Analytic deformations of group commuting squares and complex Hadamard matrices

Abstract

Let G be a finite group and denote by CG the commuting square associated to G. The defect of the group G, given by the formula d(G)=∑g∈G|G|order(g), was introduced in [9] as an upper bound for the number of linearly independent directions in which CG can be continuously deformed in the class of commuting squares. In this paper we show that this bound is actually attained, by constructing d(G) analytic families of commuting squares containing CG.In the case G=Zn, the defect d(Zn) can be interpreted as the dimension of the enveloping tangent space of the real algebraic manifold of n×n complex Hadamard matrices, at the Fourier matrix Fn (in the sense of [14,1]). The dimension of the enveloping tangent space gives a natural upper bound on the number of continuous deformations of Fn by complex Hadamard matrices, of linearly independent directions of convergence. Our result shows that this bound is reached, which is rather surprising. In particular our construction yields new analytic families of complex Hadamard matrices stemming from Fn.In the last section of the paper we use a compactness argument to prove non-equivalence (i.e. non-isomorphism as commuting squares) for dephased versions of the families of Hadamard matrices constructed throughout the paper.