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I'm stuck on understanding part of a discussion of representations and Clebsch-Gordan series in the book 'Groups, representations and Physics' by H F Jones. I'd be grateful to anyone who can help me out.

For starters, this discussion is in the SU(2) case. I don't know how to draw Young Tableau here, so I'll just go with the CG series. We have the result that [tex]3\otimes 2 = 4\oplus 2[/tex]. The 2 on the right-hand side is the part that I'll be concerned with here, and it corresponds to a Young diagram with three boxes, two in the first row, one in the second. This CG series corresponds to a product of spinors of the form [tex]\psi_a\phi_b\chi_c[/tex] (to save myself typing too much, I'll just write out the indices from now on i.e. [tex]bac=\psi_b\phi_a\chi_c[/tex]). The author writes that the explicit decomposition corresponding to this CG series is

[tex]

3 \{ab\}c = (\{ab\}c + \{bc\}a+\{ca\}b) + (\{ab\}c-\{cb\}a) + (\{ab\}c-\{ca\}b)

[/tex]

where I'm using [tex]\{ab\}[/tex] to represent the symmetric term [tex]ab+ba[/tex]. So the first term on the RHS is totally symmetric and corresponds to the 4 in the CG series above. Then there are two terms. And here's my problem; I think the Young diagram described above (first row has two boxes, second row has one) should be symmetric in a and b, antisymmetric in a and c. So why are there two extra (not fully symmetric) terms on the RHS in the equation above instead of just one? If you take them together as a single term then they're symmetric in a and b, but not antisymmetric in a and c. If you just consider the middle term its antisymmetric in a and c but not symmetric in a and b. Is there a typo in the book? Is my understanding faulty? Is it just too late at night and I've gorged myself on too much easter chocolate? Any help please? Thanks in advance