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Seok-Jin Kang

Publications and source records attributed to Seok-Jin Kang.

At least 19 recordsLinked to original sources
arXiv2026

Quantum groups of Borcherds-Cartan type and Khovanov-Lauda-Rouquier algebras

We categorify a class of quantum groups associated with quivers, possibly with loops, by constructing the corresponding Khovanov-Lauda-Rouquier algebras (KLR) algebras $R$. We prove that the indecomposable projective $R$-modules realize the canonical basis of the negative part $U^-$ of the quantum group. Moreover, for $Λ\in P^+$, the cyclotomic KLR algebra $R^Λ$ provide a categorification of the irreducible highest weight $U$-module $V(Λ)$.

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arXiv2024

Perfect basis theory for quantum Borcherds-Bozec algebras

In this paper, we develop the perfect basis theory for quantum Borcherds-Bozec algebras $U_{q}(\mathfrak g)$ and their irreducible highest weight modules $V(λ)$. We show that the lower perfect graph (resp. upper perfect graph) of every lower perfect basis (resp. upper perfect basis) of $U_{q}^{-}(\mathfrak g)$ (resp. $V(λ)$) is isomorphic to the crystal $B(\infty)$ (resp. $B(λ)$).

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arXiv2024

Crystal bases and canonical bases for quantum Borcherds-Bozec algebras

Let $U_{q}^{-}(\mathfrak g)$ be the negative half of a quantum Borcherds-Bozec algebra $U_{q}(\mathfrak g)$ and $V(λ)$ be the irreducible highest weight module with $λ\in P^{+}$. In this paper, we investigate the structures, properties and their close connections between crystal bases and canonical bases of $U_{q}^{-}(\mathfrak g)$ and $V(λ)$. We first re-construct crystal basis theory with modified Kashiwara operators. While going through Kashiwara's grand-loop argument, we prove several important lemmas, which play crucial roles in the later developments of the paper. Next, based on the theory of canonical bases on quantum Bocherds-Bozec algebras, we introduce the notion of primitive canonical bases and prove that primitive canonical bases coincide with lower global bases.

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arXiv2024

A new Young wall realization of $B(λ)$ and $B(\infty)$

Using new combinatorics of Young walls, we give a new construction of the arbitrary level highest weight crystal $B(λ)$ for the quantum affine algebras of types $A^{(2)}_{2n}$, $D^{(2)}_{n+1}$, $A^{(2)}_{2n-1}$, $D^{(1)}_n$, $B^{(1)}_n$ and $C^{(1)}_n$. We show that the crystal consisting of reduced Young walls is isomorphic to the crystal $B(λ)$. Moreover, we provide a new realization of the crystal $B(\infty)$ in terms of reduced virtual Young walls and reduced extended Young walls.

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arXiv2024

Young wall models for the level 1 highest weight and Fock space crystals of $U_q(E_6^{(2)})$ and $U_q(F_4^{(1)})$

In this paper we construct Young wall models for the level $1$ highest weight and Fock space crystals of quantum affine algebras in types $E_6^{(2)}$ and $F_4^{(1)}$. Our starting point in each case is a combinatorial realization for a certain level $1$ perfect crystal in terms of Young columns. Then using energy functions and affine energy functions we define the notions of reduced and proper Young walls, which model the highest weight and Fock space crystals respectively.

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arXiv2023

Young wall construction of level-1 highest weight crystals over $U_q(D_4^{(3)})$ and $U_q(G_2^{(1)})$

With the help of path realization and affine energy function, we give a Young wall construction of level-1 highest weight crystals $B(λ)$ over $U_{q}(G_{2}^{(1)})$ and $U_{q}(D_{4}^{(3)})$. Our construction is based on four different shapes of colored blocks, $\mathbf O$-block, $\mathbf I$-block, $\mathbf L$-block and $\mathbf{LL}$-block, obtained by cutting the unit cube in three different ways.

