# New partial symmetries from group algebras for lepton mixing

## TL;DR

This paper explores how partial symmetries derived from group algebras, specifically $S_3$ and $S_4$, can explain lepton mixing patterns and residual CP symmetries consistent with neutrino oscillation data.

## Contribution

It introduces a novel approach of using elements of group algebras to represent partial symmetries in lepton mixing, linking them to residual CP symmetries and specific mixing patterns.

## Key findings

- Lepton mixing matrices from $S_3$ group algebras exhibit trimaximal form with $- au$ reflection symmetry.
- Elements of $S_3$ group algebras are equivalent to $Z_2\times CP$ symmetry.
- Predictions for $Z_2\times CP$ broken from $S_4$ group algebras are obtained.

## Abstract

Recent stringent experiment data of neutrino oscillations induces partial symmetries such as $Z_{2}$, $Z_{2}\times CP$ to derive lepton mixing patterns. New partial symmetries expressed with elements of group algebras are studied. A specific lepton mixing pattern could correspond to a set of equivalent elements of a group algebra. The transformation which interchanges the elements could express a residual $CP$ symmetry. Lepton mixing matrices from $S_{3}$ group algebras are of the trimaximal form with the $\mu-\tau$ reflection symmetry. Accordingly, elements of $S_{3}$ group algebras are equivalent to $Z_{2}\times CP$. Comments on $S_{4}$ group algebras are given. The predictions of $Z_{2}\times CP$ broken from the group $S_{4}$ with the generalized $CP$ symmetry are also obtained from elements of $S_{4}$ group algebras.