Supersymmetric polynomials and the center of the walled Brauer algebra

We study a commuting family of elements of the walled Brauer algebra $B_{r,s}(\delta)$, called the Jucys-Murphy elements, and show that the supersymmetric polynomials in these elements belong to the center of the walled Brauer algebra. When $B_{r,s}(\delta)$ is semisimple, we show that those supersymmetric polynomials generate the center. Under the same assumption,we define a maximal commutative subalgebra of $B_{r,s}(\delta)$, 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 $B_{r,s}(\delta)$, when it is semisimple. We also give an alternative proof of a part of the classification theorem of blocks of $B_{r,s}(\delta)$ 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 $H_{r,s}(q,\rho)$ over $\mathbb C(q, \rho)$ 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 $\mathbb Z[q^{\pm1},\rho^{\pm1}]_{(q-q^{-1})}$ to $\mathbb C(q, \rho)$.Comment: Second version, Section "6. Center of the quantized walled Brauer algebra" is adde

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

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 $U_q(\mathfrak{q}(n))$.Comment: For proceedings of "Representation Theory of Algebraic Groups and Quantum Groups," Nagoya, 201

Admissible Pictures and Uq(gl(m,n))U_q(gl(m,n))-Littlewood-Richardson Tableaux

We construct a natural bijection between the set of admissible pictures and the set of $U_q(gl(m,n))$-Littlewood-Richardson tableaux.Comment: 13page

A categorification of q(2)\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.Comment: 20 pages, minor changes are made in v.2, Remark 2.3 is inserted with minor corrections in v.3, to appear in Algebras and Representation Theor

Weak Detection in the Spiked Wigner Model with General Rank

We study the statistical decision process of detecting the signal from a `signal+noise' type matrix model with an additive Wigner noise. We propose a hypothesis test based on the linear spectral statistics of the data matrix, which does not depend on the distribution of the signal or the noise. The test is optimal under the Gaussian noise if the signal-to-noise ratio is small, as it minimizes the sum of the Type-I and Type-II errors. Under the non-Gaussian noise, the test can be improved with an entrywise transformation to the data matrix. We also introduce an algorithm that estimates the rank of the signal when it is not known a priori.Comment: 35 pages, 3 figure

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.Comment: 38 pages, small changes on acknowledgemen

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 $\mathcal{O}_{int}^{\ge 0}$.Comment: 11pages, 3 figure

Crystal bases for the quantum queer superalgebra and semistandard decomposition tableaux

In this paper, we give an explicit combinatorial realization of the crystal B(\lambda) for an irreducible highest weight U_q(q(n))-module V(\lambda) 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.Comment: 38 pages, small change on acknowledgemen

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.Comment: 32 pages; minor corrections; the proof of Theorem 2.9 and the relations (5.13) - (5.15) are changed; to appear in Journal of Algebr