arXiv subjects

Se-jin Oh

Publications and source records attributed to Se-jin Oh.

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↗

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.

math.RT↗

arXiv2025

Denominators of R-matrices, higher Dorey's rules and a generalization of T-systems for quantum affine algebras

We construct a higher level analogue of Dorey's rule, which describe certain surjective morphisms between Kirillov--Reshetikhin (KR) modules over quantum affine algebras. Building on this, we establish a generalized T-system of short exact sequences and prove the denominator formula between KR modules in all nonexceptional types, except with only mild ambiguities persisting in type $C_n^{(1)}$. As a consequence, we can completely classify when a tensor product of KR modules is simple. These results have further applications to Schur positivity statements, quiver Hecke algebras, and the recently introduced $\mathfrak{d}$-invariants in monoidal categories over quantum affine algebras and quiver Hecke algebras.

math.QA↗

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.

math.RT↗

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.

math.RT↗

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 $\widehat{\mathcal{A}}$.

math.RT↗

arXiv2024

Quantum cluster algebra, braid moves and quantum virtual Grothendieck ring

In this paper, we study the quantum virtual Grothendieck ring, denoted by $\frakK_q(\g)$, which was introduced in [39], and further investigated in [26, 25]. Our approach involves examining this ring from two perspectives: first, by considering its connection to quantum cluster algebras of non-skew-symmetric types; and second, by exploring its relevance to categorification theory. We specifically focus on (i) the homomorphisms that arise from braid moves, particularly 4-moves and 6-moves, in the braid group; and (ii) the quantum Laurent positivity phenomena, which has not yet been proven for non-skew-symmetric types. As applications of our results, we derive the substitution formulas for non-skew-symmetric types discussed in [11] for skew-symmetric types, and demonstrate that any truncated element in a heart subring, denoted by $\frakK_{q,Q}(\g)$, which corresponds to a simple module over the quiver Hecke algebra $R^\g$, possesses coefficients in $\Z_{\ge 0}[q^{\pm 1/2}]$. This result is particularly interesting because it implies that each truncated Kirillov--Reshetikhin polynomial in $\frakK_{q,Q}(\g)$ and each element in the standard basis $\sfE_q(\g)$ of the entire ring $\frakK_q(\g)$ have coefficients also in $\Z_{\ge 0}[q^{\pm 1/2}]$. Since (truncated) Kirillov--Reshetikhin polynomials can be obtained using a quantum cluster algebra algorithm and appear as quantum cluster variables, they provide compelling evidence in support of the quantum Laurent positivity conjecture in non-skew-symmetric types.

math.QA↗

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.

math.QA↗

arXiv2023

Quantization of virtual Grothendieck rings and their structure including quantum cluster algebras

The quantum Grothendieck ring of a certain category of finite-dimensional modules over a quantum loop algebra associated with a complex finite-dimensional simple Lie algebra $\mathfrak{g}$ has a quantum cluster algebra structure of skew-symmetric type. Partly motivated by a search of a ring corresponding to a quantum cluster algebra of {\em skew-symmetrizable} type, the quantum {\em virtual} Grothendieck ring, denoted by $\mathfrak{K}_q(\mathfrak{g})$, is recently introduced by Kashiwara--Oh \cite{KO23} as a subring of the quantum torus based on the $(q,t)$-Cartan matrix specialized at $q=1$. In this paper, we prove that $\mathfrak{K}_q(\mathfrak{g})$ indeed has a quantum cluster algebra structure of skew-symmetrizable type. This task essentially involves constructing distinguished bases of $\mathfrak{K}_q(\mathfrak{g})$ that will be used to make cluster variables and generalizing the quantum $T$-system associated with Kirillov--Reshetikhin modules to establish a quantum exchange relation of cluster variables. Furthermore, these distinguished bases naturally fit into the paradigm of Kazhdan--Lusztig theory and our study of these bases leads to some conjectures on quantum positivity and $q$-commutativity.

math.RT↗

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

math.RT↗

arXiv2023

Braid group action on quantum virtual Grothendieck ring through constructing presentations

As a continuation of \cite{JLO1}, we investigate the quantum virtual Grothendieck ring $\frakK_q(\g)$ associated with a finite dimensional simple Lie algebra $\g$, especially of non-simply-laced type. We establish an isomorphism $\Uppsi_Q$ between the heart subring $\frakK_{q,Q}(\g)$ of $\frakK_q(\g)$ associated with a Dynkin quiver $Q$ of type $\g$ and the unipotent quantum coordinate algebra $\calA_q(\n)$ of type $\g$. This isomorphism and the categorification theory via quiver Hecke algebras enable us to obtain a presentation of $\frakK_q(\g)$, which reveals that $\frakK_q(\g)$ can be understood as a boson-extension of $\calA_q(\n)$. Then we show that the automorphisms, arising from the reflections on Dynkin quivers and the isomorphisms $\Uppsi_Q$, preserve the canonical basis $\sfL_q$ of $\frakK_q(\g)$. Finally, we prove that such automorphisms produce a braid group $B_\g$ action on $\frakK_q(\g)$.

math.RT↗

arXiv2023

Isomorphisms among quantum Grothendieck rings and cluster algebras

We establish a cluster theoretical interpretation of the isomorphisms of [F.-H.-O.-O., J. Reine Angew. Math., 2022] among quantum Grothendieck rings of representations of quantum loop algebras. Consequently, we obtain a quantization of the monoidal categorification theorem of [Kashiwara-Kim-Oh-Park, arXiv:2103.10067]. We establish applications of these new ingredients. First we solve long-standing problems for any non-simply-laced quantum loop algebras: the positivity of $(q,t)$-characters of all simple modules, and the analog of Kazhdan-Lusztig conjecture for all reachable modules (in the cluster monoidal categorification). We also establish the conjectural quantum $T$-systems for the $(q,t)$-characters of Kirillov-Reshetikhin modules. Eventually, we show that our isomorphisms arise from explicit birational transformations of variables, which we call substitution formulas. This reveals new non-trivial relations among $(q, t)$-characters of simple modules.

math.RT↗

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↗

arXiv2023

The $(q,t)$-Cartan matrix specialized at $q=1$

The $(q,t)$-Cartan matrix specialized at $t=1$, usually called the quantum Cartan matrix, has deep connections with (i) the representation theory of its untwisted quantum affine algebra, and (ii) quantum unipotent coordinate algebra, root system and quantum cluster algebra of kew-symmetric type. In this paper, we study the $(q,t)$-Cartan matrix specialized at $q=1$, called the $t$-quantized Cartan matrix, and investigate the relations with (ii') its corresponding quantum unipotent coordinate algebra, root system and quantum cluster algebra of skew-symmetrizable type.

math.QA↗

arXiv2023

$t$-quantized Cartan matrix and R-matrices for cuspidal modules over quiver Hecke algebras

As every simple module of a quiver Hecke algebra appears as the image of the R-matrix defined on the convolution product of certain cuspidal modules, knowing the $\mathbb{Z}$-invariants of the R-matrices between cuspidal modules is quite significant. In this paper, we prove that the $(q,t)$-Cartan matrix specialized at $q=1$ of an arbitrary finite type, called the $t$-quantized Cartan matrix, informs us of the invariants of R-matrices. To prove this, we use combinatorial AR-quivers associated with Dynkin quivers and their properties as crucial ingredients.

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↗