Group-Adapted Irreducible Matrix Units for the Walled Brauer Algebra

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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.