Neutrino masses and mixing from flavour antisymmetry

We discuss consequences of assuming ($i$) that the (Majorana) neutrino mass matrix $M_\nu$ displays flavour antisymmetry, $S_\nu^T M_\nu S_\nu=-M_\nu$ with respect to some discrete symmetry $S_\nu$ contained in $SU(3)$ and ($ii$) $S_\nu$ together with a symmetry $T_l$ of the Hermitian combination $M_lM_l^\dagger$ of the charged lepton mass matrix forms a finite discrete subgroup $G_f$ of $SU(3)$ whose breaking generates these symmetries. Assumption ($i$) leads to at least one massless neutrino and allows only four textures for the neutrino mass matrix in a basis with a diagonal $S_\nu$ if it is assumed that the other two neutrinos are massive. Two of these textures contain a degenerate pair of neutrinos.Assumption ($ii$) can be used to determine the neutrino mixing patterns. We work out these patterns for two major group series $\Delta(3 N^2)$ and $\Delta(6 N^2)$ as $G_f$. It is found that all $\Delta(6 N^2)$ and $\Delta(3 N^2)$ groups with even $N$ contain some elements which can provide appropriate $S_\nu$. Mixing patterns can be determined analytically for these groups and it is found that only one of the four allowed neutrino mass textures is consistent with the observed values of the mixing angles $\theta_{13}$ and $\theta_{23}$. This texture corresponds to one massless and a degenerate pair of neutrinos which can provide the solar pair in the presence of some perturbations. The well-known groups $A_4$ and $S_4$ provide examples of the groups in respective series allowing correct $\theta_{13}$ and $\theta_{23}$. An explicit example based on $A_4$ and displaying a massless and two quasi degenerate neutrinos is discussed.