Gelfand-Tsetlin basis for partially transposed permutations, with applications to quantum information

Understand

We study representation theory of the partially transposed permutation matrix algebra, a matrix representation of the diagrammatic walled Brauer algebra.

• This algebra plays a prominent role in mixed Schur-Weyl duality that appears in various contexts in quantum information.
• Our main technical result is an explicit formula for the action of the walled Brauer algebra generators in the Gelfand-Tsetlin basis.
• It generalizes the well-known Gelfand-Tsetlin basis for the symmetric group (also known as Young's orthogonal form or Young-Yamanouchi basis).