A 6-dimensional ( fundamental) representation of $SU(6)$ becomes (3,2) representation in $SU(3)\times SU(2)$. We can decompose $6\times 6$ of $SU(6)$ into 21-dimensional symmetric and 15-dimensional anti-symmetric representations. What can be the symmetric and anti-symmetric parts of that representation for $SU(3)\times SU(2)$? Or is it not possible to decompose in this way?


To find answer I proceed as follows:
$(3,2)\times(3,2)=(1+8,1,3)=(1,1)+(1,3)+(8,1)+(8,3)$. But I failed to identify all the symmetric and antisymmetric representations out of them. Alone $(8,3)$ is 24 dimensional and bigger than the 21-dymensional symmetric representation.



What are the symmetric and anti-symmetric representations of $6\times6$ of $SU(6)$ in $SU(3)\times SU(2)$?
First we have $3\times 3 = 6+3$ and $2\times 2 = 3+1$. Then combining this we have


$(3,2)\times (3,2) = [ (3,1)+(6,3) ] + [ (3,3)+(6,1) ]$


