A Built-in Horizontal Symmetry of $SO(10)$

Understand

In a renormalizable $SO(10)$ theory, all fermion mass matrices are linear combinations of three fundamental types, $M^{10}, M^{\overline{126}}$, and $M^{120}$, whose superscripts indicate their $SO(10)$ transformation properties.

• We point out that each of these fundamental mass matrices possesses a natural symmetry that can be used to generate an unbroken horizontal symmetry $\G$, if the natural symmetry is taken to be the residual symmetry.
• This built-in symmetry is a Coxeter group.
• If it is finite, it must be one of five groups, $S_4,\ Z_2\x S_4$,\ $Z_2\x A_5$, plus two `rank-4' groups.