Eightfold way

The eightfold way organizes light hadrons into approximate $SU(3)$ flavor multiplets built from the up, down, and strange quarks.

The antisymmetric flavour representation ${\bigwedge }^{3}3$ has dimension one and consists of the alternating $uds$ combination. Its sole weight is $(I_3,Y)=(0,0)$, distinct as a representation from the octet $\Lambda$ at the same weight. The Pauli constraint on three-quark flavour multiplets requires orbital excitation for a three-quark colour-singlet realization.

The completely symmetric three-quark representation of flavour symmetry has dimension ten. Its weight diagram has rows $(Y,I)=(1,3/2),(0,1),(-1,1/2),(-2,0)$. It includes $uuu$ at the ${\Delta }^{++}$ weight and $sss$ at the ${\Omega }^{-}$ weight. A spatially symmetric three-quark colour singlet realizes this multiplet with spin $3/2$.

The pseudoscalar octet of flavor symmetry contains three pions, four kaons and the Eta octet state. In its weight diagram, the neutral pion and ${\eta }_8$ are separate states at the origin.

The zero-isospin, zero-flavor hypercharge state ${\eta }_8=(u\bar{u}+d\bar{d}-2s\bar{s})/\sqrt{6}$ in the pseudoscalar meson octet. It is a pure octet basis state; the physical eta may mix it with the flavor-singlet state.

The spin-one-half octet of flavor symmetry contains the nucleons, Sigma baryons, Lambda baryon, and Xi baryons. Its weight diagram has six peripheral states and two states at $(I_3,Y)=(0,0)$: ${\Sigma }^0$ and ${\Lambda }^0$, with different total isospin.