Searcharxiv⌕ Search

arXiv subjects

Ji Hye Jung

Publications and source records attributed to Ji Hye Jung.

10 recordsLinked to original sources
arXiv2017

Supersymmetric polynomials and the center of the walled Brauer algebra

We study a commuting family of elements of the walled Brauer algebra $B_{r,s}(δ)$, called the Jucys-Murphy elements, and show that the supersymmetric polynomials in these elements belong to the center of the walled Brauer algebra. When $B_{r,s}(δ)$ is semisimple, we show that those supersymmetric polynomials generate the center. Under the same assumption,we define a maximal commutative subalgebra of $B_{r,s}(δ)$, called the \emph{Gelfand-Zetlin subalgebra}, and show that it is generated by the Jucys-Murphy elements. As an application, we construct a complete set of primitive orthogonal idempotents of $B_{r,s}(δ)$, when it is semisimple. We also give an alternative proof of a part of the classification theorem of blocks of $B_{r,s}(δ)$ in non-semisimple cases, which appeared in the work of Cox-De~Visscher-Doty-Martin.Finally, we present an analogue of Jucys-Murpy elements for the quantized walled Brauer algebra $H_{r,s}(q,ρ)$ over $\mathbb C(q, ρ)$ and by taking the classical limit we show that the supersymmetric polynomials in these elements generates the center. It follows that H. Morton conjecture, which appeared in the study of the relation between the framed HOMFLY skein on the annulus and that on the rectangle with designated boundary points, holds if we extend the scalar from $\mathbb Z[q^{\pm1},ρ^{\pm1}]_{(q-q^{-1})}$ to $\mathbb C(q, ρ)$.

math.RT↗
arXiv2016

A categorification of $\mathfrak{q}(2)$-crystals

We provide a categorification of $\mathfrak{q}(2)$-crystals on the singular $\mathfrak{gl}_{n}$-category ${\mathcal O}_{n}$. Our result extends the $\mathfrak{gl}_{2}$-crystal structure on ${\rm Irr} ({\mathcal O}_{n})$ defined by Bernstein-Frenkel-Khovanov. Further properties of the ${\mathfrak q}(2)$-crystal ${\rm Irr}({\mathcal O}_{n})$ are also discussed.

math.RT↗
arXiv2016

Quantum walled Brauer-Clifford superalgebras

We introduce a new family of superalgebras, the quantum walled Brauer-Clifford superalgebras ${\mathsf {BC}}_{r,s}(q)$. The superalgebra ${\mathsf {BC}}_{r,s}(q)$ is a quantum deformation of the walled Brauer-Clifford superalgebra ${\mathsf {BC}}_{r,s}$ and a super version of the quantum walled Brauer algebra. We prove that ${\mathsf {BC}}_{r,s}(q)$ is the centralizer superalgebra of the action of ${\mathfrak U}_{q}({\mathfrak q}(n))$ on the mixed tensor space $\mathbf{V}_{q}^{r,s}=\mathbf{V}_{q}^{\otimes r} \otimes (\mathbf{V}_q^*)^{\otimes s}$ when $n \ge r+s$, where ${\mathbf V}_{q}=\mathbb{C}(q)^{(n|n)}$ is the natural representation of the quantum enveloping superalgebra ${\mathfrak U}_{q}({\mathfrak q}(n))$ and $\mathbf{V}_q^*$ is its dual space. We also provide a diagrammatic realization of ${\mathsf {BC}}_{r,s}(q)$ as the $(r,s)$-bead tangle algebra ${\mathsf {BT}}_{r,s}(q)$. Finally, we define the notion of $q$-Schur superalgebras of type $\mathsf{Q}$ and establish their basic properties.

math.RT↗
arXiv2013

Quantum queer superalgebras

We give a brief survey of recent developments in the highest weight representation theory and the crystal basis theory of the quantum queer superalgebra $U_q(\mathfrak{q}(n))$.

math.RT↗
arXiv2013

Crystal bases for the quantum queer superalgebra and semistandard decomposition tableaux

In this paper, we give an explicit combinatorial realization of the crystal B(λ) for an irreducible highest weight U_q(q(n))-module V(λ) in terms of semistandard decomposition tableaux. We present an insertion scheme for semistandard decomposition tableaux and give algorithms of decomposing the tensor product of q(n)-crystals. Consequently, we obtain explicit combinatorial descriptions of the shifted Littlewood-Richardson coefficients.

math.RT↗
arXiv2013

Crystal bases for the quantum queer superalgebra

In this paper, we develop the crystal basis theory for the quantum queer superalgebra $U_q(\mathfrak q(n))$. We define the notion of crystal bases and prove the tensor product rule for $U_q(\mathfrak q(n))$-modules in the category $O_int^{\geq 0}$. Our main theorem shows that every $U_q(\mathfrak q(n))$-module in the category $O_int^{\geq 0}$ has a unique crystal basis.

math.RT↗
arXiv2012

Mixed Schur-Weyl-Sergeev duality for queer Lie superalgebras

We introduce a new family of superalgebras $\overrightarrow{B}_{r,s}$ for $r, s \ge 0$ such that $r+s>0$, which we call the walled Brauer superalgebras, and prove the mixed Scur-Weyl-Sergeev duality for queer Lie superalgebras. More precisely, let $\mathfrak{q}(n)$ be the queer Lie superalgebra, ${\mathbf V} =\mathbb{C}^{n|n}$ the natural representation of $\mathfrak{q}(n)$ and ${\mathbf W}$ the dual of ${\mathbf V}$. We prove that, if $n \ge r+s$, the superalgebra $\overrightarrow{B}_{r,s}$ is isomorphic to the supercentralizer algebra $_{\mathfrak{q}(n)}({\mathbf V}^{\otimes r} \otimes {\mathbf W}^{\otimes s})^{\op}$ of the $\mathfrak{q}(n)$-action on the mixed tensor space ${\mathbf V}^{\otimes r} \otimes {\mathbf W}^{\otimes s}$. As an ingredient for the proof of our main result, we construct a new diagrammatic realization $\overrightarrow{D}_{k}$ of the Sergeev superalgebra $Ser_{k}$. Finally, we give a presentation of $\overrightarrow{B}_{r,s}$ in terms of generators and relations.

math.RT↗
arXiv2011

Quantum Queer Superalgebra and Crystal Bases

In this paper, we develop the crystal basis theory for the quantum queer superalgebra $\Uq$. We define the notion of crystal bases, describe the tensor product rule, and present the existence and uniqueness of crystal bases for finite-dimensional $\Uq$-modules in the category $\mathcal{O}_{int}^{\ge 0}$.

math.QA↗
arXiv2011

Highest weight modules over quantum queer Lie superalgebra U_q(q(n))

In this paper, we investigate the structure of highest weight modules over the quantum queer superalgebra $U_q(q(n))$. The key ingredients are the triangular decomposition of $U_q(q(n))$ and the classification of finite dimensional irreducible modules over quantum Clifford superalgebras. The main results we prove are the classical limit theorem and the complete reducibility theorem for $U_q(q(n))$-modules in the category $O_q^{\geq 0}$.

math.RT↗