with kernel formula_5, hence isomorphic to the standard model's true gauge group formula_6. For the zeroth power formula_7, this acts trivially, matching a left-handed neutrino, formula_8. For the first exterior power formula_9, the standard model's group action preserves the splitting formula_10. The formula_11 transforms trivially in , as a doublet in , and under the representation of (as weak hypercharge is conventionally normalized as ); this matches a right-handed anti-lepton, formula_12 (as formula_13 in SU(2)). The formula_14 transforms as a triplet in SU(3), a singlet in SU(2), and under the Y = − representation of U(1) (as ); this matches a right-handed down quark, formula_15.