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Il-Seung Jang

Publications and source records attributed to Il-Seung Jang.

14 recordsLinked to original sources
arXiv2026

Special Kirillov-Reshetikhin crystals

We give a uniform realization of Kirillov-Reshetikhin crystals $B^{r,s}$, in terms of PBW crystals, for all affine types, $s \geq 1$, and nodes $r$ in the orbit of the $0$ node in the Dynkin diagram. Our proof is almost uniform except we prove some technical lemmas by first proving the case when the $r$-th fundamental weight $Λ_r$ is minuscule by using properties of minuscule crystal and then extend this to the case when $Λ_r$ is cominuscule using virtual crystals.

math.CO↗
arXiv2026

Unipotent quantum coordinate ring and minuscule prefundamental representations: twisted case

We study the prefundamental modules $L_{s,a}^{\pm}$ over the Borel subalgebras of the twisted quantum loop algebras, which are introduced by Wang. A character formula for $L_{s,a}^{\pm}$ is obtained from that for the prefundamental modules over the untwisted quantum loop algebras by applying a character folding map. This allows us to realize minuscule prefundamental modules $L_{s,a}^{\pm}$ for types $A_{2n-1}^{(2)}$ and $D_{n+1}^{(2)}$ in terms of the unipotent quantum coordinate ring associated with the $s$-th level $0$ fundamental weight, where $s = 1$ for type $A_{2n-1}^{(2)}$ and $s = n$ for type $D_{n+1}^{(2)}$. This result is a continuation of the realization of (co)minuscule prefundamental modules established by earlier works [J-Kwon-Park, Int. Math. Res. Not., 2023] and [J-Kwon-Park, J. Algebra, 2025].

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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 $η_r \in \mathbb{C}^\times$.

math.RT↗
arXiv2025

Quantum nilpotent subalgebras of classical quantum groups and affine crystals

We study the crystal of quantum nilpotent subalgebra of $U_q(D_n)$ associated to a maximal Levi subalgebra of type $A_{n-1}$. We show that it has an affine crystal structure of type $D_n^{(1)}$ isomorphic to a limit of perfect Kirillov-Reshetikhin crystal $B^{n,s}$ for $s\geq 1$, and give a new polytope realization of $B^{n,s}$. We show that an analogue of RSK correspondence for type $D$ due to Burge is an isomorphism of affine crystals and give a generalization of Greene's formula for type $D$.

math.QA↗
arXiv2025

Crystal base of the negative half of quantum orthosymplectic superalgebra

We construct a crystal base of the negative half of a quantum orthosymplectic superalgebra. It can be viewed as a limit of the crystal bases of $q$-deformed irreducible oscillator representations. We also give a combinatorial description of the embedding from the crystal of a $q$-oscillator representation to that of the negative half subalgebra given in terms of a PBW type basis. It is given as a composition of embeddings into the crystals of intermediate parabolic Verma modules, where the most non-trivial one is from an oscillator module to a maximally parabolic Verma module with respect to a quantum subsuperalgebra for $\mathfrak{gl}_{m|n}$. A new crystal theoretic realization of Burge correspondence of orthosymplectic type plays an important role for the description of this embedding.

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.

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arXiv2023

Path description for $q$-characters of fundamental modules in type $C$

In this paper, we investigate the behavior of monomials in the $q$-characters of the fundamental modules over a quantum affine algebra of untwisted type C. As a result, we give simple closed formulae for the $q$-characters of the fundamental modules in terms of sequences of vertices in $\mathbb{R}^2$, so-called paths, with an admissible condition. This may be viewed as a type C analog of the path description of $q$-characters in types A and B due to Mukhin--Young.

math.QA↗
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 $\frakK_q(\g)$.

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arXiv2022

Crystal base of the negative half of the quantum superalgebra $U_q(\mathfrak{gl}(m|n))$

We construct a crystal base of $U_q(\mathfrak{gl}(m|n))^-$, the negative half of the quantum superalgebra $U_q(\mathfrak{gl}(m|n))$. We give a combinatorial description of the associated crystal $\mathscr{B}_{m|n}(\infty)$, which is equal to the limit of the crystals of the ($q$-deformed) Kac modules $K(λ)$. We also construct a crystal base of a parabolic Verma module $X(λ)$ associated with the subalgebra $U_q(\mathfrak{gl}_{0|n})$, and show that it is compatible with the crystal base of $U_q(\mathfrak{gl}(m|n))^-$ and the Kac module $K(λ)$ under the canonical embedding and projection of $X(λ)$ to $U_q(\mathfrak{gl}(m|n))^-$ and $K(λ)$, respectively.

math.QA↗
arXiv2022

A combinatorial realization of Kirillov-Reshetikhin crystals for type E arising from translations

The main purpose of this paper is to give a combinatorial realization of Kirillov-Reshetikhin (KR simply) crystals $B^{r, s}$ for type $\text{E}_n^{(1)}$ with a minuscule node $r$ and $s \ge 1$. To do this, we describe explicitly the crystal of the quantum nilpotent subalgebra associated with the translation by the negative of the $r$-th fundamental weight. Then the crystal can be extended as an affine crystal, in which a certain subcrystal characterized by the $\varepsilon_r^*$-statistic is isomorphic to $B^{r,s}$ as an affine crystal, where $\varepsilon_r^*$ is also realized precisely in terms of triple and quadruple paths.

math.QA↗
arXiv2021

Unipotent quantum coordinate ring and prefundamental representations for types $A_n^{(1)}$ and $D_n^{(1)}$

We give a new realization of the prefundamental representations $L^\pm_{r,a}$ introduced by Hernandez and Jimbo, when the quantum loop algebra $U_q(\mathfrak{g})$ is of types $A_n^{(1)}$ and $D_n^{(1)}$, and the $r$-th fundamental weight $\varpi_r$ for types $A_n$ and $D_n$ is minuscule. We define an action of the Borel subalgebra $U_q(\mathfrak{b})$ of $U_q(\mathfrak{g})$ on the unipotent quantum coordinate ring associated to the translation by $-\varpi_r$, and show that it is isomorphic to $L^\pm_{r,a}$. We then give a combinatorial realization of $L^+_{r,a}$ in terms of the Lusztig data of the dual PBW vectors.

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arXiv2020

Flagged Littlewood-Richardson tableaux and branching rule for classical groups

We give a new formula for the branching rule from ${\rm GL}_n$ to ${\rm O}_n$ generalizing the Littlewood's restriction formula. The formula is given in terms of Littlewood-Richardson tableaux with certain flag conditions which vanish in a stable range. As an application, we give a combinatorial formula for the Lusztig $t$-weight multiplicity $K_{μ0}(t)$ of type $B_n$ and $D_n$ with highest weight $μ$ and weight $0$.

math.RT↗
arXiv2020

Lusztig data of Kashiwara-Nakashima tableaux in type D

We describe the embedding from the crystal of Kashiwara-Nakashima tableaux in type $D$ of an arbitrary shape into that of $\mathbf{i}$-Lusztig data associated to a family of reduced expressions $\mathbf{i}$ which are compatible with the maximal Levi subalgebra of type $A$. The embedding is described explicitly in terms of well-known combinatorics of type $A$ including the Schützenberger's jeu de taquin and an analog of RSK algorithm.

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