Three generations and a trio of trialities

Abstract

We identify the Standard Model's $\mathrm{su}\left(3\right)\oplus \mathrm{su}\left(2\right)\oplus u\left(1\right)$ internal symmetries within the triality symmetries $\mathrm{tri}\left(C\right)\oplus \mathrm{tri}\left(H\right)\oplus \mathrm{tri}\left(O\right)$. From here, the corresponding Standard Model group action is applied to the triality triple $({\Psi }_{+},{\Psi }_{-},V)$ for ${\Psi }_{+},{\Psi }_{-},V\in C\otimes H\otimes O$. Together, ${\Psi }_{+}$ and ${\Psi }_{-}$ provide the correct irreducible representations for two generations. Owing to a certain Cartan factorization, which we define, V provides the irreducible representations for a third generation. Said more explicitly in another way, division algebraic multiplication merges a third generation of spinor representations into a set of scalar bosons. This set of scalar bosons includes the familiar Standard Model Higgs representation.