I have a particular problem, the following.
$T^{a_1 \dots a_p;b_1 \dots b_p}$ is a tensor with the following symmetries.
1) $a_i$'s and $b_i$'s are completely antisymmetric, ie restricted to either$a_i's $ or $ b_i's$
2) Total interchange of $a_i$'s with $b_i$'s are symmetric, that is $T^{a_1 \dots a_p;b_1 \dots b_p} = T^{b_1 \dots b_p;a_1 \dots a_p}$
Is this an irreducible representation of the full $S_{2p}$ group? if not how to decompose it? And what are/is the associated young tableau.
In general I would be grateful if anyone can provide me with a reference on how to go about finding the associated tableau given arbitrary symmetries, such as cyclic symmetry etc..
I am guessing the usual procedure of the Littlewood-Richardson rule can be constrained further when additional symmetries are present?
Thanks for you help.
The problem you are describing is strongly related the "plethysm" problem.
$\mathbb S_\lambda(V)$ denotes the Schur functor. We find that the irreducible polynomial representations of $\mathrm{GL}_n(\mathbb C)$ are $\mathbb S_\lambda(\mathbb C^k)$ (with $\lambda$ having at most $k$ rows). The question is how does $$\mathbb S_\lambda(\mathbb S_\mu(\mathbb C^k))$$ split into irreducibles. This problem is in general hard and still open. I do not know about the progress on your very concrete and probably relatively easy problem, that is how does $$\mathbb S_{2^1}(\mathbb S_{1^p}(V))$$ split into irreducible.
I don't know about many references but googling "plethysm" should help. I learned about the concept in Fulton-Harris: Representation theory p.82 Exercise 6.17.