Young Tableaux for SU(3) representations vs. j=1 objects.

[Ganesh Ujwal]

I'm working through Sakurai's Modern Quantum Mechanics and in the section on Permutation Symmetry and Young Tableaux, he mentions that a tableau constructed of [latex]\square = \boxed{1},\boxed{2},\boxed{3}[/latex] corresponds to a irrep of [latex]SU(3)[/latex], but if each box is instead a [latex]j=1[/latex] object, a tableau is not a irrep of the rotation group.

He then goes on to discuss this in detail, although I cannot follow his argument other than taking a guess that the decomposition of the mixed symmetry tableau leads to some incompatibilities in representing the rotation group.

Can anyone clear up why a tableau composed of a [latex]\square = \boxed{1},\boxed{2},\boxed{3}[/latex] can correspond to a definite representation of [latex]SU(3)[/latex] but not to a [latex]3[/latex]-dimensional representation of [latex]SU(2)[/latex], or the rotation group?

Edited December 18, 2014 by Ganesh Ujwal