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arXiv2021

Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras

Let $J$ be a set of pairs consisting of good modules over an affine quantum algebra and invertible elements. The distribution of poles of the normalized R-matrices yields Khovanov-Lauda-Rouquier algebras $R^J$. We define a functor $F$ from the category $S_J$ of finite-dimensional graded $R^J$-modules to the category of finite-dimensional integrable $U_q(g)$-modules. The functor $F$ sends convolution products of $R^J$-modules to tensor products of $U_q(g)$-modules. It is exact if $R^J$ is of finite type A,D,E. When $J$ is the vector representation of $A^{(1)}_{n-1}$, we recover the affine Schur-Weyl duality. Focusing on this case, we obtain an abelian rigid graded tensor category $T_J$ by localizing the category $S_J$. The functor $F$ factors through $T_J$. Moreover, the Grothendieck ring of the category $C_J$, the image of $F$, is isomorphic to the Grothendieck ring of $T_J$ at $q=1$.

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arXiv2020

Abstract crystals for quantum Borcherds-Bozec algebras

In this paper, we develop the theory of abstract crystals for quantum Borcherds-Bozec algebras. Our construction is different from the one given by Bozec. We further prove the crystal embedding theorem and provide a characterization of ${B}(\infty)$ and ${B}(λ)$ as its application, where ${B}(\infty)$ and ${B}(λ)$ are the crystals of the negative half part of the quantum Borcherds-Bozec algebra $U_q(\mathfrak g)$ and its irreducible highest weight module $V(λ)$, respectively.

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arXiv2020

Classical limit of quantum Borcherds-Bozec algebras

Let $\mathfrak{g}$ be a Borcherds-Bozec algebra, $U(\mathfrak{g})$ be its universal enveloping algebra and $U_{q}(\mathfrak{g})$ be the corresponding quantum Borcherds-Bozec algebra. We show that the classical limit of $U_{q}(\mathfrak{g})$ is isomorphic to $U(\mathfrak{g})$ as Hopf algebras. Thus $U_{q}(\mathfrak{g})$ can be regarded as a quantum deformation of $U(\mathfrak{g})$. We also give explicit formulas for the commutation relations among the generators of $U_{q}(\mathfrak{g})$.

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arXiv2019

Representation theory of symmetric groups and the strong Lefschetz property

We investigate the structure and properties of an Artinian monomial complete intersection quotient $A(n,d)=\mathbf{k} [x_{1}, \ldots, x_{n}] \big / (x_{1}^{d}, \ldots, x_{n}^d)$. We construct explicit homogeneous bases of $A(n,d)$ that are compatible with the $S_{n}$-module structure for $n=3$, all exponents $d \ge 3$ and all homogeneous degrees $j \ge 0$. Moreover, we derive the multiplicity formulas, both in recursive form and in closed form, for each irreducible component appearing in the $S_{3}$-module decomposition of homogeneous subspaces. 4, 5$.

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arXiv2018

Borcherds-Bozec algebras, root multiplicities and the Schofield construction

Using the twisted denominator identity, we derive a closed form root multiplicity formula for all symmetrizable Borcherds-Bozec algebras and discuss its applications including the case of Monster Borcherds-Bozec algebra. In the second half of the paper, we provide the Schofield constuction of symmetric Borcherds-Bozec algebras.

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arXiv2018

Monoidal categorification of cluster algebras (merged version)

We prove that the quantum cluster algebra structure of a unipotent quantum coordinate ring $A_q(\mathfrak{n}(w))$, associated with a symmetric Kac-Moody algebra and its Weyl group element $w$, admits a monoidal categorification via the representations of symmetric Khovanov-Lauda- Rouquier algebras. In order to achieve this goal, we give a formulation of monoidal categorifications of quantum cluster algebras and provide a criterion for a monoidal category of finite-dimensional graded $R$-modules to become a monoidal categorification, where $R$ is a symmetric Khovanov-Lauda-Rouquier algebra. Roughly speaking, this criterion asserts that a quantum monoidal seed can be mutated successively in all the directions, once the first-step mutations are possible. Then, we show the existence of a quantum monoidal seed of $A_q(\mathfrak{n}(w))$ which admits the first-step mutations in all the directions. As a consequence, we prove the conjecture that any cluster monomial is a member of the upper global basis up to a power of $q^{1/2}$. In the course of our investigation, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.

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arXiv2017

Young wall model for $A_2^{(2)}$-type adjoint crystals

We construct a Young wall model for higher level $A_2^{(2)}$-type adjoint crystals. The Young walls and reduced Young walls are defined in connection with affin energy function. We prove that the affine crystal consisiting of reduced Young walls provides a realization of highest weight crystals $B(λ)$ and $B(\infty)$.

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