Algebraic invariants of truncated Fourier matrices

Abstract

A partial Hadamard matrix $H\in M_{M\times N}(\mathbb{C})$ is called of "classical type" if the associated quantum semigroup $G\subset {\tilde{S}}_M^{+}$ is classical. In combinatorial terms, if $H_1,\dots ,H_M\in {\mathbb{T}}^N$ 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.

Withdrawn by the author - the main findings in this paper are now part of arXiv:1411.0577

References in corpus (2)

• The Gale-Berlekamp game for complex Hadamard matrices
• Quantum invariants of deformed Fourier matrices