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Chebotar\"ev proved that every minor of a discrete Fourier matrix of primeorder is nonzero. We prove a generalization of this result that includesanalogues for discrete cosine and discrete sine matrices as special cases. Weestablish these results via a generalization of the Bir\'o-Meshulam-Taouncertainty principle to functions with symmetries that arise from certaingroup actions, with some of the simplest examples being even and odd functions.We show that our result is best possible and in some cases is stronger thanthat of Bir\'o-Meshulam-Tao. Some of these results hold in certaincircumstances for non-prime fields; Gauss sums play a central role in suchinvestigations.

An improved uncertainty principle for functions with symmetry