The irreducible representations of the Symmetric group $S_5$ are classified by the partitions of $5$. For the standard representation which corresponds to the partition (4,1) the ring of invariants is generated by the elementary symmetric polynomials and hence is a polynomial ring. For other irreducible representations except for the trivial and sign representation, $S_5$ does not represent as a pseudo-reflection group, hence for these representations the ring of invariants is not a polynomial ring any more. I also learnt that a minimal generating set for these rings is not possible (or hard) to compute using the programs available till date. Do we have any combinatorial technique to compute them explicitly ?

For example if we take the tensor product of standard and sign representation (which corresponds to the partition 2+1+1+1) is there any way to get a minimal generating set for the ring of invariants ?

Further it would be a great help if someone could provide me an example of a representation of $S_5$ (not necessarily irreducible and except standard, natural, sign and trivial) whose ring of invariants in terms of generators and relations is known.