Walled Brauer algebras as idempotent truncations of level 2 cyclotomic quotients

TL;DR

This paper establishes an explicit isomorphism between the walled Brauer algebra and an idempotent truncation of a level two cyclotomic degenerate affine walled Brauer algebra, linking algebraic structures with Lie theory.

Contribution

It provides a new realization of the walled Brauer algebra as a truncation of a cyclotomic algebra, enabling construction of central elements and proving Koszulity under certain conditions.

Findings

Explicit isomorphism between walled Brauer algebra and cyclotomic algebra

Method to construct central elements in walled Brauer algebra

Proof of Koszulity when delta is non-zero

Abstract

We realize (via an explicit isomorphism) the walled Brauer algebra for an arbitrary integral parameter delta 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. This provides us a method to construct central elements in the walled Brauer algebras and can be applied to establish the Koszulity of the walled Brauer algebra if delta is not zero.