For quite a time I wrap my mind about the following:
Can (complex) Hadamard matrices of order $n!$ be related to symmetric groups?
What I tried so far is some numerics and a lot of unfruitful thinking.
Thanks for every reply...
Reply to the comments:
I know that the matrix of the discrete Fourier Transform (a special representative of complex Hadamard matrices) is related to abelian groups. I hoped that giving the matrix some more degrees of freedom could extend her (let's say) applicability to non-abelian, i.e. symmetric groups.
EDIT
I found some papers dealing with the connection of complex Hadamard matrices and Quantum Permutation Groups (e.g. QUANTUM PERMUTATION GROUPS: A SURVEY by TEODOR BANICA, JULIEN BICHON, AND BENOIT COLLINS), but to admit I didn't get what quantum groups are yet.
So if anybody guide me along the path (if there exists one) from complex Hadamard matrices to Symmetric groups via Quantum Permutation Groups I would be ever so happy.
Thanks,