Symmetry of Young's Tableaux with conjugate representation

[joex444]

This isn't an assigned problem, just a popular forum I was hoping someone here would be able to help or move it to where it should be...

Homework Statement

I was working out the Young's tableaux for two SU(3) representations where
$$3 \otimes 3 = 6 \oplus \bar{3}$$, where the 6 is symmetric and the 3 bar is anti-symmetric.

Homework Equations

The Attempt at a Solution

Now, we can also do $$\bar{3} \otimes \bar{3} = \bar{6} \oplus 3$$ but the diagrams are not pure row or columns. Here the 6 bar representation is
X X
X X

and the 3 is
X X
X
X

So, in the case of $$3 \otimes 3$$ we have (if you refer to Sakurai p.~370 where he gives the definition of p,q) the 6 representation as (p,q) = (2,0) and the 3 bar as (0,1). In the case of $$\bar{3} \otimes \bar{3}$$ we obtain a (0,2) and (1,0), as expected (?).

Anyways, would the (0,2) representation have the same symmetry as the (2,0) rep?

 

[NateD]

Would the (1,0) rep have the same symmetry as the (0,1)? Or do they both just have no symmetry?The (2,0) and (0,1) representations are both symmetric under exchange of the two indices, while the (0,2) and (1,0) are both anti-symmetric.