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Se-jin Oh

Publications and source records attributed to Se-jin Oh.

61 records · Page 4Linked to original sources
arXiv2014

Young walls and graded dimension formulas for finite quiver Hecke algebras of type $A^{(2)}_{2\ell}$ and $D^{(2)}_{\ell+1}$

We study graded dimension formulas for finite quiver Hecke algebras $R^{Λ_0}(β)$ of type $A^{(2)}_{2\ell}$ and $D^{(2)}_{\ell+1}$ using combinatorics of Young walls. We introduce the notion of standard tableaux for proper Young walls and show that the standard tableaux form a graded poset with lattice structure. We next investigate Laurent polynomials associated with proper Young walls and their standard tableaux arising from the Fock space representations consisting of proper Young walls. Then we prove the graded dimension formulas described in terms of the Laurent polynomials. When evaluating at $q=1$, the graded dimension formulas recover the dimension formulas for $R^{Λ_0}(β)$ described in terms of standard tableaux of strict partitions.

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arXiv2013

Supercategorification of quantum Kac-Moody algebras II

In this paper, we investigate the supercategories consisting of supermodules over quiver Hecke superalgebras and cyclotomic quiver Hecke superalgebras. We prove that these supercategories provide a supercategorification of a certain family of quantum superalgebras and their integrable highest weight modules. We show that, by taking a specialization, we obtain a supercategorification of quantum Kac-Moody superalgebras and their integrable highest weight modules.

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arXiv2012

Construction of Irreducible Representations over Khovanov-Lauda-Rouquier Algebras of Finite Classical Type

We give an explicit construction of irreducible modules over Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^λ$ for finite classical types using a crystal basis theoretic approach. More precisely, for each element $v$ of the crystal $B(\infty)$ (resp. $B(λ)$), we first construct certain modules $\nabla(\mathbf{a};k)$ labeled by the adapted string $\mathbf{a}$ of $v$. We then prove that the head of the induced module $\ind \big(\nabla(\mathbf{a};1) \boxtimes...\boxtimes \nabla(\mathbf{a};n)\big)$ is irreducible and that every irreducible $R$-module (resp. $R^λ$-module) can be realized as the irreducible head of one of the induced modules $\ind (\nabla(\mathbf{a};1) \boxtimes...\boxtimes \nabla(\mathbf{a};n))$. Moreover, we show that our construction is compatible with the crystal structure on $B(\infty)$ (resp. $B(λ)$).

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arXiv2012

Categorification of Quantum Generalized Kac-Moody Algebras and Crystal Bases

We construct and investigate the structure of the Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^λ$ which give a categrification of quantum generalized Kac-Moody algebras. Let $U_\A(\g)$ be the integral form of the quantum generalized Kac-Moody algebra associated with a Borcherds-Cartan matrix $A=(a_{ij})_{i,j \in I}$ and let $K_0(R)$ be the Grothedieck group of finitely generated projective graded $R$-modules. We prove that there exists an injective algebra homomorphism $Φ: U_\A^-(\g) \to K_0(R)$ and that $Φ$ is an isomorphism if $a_{ii}\ne 0$ for all $i\in I$. Let $B(\infty)$ and $B(λ)$ be the crystals of $U_q^-(\g)$ and $V(λ)$, respectively, where $V(λ)$ is the irreducible highest weight $U_q(\g)$-module. We denote by $\mathfrak{B}(\infty)$ and $\mathfrak{B}(λ)$ the isomorphism classes of irreducible graded modules over $R$ and $R^λ$, respectively. If $a_{ii}\ne 0$ for all $i\in I$, we define the $U_q(\g)$-crystal structures on $\mathfrak{B}(\infty)$ and $\mathfrak{B}(λ)$, and show that there exist crystal isomorphisms $\mathfrak{B}(\infty) \simeq B(\infty)$ and $\mathfrak{B}(λ) \simeq B(λ)$. One of the key ingredients of our approach is the perfect basis theory for generalized Kac-Moody algebras.

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arXiv2011

Young Walls of Type $D^(2)_n+1$ and Strict Partitions

We show that the number of reduced Young walls of type $D_{n+1}^{(2)}$ with $m$ blocks is independent of $n$ and the same as the number of strict partitions of $m$. Thus the principally specialized character $χ_n^{Λ_0}(t)$ of $V(Λ_0)$ over $U_q(D_{n+1}^{(2)})$ can be interpreted as a generating function for strict partitions. Hence we obtain an infinite family of generalizations of Euler's partition identity.

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