 Symmetric spin-flavour SU6 representation

Three light quarks have a six-dimensional single-quark spin-flavour space ${\mathbb{C}}^3\otimes {\mathbb{C}}^2$. A symmetric spatial ground state and the antisymmetric three-quark colour singlet require the symmetric cube of that space. Its dimension is $\left(\genfrac{}{}{0ex}{}{8}{3}\right)=56$, and restriction to flavour $SU(3)$ and spin $SU(2)$ gives a baryon decuplet of spin $3/2$ and a baryon octet of spin $1/2$. The two mixed-permutation multiplicity spaces pair once in the symmetric sector, explaining why the two flavour-octet copies in the unconstrained tensor cube do not represent two distinct ground-state octets.