Traceless projection of mixed tensor products, and walled Brauer algebras

TL;DR

The paper develops a universal method to project mixed tensors $V^{m,n}=V^{\otimes m}\otimes V^{*\otimes n}$ onto the traceless subspace $V^{m,n}_0$ under the diagonal $GL(N)$ action by constructing a central idempotent in the centraliser algebra $C_{m,n}(N)$. Central to the method is the operator \mathscr{A}_{m,n} defined from trace contractions, whose kernel equals the traceless subspace; the projector is built as a product over nonzero eigenvalues using the spectrum of \mathscr{A}_{m,n}$, with eigenvalues explicitly described via Schur-Weyl duality and Littlewood-Richardson coefficients: a = $rN$ + $c(\rho/\mu)$ + $c(\sigma/\nu)$. A detailed algorithm expresses the eigenvalues in terms of partitions, enabling a factorised, GL(N)-invariant projector that commutes with the $S_m\times S_n$ action; the projector arises as a splitting idempotent in $C_{m,n}(N)$, and the paper also derives an analogue for the walled Brauer algebra $B_{m,n}(\delta)$ in the semisimple regime. The construction extends to real vector spaces, unitary groups, and manifolds with affine connections, and suggests further applications to trace decompositions of mixed tensors and refined decompositions (e.g., doubly-traceless components) via central idempotents. Overall, the work provides a unified, representation-theoretic framework for explicitly isolating traceless components in mixed-tensor spaces with broad algebraic and geometric relevance.

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.

Theorems & Definitions (31)

• Lemma 1.1
• proof
• Lemma 2.1
• Theorem 2.2
• Theorem 2.3
• Proposition 2.4
• Lemma 2.5
• Proposition 2.6
• proof
• Proposition 2.7