SL(2,\xa07) representations and their relevance to neutrino physics

G Aliferis,N D Vlachos,G K Leontaris

doi:10.1140/epjc/s10052-017-4919-1

Abstract

The investigation of the rôle of finite groups in flavor physics and, particularly, in the interpretation of the neutrino data has been the subject of intensive research. Motivated by this fact, in this work we derive the three-dimensional unitary representations of the projective linear group PSL_2(7). Based on the observation that the generators of the group exhibit a Latin square pattern, we use available computational packages on discrete algebra to determine the generic properties of the group elements. We present analytical expressions and discuss several examples which reproduce the neutrino mixing angles in accordance with the experimental data.

Highlights

The explicit form of the three-dimensional unitary representations would be a very useful tool in many physics applications including their possible relevance to the structure of the neutrino mass matrix and lepton mixing angles
Taking for example the SU (5) gauge theory, in some F-theory framework, the trilinear Yukawa couplings for the various types of fields are realized at different points of the internal manifold and they correspond to different symmetry enhancements of the SU (5) singularity [12]
In the present section we describe the basic steps for the construction of the unitary representations of the P S L2( p) group

Summary

On the properties of the representation matrices

We investigate useful properties of Latin square matrices in view of their relation to the P S L2(7) elements. Since P S L2 (7) contains four conjugacy classes characterized by elements of order 2 (el2), order 3 (el3), order 4 (el4), and 7 (el7) (see the appendix for notation), in order to construct the mixing matrices one has to combine the diagonalizing matrices in all possible ways. This search can only be done numerically. Some comments concerning the order 2 elements are here in order The eigenvalues of these matrices are (1, −1, −1), respectively, i.e. there exists a degenerate two-dimensional subspace implying that the eigenvectors related to the degenerate eigenvalue cannot be uniquely defined. This matrix defines a U (2) rotation for v2 and v3 that leaves the representation matrices invariant

Working examples

Summary and conclusions