arXiv subjects

Myungho Kim

Publications and source records attributed to Myungho Kim.

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 $M(w_{\le k } \Lambda_{i_k}, v_{\le k} \Lambda_{i_k} )$. We prove that the Grothendieck ring $K(\mathcal{C}_{wv}) $ lies between the cluster algebra and the upper cluster algebra.

math.RT↗

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.

math.AG↗

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 $\mathbb{C}[X({\underline{\Delta}} {\boldsymbol{i}})]$ of the braid variety $X({\underline{\Delta}} {\boldsymbol{i}})$ equipped with an initial seed arising from a Demazure weave $\mathfrak{W}$, where ${\boldsymbol{i}}$ and ${\underline{\Delta}}$ are expression sequences of $\mathtt{b}$ and the half twist $\Delta$, respectively. We explicitly construct a Demazure weave $\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})$ for each admissible chain $\mathfrak{C}$ associated with ${\boldsymbol{i}}$, and prove that there exists an algebra isomorphism $\varphi_{{\boldsymbol{i}}}\colon {\widetilde{\mathbb{A}}}_\mathbb{C}(\mathtt{b})\to\mathfrak{C}[X({\underline{\Delta}} {\boldsymbol{i}})]$ which is compatible with the two seeds arising from $\mathfrak{C}$ and $\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})$. Moreover, the isomorphism $\varphi_{{\boldsymbol{i}}}$ sends the PBW vectors ${\overline{\mathsf{p}}}_{{\boldsymbol{i}},k} \in {\widetilde{\mathbb{A}}}_\mathbb{C}(\mathtt{b})$ to the coordinates $z_k \in \mathfrak{C}[X({\underline{\Delta}} {\boldsymbol{i}})]$ 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 $\varphi_{{\boldsymbol{i}}}$ from the viewpoint of monoidal categorification using Hernandez--Leclerc categories.

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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

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

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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arXiv2024

Exchange matrices of I-boxes

Admissible chains of i-boxes are important combinatorial tools in the monoidal categorification of cluster algebras, as they provide seeds of the cluster algebra. In this paper, we explore the properties of maximal commuting families of i-boxes in a more general setting, and define a certain matrix associated with such a family, which we call the exchange matrix. It turns out that, when considering the cluster algebra structure on the Grothendieck rings, this matrix is indeed the exchange matrix of the seed associated with the family, both in certain categories of modules over quantum affine algebras and over quiver Hecke algebras. We prove this by constructing explicit short exact sequences that represent the mutation relations.

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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,

math.RT↗

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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arXiv2023

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 $\Delta$ in the family produces a completely integrable system of $X$ over $\Delta$. 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.

math.SG↗

arXiv2023

Modified Ariki-Koike algebra and Yokounuma-Hecke like relations

We find new presentations of the modified Ariki-Koike algebra (known also as Shoji's algebra) $\mathcal H_{n,r}$ over an integral domain $R$ associated with a set of parameters $q,u_1,\ldots,u_r$ in $R$. It turns out that the algebra $\mathcal H_{n,r}$ has a set of generators $t_1,\ldots,t_n$ and $g_1,\ldots g_{n-1}$ subject to a set of defining relations similar to the relations of Yokonuma-Hecke algebra. We also obtain a presentation of $\mathcal H_{n,r}$ which is independent to the choice of $u_1,\ldots u_r$. Hence the algebras associated with the parameters $q, u_1,\ldots u_r$ and $q, u'_1,\ldots u'_r$ are isomorphic even in the case that $(u_1,\ldots u_r)$ and $(u'_1,\ldots u'_r)$ are different. As applications of the presentations, we find an explicit trace form on the algebra $\mathcal H_{n,r}$ which is symmetrising provided the parameters $u_1,\ldots, u_r$ are invertible in $R$. We also show that the symmetric group $\mathfrak S(r)$ acts on the algebra $\mathcal H_{n,r}$, and find a basis and a set of generators of the fixed subalgebra $\mathcal H_{n,r}^{\mathfrak S(r)}$.

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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.

math.RT↗

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 $\Lambda$-invariants of R-matrices in the quiver Hecke algebra theory.

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arXiv2023

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.

math.RT↗

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.

math.RT↗

arXiv2021

Monoidal categorification and quantum affine algebras II

We introduce a new family of real simple modules over the quantum affine algebras, called the affine determinantial modules, which contains the Kirillov-Reshetikhin (KR)-modules as a special subfamily, and then prove T-systems among them which generalize the T-systems among KR-modules and unipotent quantum minors in the quantum unipotent coordinate algebras simultaneously. We develop new combinatorial tools: admissible chains of i-boxes which produce commuting families of affine determinantial modules, and box moves which describe the T-system in a combinatorial way. Using these results, we prove that various module categories over the quantum affine algebras provide monoidal categorifications of cluster algebras. As special cases, Hernandez-Leclerc categories provide monoidal categorifications of the cluster algebras for an arbitrary quantum affine algebra.

math.QA↗

arXiv2020

PBW theory for quantum affine algebras

Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of arbitrary type and let $\mathcal{C}_{\mathfrak{g}}$ be Hernandez-Leclerc's category. We can associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality datum $D$ in $\mathcal{C}_{\mathfrak{g}}$. We introduce the notion of a strong (complete) duality datum $D$ and prove that, when $D$ is strong, the induced duality functor $F_D$ sends simple modules to simple modules and preserves the invariants $\Lambda$ and $\Lambda^\infty$ introduced by the authors. We next define the reflections $\mathcal{S}_k$ and $\mathcal{S}^{-1}_k$ acting on strong duality data $D$. We prove that if $D$ is a strong (resp.\ complete) duality datum, then $\mathcal{S}_k(D)$ and $\mathcal{S}_k^{-1}(D)$ are also strong (resp.\ complete ) duality data. We finally introduce the notion of affine cuspidal modules in $\mathcal{C}_{\mathfrak{g}}$ by using the duality functor $F_D$, and develop the cuspidal module theory for quantum affine algebras similarly to the quiver Hecke algebra case.

math.RT↗

arXiv2020

Polynomial invariants on matrices and partition, Brauer algebra

We identify the dimension of the centralizer of the symmetric group $\mathfrak{S}_d$ in the partition algebra $\mathcal{A}_d(\delta)$ and in the Brauer algebra $\mathcal{B}_d(\delta)$ with the number of multidigraphs with $d$ arrows and the number of disjoint union of directed cycles with $d$ arrows, respectively. Using Schur-Weyl duality as a fundamental theory, we conclude that each centralizer is related with the $G$-invariant space $P^d(M_n(\mathbf{k}))^G$ of degree $d$ homogeneous polynomials on $n \times n$ matrices, where $G$ is the orthogonal group and the group of permutation matrices, respectively. Our approach gives a uniform way to show that the dimensions of $P^d(M_n(\mathbf{k}))^G$ are stable for sufficiently large $n$.

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