Tensor cube of the defining SU(3) representation ($3^{\otimes 3}=10\oplus 8\oplus 8\oplus 1$)

The tensor product of Lie algebra representations $3\otimes 3$ splits into the symmetric $6$ and antisymmetric $\overline{3}$. Then $6\otimes 3=10\oplus 8$ and $\overline{3}\otimes 3=8\oplus 1$, as highest-weight character subtraction verifies. The decuplet is fully symmetric under quark permutations, the singlet fully antisymmetric, and the two octets constitute the two-dimensional mixed permutation multiplicity space. Applying the Pauli constraint on three-quark flavour multiplets gives one ground-state spin-one-half octet and one spin-three-halves decuplet.

// Target: physics.bigb