Discrete symmetries, roots of unity, and lepton mixing

Understand

We investigate the possibility that the first column of the lepton mixing matrix U is given by u_1 = (2,-1,-1)^T/sqrt{6}.

• In a purely group-theoretical approach, based on residual symmetries in the charged-lepton and neutrino sectors and on a theorem on vanishing sums of roots of unity, we discuss the finite groups which can enforce this.
• Assuming that there is only one residual symmetry in the Majorana neutrino mass matrix, we find the almost unique solution Z_q x S_4 where the cyclic factor Z_q with q = 1,2,3,...
• is irrelevant for obtaining u_1 in U.