Second order deformations of group commuting squares and Hadamard matrices

Abstract

In [Indiana Univ. Math. J. 60 (2011), pp. 847–857] the first author introduced second order necessary conditions for a commuting square to admit sequential deformations in the moduli space of non-isomorphic commuting squares. In this paper we investigate these conditions for commuting squares C G \mathfrak {C}_G constructed from finite groups G G . We are especially interested in the case G = Z n G=\mathbb {Z}_n , since deformations of C Z n \mathfrak {C}_{\mathbb {Z}_n} correspond to deformations of the Fourier matrix F n F_n in the moduli space of non-equivalent complex Hadamard matrices. We show that for G = Z n G=\mathbb {Z}_n the second order conditions follow automatically from the first order conditions, but this is not necessarily true for other finite abelian groups G G . Our result gives a complete description of the second order deformations of the Fourier matrix F n F_n in the moduli space of non-equivalent complex Hadamard matrices.