# Counting operators with group theory

1. Apr 6, 2017

### Luca_Mantani

1. The problem statement, all variables and given/known data
I have an exercise that I do not know how to solve. $N$ is a nucleon field, in the fundamental representation of $SU(4)$. We want to classify operators by their $SU(4)$ transformation properties, bearing in mind that the nucleon is a fermion and we need antisymmetric products of fields in our operators. The problem asks to combine the representations for two $N$'s and two $N^\dagger$'s and count the number of operators (with no derivatives) from counting the representations. The operator to consider is:
$$(N^\dagger N)(N^\dagger N)$$

2. Relevant equations
The problem provide these two helpful equations:
$$4\otimes \bar{4}=1\oplus 15$$
$$4\otimes 4 = 6\oplus 10$$

3. The attempt at a solution
I don't know where to start, I only know that the solution has to be of the form $x\oplus y \oplus z$. Also as a follow up question, it asks to find out if each of these representations has a $SU(2)$ singlet subgroup or not.

It would be helpful if someone know hot to tackle this kind of problem and is willing to explain to me the procedure.