2023]. We show that the ordinary character of $L_{r,a}^{\pm}$ is equal to that of the unipotent quantum coordinate ring $U_q^-(w_r)$ associated to fundamental $r$-th coweight. When $r$ is cominuscule, we prove that there exists a $U_q(\mathfrak{b})$-module structure on $U_q^-(w_r)$, which is isomorphic to $L_{r,aη_r}^\pm$ for some $η_r \in \mathbb{C}^\times$.

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arXiv2026

A Comparison of cluster algebra structures arising from $i$-boxes and Demazure weaves

We compare two cluster algebras related to a positive element $\mathtt{b}$ in the braid group of finite $ADE$ type. One is the localized bosonic extension ${\widetilde{\mathbb{A}}}_\mathbb{C}(\mathtt{b})$ equipped with an initial seed arising from an admissible chain $\mathfrak{C}$ of $i$-boxes, which is deeply connected to monoidal categorification. The other is the coordinate ring $\mathbb{C}[X({\underlineΔ} {\boldsymbol{i}})]$ of the braid variety $X({\underlineΔ} {\boldsymbol{i}})$ equipped with an initial seed arising from a Demazure weave $\mathfrak{W}$, where ${\boldsymbol{i}}$ and ${\underlineΔ}$ are expression sequences of $\mathtt{b}$ and the half twist $Δ$, respectively. We explicitly construct a Demazure weave $\mathfrak{W}_{\underlineΔ}(\mathfrak{C})$ for each admissible chain $\mathfrak{C}$ associated with ${\boldsymbol{i}}$, and prove that there exists an algebra isomorphism $φ_{\boldsymbol{i}}\colon {\widetilde{\mathbb{A}}}_\mathbb{C}(\mathtt{b})\to\mathfrak{C}[X({\underlineΔ} {\boldsymbol{i}})]$ which is compatible with the two seeds arising from $\mathfrak{C}$ and $\mathfrak{W}_{\underlineΔ}(\mathfrak{C})$. Moreover, the isomorphism $φ_{\boldsymbol{i}}$ sends the PBW vectors ${\overline{\mathsf{p}}}_{\boldsymbol{i},k} \in {\widetilde{\mathbb{A}}}_\mathbb{C}(\mathtt{b})$ to the coordinates $z_k \in \mathfrak{C}[X({\underlineΔ} {\boldsymbol{i}})]$ indexed by the letters of ${\boldsymbol{i}}$. As applications, we investigate a connection between Demazure weaves and signed words via the $i$-boxes and interpret the isomorphism $φ_{\boldsymbol{i}}$ from the viewpoint of monoidal categorification using Hernandez--Leclerc categories.

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arXiv2026

Newton--Okounkov bodies of partial flag varieties via cluster algebras

We construct Newton--Okounkov polytopes of Schubert varieties in partial flag varieties of arbitrary type using the cluster structure on a unipotent cell. When the governing cluster algebra is of infinite type, we prove that for any very ample homogeneous line bundle over a simply laced partial flag variety, the resulting family of Newton--Okounkov polytopes contains infinitely many pairwise nonequivalent polytopes up to integral affine transformation. As an application to symplectic geometry, we construct infinitely many distinct monotone Lagrangian tori in a broad class of simply laced partial flag varieties.

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arXiv2025

Faithful action of braid group on bosonic extensions

The braid group action on the bosonic extension of the quantum group has been introduced in recent works, and it can be regarded as a generalization of Lusztig's symmetries on the quantum group. In this notes, we prove the faithfulness of this braid group action.

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arXiv2025

Reflection functors on quiver Hecke algebras

We construct the reflection functors for quiver Hecke algebras of an arbitrary symmetrizable Kac-Moody type. These reflection functors categorify Lusztig's braid symmetries.

