Walled Brauer algebras as idempotent truncations of level 2 cyclotomic quotients

Abstract

We realize (via an explicit isomorphism) the walled Brauer algebra Br,t(δ) for arbitrary integral parameter δ as an idempotent truncation of a level two cyclotomic degenerate affine walled Brauer algebra. The latter arises naturally in Lie theory as the endomorphism ring of so-called mixed tensor products, i.e. of a parabolic Verma module tensored with some copies of the natural representation and its dual. The result provides a method to construct central elements in the walled Brauer algebra, and also implies the Koszulity of the walled Brauer algebras if δ≠0.