A state-space approach to quantum permutations

The Frucht property in the quantum group setting

“…In what regards the higher orbitals, things do not seem to work here, as noticed in [22]. As explained in [24], the 3-orbital can fail to be transitive when the diagonal elements of the magic unitary generate a commutative algebra. We have now a precise counterexample for this, our result being as follows:…”

“…(3) At N = 3 now, by using the same idea as in the N = 2 case, we must prove that the entries of any 3 × 3 magic matrix commute. This is something quite tricky, and there are 4 known proofs here [1], [22], [24], [35]. According to the proof in [24], which is the most recent, it suffices to show that u 11 u 22 = u 22 u 11 , by showing:…”

“…This is something quite tricky, and there are 4 known proofs here [1], [22], [24], [35]. According to the proof in [24], which is the most recent, it suffices to show that u 11 u 22 = u 22 u 11 , by showing:…”

“…When G × is finite and so C(G × ) is a multi-matrix algebra, the one-dimensional summands correspond precisely to the characters. See [24] for more.…”

The Frucht property in the quantum group setting

“…When G × is finite and so C(G × ) is a multi-matrix algebra, the one-dimensional summands correspond precisely to the characters. See [24] for more. We recall that for any finite group G we have a Peter-Weyl decomposition, as follows:…”