Traceless projection of mixed tensor products, and walled Brauer algebras

TL;DR

This paper presents a method for constructing traceless projections of mixed tensor products using Schur-Weyl duality and explores their algebraic properties within the framework of walled Brauer algebras, with applications to hermitian spaces.

Contribution

It introduces a self-contained procedure for the traceless projection in mixed tensor products and extends the concept to walled Brauer algebras in the semisimple case.

Findings

Constructed the traceless projector as a unique idempotent in the centraliser algebra.

Extended the traceless projector concept to walled Brauer algebras.

Applied the results to mixed tensor products of hermitian spaces and their conjugates.

Abstract

We describe a self-contained procedure for constructing the traceless projection of mixed tensor products (built out of a finite-dimensional complex vector space and its dual). The construction relies on the Schur-Weyl duality for the general linear group and regards rational representations thereof. By identifying the traceless subspace as a particular rational representation, the traceless projector which commutes with the group action can be understood as a uniquely defined idempotent in the centraliser algebra. We also identify and construct the analogue of the traceless projector in the walled Brauer algebras when the latter are semisimple. Among possible applications of the traceless projector, we show how the result applies to mixed tensor products built out of a finite-dimensional hermitian space and its complex conjugate.