Supersymmetric polynomials and the center of the walled Brauer algebra

Abstract

We study a commuting family of elements of the walled Brauer algebra Br,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 Br,s(δ) is semisimple, we show that those supersymmetric polynomials generate the center. Under the same assumption, we define a maximal commutative subalgebra of Br,s(δ), called the 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 Br,s(δ), when it is semisimple. We also give an alternative proof of a part of the classification theorem of blocks of Br,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 Hr,s(q, ρ) over 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 HOM-FLY skein on the annulus and that on the rectangle with designated boundary points, holds if we extend the scalar from Z[q ±1 , ρ ±1 ] (q−q −1 ) to C(q, ρ).

Representations of the oriented skein category

“…where the interesting strand is the ith or (r + j)th from the right, respectively. These elements were also introduced by Morton (extending an observation from [Ra] in the case of the Iwahori-Hecke algebra): up to an obvious symmetry and rescaling they are the elements T and U from the proof of [M,Theorem 1]; see [JK,Remark 6.7]. (Jung and Kim also prove a version of [SS,Conjecture 7.4] in the degenerate case.)…”

The affine VW supercategory

“…[17], or walled Brauer algebras, see e.g. [22], [36]. Compare also with [14], where the center of the Brauer superalgebra sBr a is described in a similar way.…”