Generating Sets for the Kernels of Frobenius Polarization Operators

Abstract

We study generating sets for the kernels of the Frobenius polarization operators associated with elements of finite groups. By applying the discrete Fourier transform on each coset, we describe these kernels in terms of additive semigroups defined by zero-sum relations among roots of unity. We show that the monomials corresponding to the minimal nonzero elements of these semigroups form generating sets of minimum cardinality. We further describe the minimal nonzero elements in the general case in terms of those in the cyclic case and obtain a formula for the minimum number of generators. For the operator associated with a generator of a cyclic group, we also show, as a consequence of known results on integer zero-sum relations and circulant determinant representations, that its kernel and the subalgebra generated by these circulant determinants have the same field of fractions.