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Algebraic invariants of truncated Fourier matrices

A partial Hadamard matrix $H\in M_{M\times N}(\mathbb C)$ is called of "classical type" if the associated quantum semigroup $G\subset\widetilde{S}_M^+$ is classical. In combinatorial terms, if $H_1,\ldots,H_M\in\mathbb T^N$ are the rows of the matrix, the vectors $H_i/H_j\in\mathbb T^N$ must be pairwise proportional, or orthogonal. We propose here of definition for the algebraic (or quantum) invariants of such matrices. For the truncated Fourier matrices, which are all of classical type, we obtain certain Bernoulli laws, that we compute in the $N>>M$ regime.