Observe how conjugation by the Pauli group operations recovers all $B_3$ generators from a single one.
Recall that the three Pauli operations give a set of MUBs for qubits. They are also used to create a simple unknot quandle. In M theory, triality places $2 \times 2$ matrices inside $3 \times 3$ matrices in three (or six) distinct ways. This is a tripling of the massless particle spectrum of the standard model (allowing for mirror neutrinos). Recall that the braid group $B_3$ extends the $2 \times 2$ modular group, notoriously important in $S$ duality.
6 years ago