A Multigraph Approach for Performing the Quantum Schur Transform
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We take inspiration from the Okounkov-Vershik approach to the representation theory of the symmetric groups to develop a new way of understanding how the Schur-Weyl duality can be used to perform the Quantum Schur Transform. The Quantum Schur Transform is a unitary change of basis transformation between the computational basis of (ℂ^d)^⊗ n and the Schur-Weyl basis of (ℂ^d)^⊗ n. We describe a new multigraph, which we call the Schur-Weyl-Young graph, that represents both standard Weyl tableaux and standard Young tableaux in the same diagram. We suggest a major improvement on Louck's formula for calculating the transition amplitudes between two standard Weyl tableaux appearing in adjacent levels of the Schur-Weyl-Young graph for the case d=2, merely by looking at the entries in the two tableaux. The key theoretical component that underpins our results is the discovery of a branching rule for the Schur-Weyl states, which we call the Schur-Weyl branching rule. This branching rule allows us to perform the change of basis transformation described above in a straightforward manner for any n and d.
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