Group-Adapted Irreducible Matrix Units for the Walled Brauer Algebra

This paper investigates the representation theory of the algebra of partially transposed permutation operators, $\mathcal{A}^d_{p,p}$, which provides a matrix representation for the abstract walled Brauer algebra. This algebra has recently gained significant attention due to its relevance in quantum information theory, particularly in the efficient quantum circuit implementation of the mixed Schur-Weyl transform. In contrast to previous Gelfand-Tsetlin type approaches, our main technical contribution is the explicit construction of irreducible matrix units in the second-highest ideal that are group-adapted to the action of $\mathbb{C}[S_p]\times \mathbb{C}[S_p]$ subalgebra, where $S_p$ is the symmetric group. This approach suggests a recursive method for constructing irreducible matrix units in the remaining ideals of the algebra. The framework is general and applies to systems with arbitrary numbers of components and local dimensions. In addition, we present a complementary construction method based on tensor networks of Clebsch-Gordan coefficients of the unitary group. This approach enables the construction of all group-adapted irreducible matrix units, but requires knowledge of certain Littlewood-Richardson coefficients. This method can be successfully applied for a reasonably small number of particles with the support of dedicated software. The obtained results are applied to a special class of operators motivated by the mathematical formalism appearing in all variants of the port-based teleportation protocols through the mixed Schur-Weyl duality. We demonstrate that the given irreducible matrix units are, in fact, eigenoperators for the considered class.

💡 Research Summary

This paper studies the representation theory of the algebra of partially transposed permutation operators, denoted 𝔄ᵈ_{p,p}, which furnishes a concrete matrix realization of the abstract walled Brauer algebra B_{d,p,p}. The motivation stems from quantum information theory, where the mixed Schur–Weyl duality—an extension of the classic Schur–Weyl duality that incorporates partial transposition—plays a central role in the efficient implementation of mixed Schur transforms and in the analysis of port‑based teleportation protocols.

The authors first recall the necessary combinatorial background: partitions, Young diagrams, standard and semistandard Young tableaux, hook‑length and hook‑content formulas, and the Schur–Weyl duality linking irreducible representations of the symmetric group Sₚ and the unitary group U(d). They then introduce the walled Brauer algebra as the algebra generated by partially transposed permutation operators acting on the Hilbert space (ℂᵈ)^{⊗(p+q)}. In the special case q = p, the algebra 𝔄ᵈ_{p,p} decomposes into two distinguished two‑sided ideals: the maximal ideal M(p) and the second‑highest ideal M(p‑1).

A major technical contribution is the explicit construction of irreducible matrix units inside M(p‑1) that are group‑adapted to the subalgebra ℂ