Connect discreteness to continuousness in leptonic flavor symmetries

Discrete groups are widely used in the expression of flavor symmetries of leptons. In this paper, we employ a novel mathematical object called group-algebra (GA) to describe symmetries of the leptonic mass matrices. A GA element is constructed by a discrete group with continuous parameters. For a GA element, there is an equivalent symmetry which can be continuous, discrete, and hybrid. According to the equivalence between a GA element and other symmetries, we perform a classification of 198 nontrivial elements of the GA generated by the group $S_{4}$. Based on the results of the classification, the phenomenological consequences of the $S_{4}$ GA are illustrated.