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arXiv2025

Monoidal categorification and quantum affine algebras III

Let $U_q'(\mathfrak{g})$ be an arbitrary quantum affine algebra of either untwisted or twisted type, and let $\mathscr{C}_{\mathfrak{g}}^0$ be its Hernandez-Leclerc category. We denote by $\mathsf{B}$ the braid group determined by the simply-laced finite type Lie algebra $ \mathsf{g}$ associated with $U_q'(\mathfrak{g})$. For any complete duality datum $\mathbb{D}$ and any sequence of simple roots of $\mathsf{g}$, we construct the corresponding affine cuspidal modules and affine determinantial modules and study their key properties including T-systems. Then, for any element $b$ of the positive braid monoid $\mathsf{B}^+$, we introduce a distinguished subcategory $\mathscr{C}_{\mathfrak{g}}^{\mathbb{D}}(b)$ of $\mathscr{C}_{\mathfrak{g}}^0$ categorifying the specialization of the bosonic extension $\widehat{\mathcal{A}}(b)$ at $q^{1/2}=1$ and investigate its properties including the categorical PBW structure. We finally prove that the subcategory $\mathscr{C}_{\mathfrak{g}}^{\mathbb{D}}(b)$ provides a monoidal categorification of the (quantum) cluster algebra $\widehat{\mathcal{A}}(b)$, which significantly generalizes the earlier monoidal categorification developed by the authors.

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arXiv2025

Cluster algebras and monotone Lagrangian tori

Motivated by the construction of Newton--Okounkov bodies and toric degenerations via cluster algebras in [GHKK18, FO25], we consider a family of Newton--Okounkov polytopes of a complex smooth Fano variety $X$ related by a composition of tropicalized cluster mutations. According to the work of [HK15], the toric degeneration associated with each Newton--Okounkov polytope $Δ$ in the family produces a completely integrable system of $X$ over $Δ$. We investigate circumstances in which each completely integrable system possesses a monotone Lagrangian torus fiber. We provide a sufficient condition, based on the data of tropical integer points and exchange matrices, for the family of constructed monotone Lagrangian tori to contain infinitely many monotone Lagrangian tori, no two of which are related by any symplectomorphism. By employing this criterion and exploiting the correspondence between the tropical integer points and the dual canonical basis elements, we generate infinitely many distinct monotone Lagrangian tori on flag manifolds of arbitrary type except in a few cases.

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arXiv2025

Crystals and quantum twist automorphisms

Let $η_w$ be the quantum twist automorphism for the quantum unipotent coordinate ring $\mathrm{A}_q(\mathfrak{n}(w))$ introduced by Kimura and Oya. In this paper, we study the quantum twist automorphism $η_w$ in the viewpoint of the crystal bases theory and provide a crystal-theoretic description of $η_w$. In the case of the $*$-twisted minuscule crystals of classical finite types, we provide a combinatorial description of $η_w$ in terms of (shifted) Young diagrams. We further investigate the periodicity of $η_w$ up to a multiple of frozen variables in various setting.

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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

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

Braid symmetries on bosonic extensions

We introduce a family of automorphisms on the bosonic extension of arbitrary type and show that they satisfy the braid relations. They preserve the global basis and the crystal basis. Using this braid group action, we define a subalgebra for each positive braid word, which possesses the PBW type basis. As an application, we show that the tensor product decomposition of the positive bosonic extionsion,

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arXiv2024

Global bases for Bosonic extensions of quantum unipotent coordinate rings

In the paper, we establish the global basis theory for the bosonic extension $\widehat{\mathcal{A}}$ associated with an arbitrary generalized Cartan matrix. When $\widehat{\mathcal{A}}$ is of simply-laced finite type, it is isomorphic to the quantum Grothendieck ring of the Hernandez-Leclerc category over a quantum affine algebra. In this case, we show that the $(t,q)$-characters of simple modules in the Hernandez-Leclerc category correspond to the normalized global basis of $\widehat{\mathcal{A}}$.

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arXiv2024

PBW theory for Bosonic extensions of quantum groups

In this paper, we develop the PBW theory for the bosonic extension $\qbA{\g}$ of a quantum group $\mathcal{U}_q(\g)$ of \emph{any} finite type. When $\g$ belongs to the class of \emph{simply-laced type}, the algebra $\qbA{\g}$ arises from the quantum Grothendieck ring of the Hernandez-Leclerc category over quantum affine algebras of untwisted affine types. We introduce and investigate a symmetric bilinear form $\pair{\ , \ }$ on $\qbA{\g}$ which is invariant under the braid group actions $\bT_i$ on $\qbA{\g}$, and study the adjoint operators $\Ep_{i,p}$ and $\Es_{i,p}$ with respect to $\pair{\ , \ }$. It turns out that the adjoint operators $\Ep_{i,p}$ and $\Es_{i,p}$ are analogues of the $q$-derivations $e_i'$ and $\es_i$ on the negative half $\calU_q^-(\g)$ of $\calU_q(\g)$. Following this, we introduce a new family of subalgebras denoted as $\qbA{\mathfrak{g}}(\ttb)$ in $\qbA{\mathfrak{g}}$. These subalgebras are defined for any elements $\ttb$ in the positive submonoid $\bg^+$ of the (generalized) braid group $\ttB$ of $\g$. We prove that $\qbA{\mathfrak{g}}(\ttb)$ exhibits PBW root vectors and PBW bases defined by $\bT_\ii$ for any sequence $\ii$ of $\ttb$. The PBW root vectors satisfy a Levendorskii-Soibelman formula and the PBW bases are orthogonal with respect to $\pair{\ , \ }$. The algebras $\qbA{\g} (\ttb)$ can be understood as a natural extension of quantum unipotent coordinate rings.

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arXiv2024

Affinizations, R-matrices and reflection functors

In this paper we establish affinizations and R-matrices in the language of pro-objects, and as an application, we construct reflection functors over the localizations of quiver Hecke algebras of arbitrary finite types. This reflection functor categorifies the braid group action on the half of a quantum group and the Saito reflection.

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arXiv2023

Localizations for quiver Hecke algebras III

Let $R$ be a quiver Hecke algebra, and let $\mathcal{C}_{w,v}$ be the category of finite-dimensional graded $R$-module categorifying a $q$-deformation of the doubly-invariant algebra $^{N'(w)} \mathbb{C}[N] ^{N(v)} $. In this paper, we prove that the localization $\tilde{\mathcal{C}}_{w,v}$ of the category $\mathcal{C}_{w,v}$ can be obtained as the localization by right braiders arising from determinantial modules. As its application, we show several interesting properties of the localized category $\tilde{\mathcal{C}}_{w,v} $ including the right rigidity.

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arXiv2023

Laurent family of simple modules over quiver Hecke algebra

We introduce the notions of quasi-Laurent and Laurent families of simple modules over quiver Hecke algebras of arbitrary symmetrizable types. We prove that such a family plays a similar role of a cluster in the quantum cluster algebra theory and exhibits a quantum Laurent positivity phenomenon for the basis of the quantum unipotent coordinate ring $\mathcal{A}_q(\mathfrak{n}(w))$, coming from the categorification. Then we show that the families of simple modules categorifying GLS-clusters are Laurent families by using the PBW-decomposition vector of a simple module $X$ and categorical interpretation of (co-)degree of $[X]$. As applications of such $\mathbb{Z}$-vectors, we define several skew symmetric pairings on arbitrary pairs of simple modules, and investigate the relationships among the pairings and $Λ$-invariants of R-matrices in the quiver Hecke algebra theory.

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arXiv2022

Localizations for quiver Hecke algebras II

We prove that the localization of the monoidal category $\mathcal{C}_w$ is rigid, and the category $\mathcal{C}_{w,v}$ admits a localization via a real commuting family of central objects. Note that the localization of $\mathcal{C}_{w,v}$ categorifies the open Richardson variety.

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