I understand that given an integer partition we can define an irreducible representation of the general linear group by a Schur functor. Given an irreducible representation $W$ of $GL\left(V\right)$ constructed from the partition $\lambda$ I would like to know how to decompose $S^2W$ into irreducible components. e.g. if $W=V$ then $S^2W$ is already irreducible. How do I work out the decomposition in general?
2026-03-25 15:58:49.1774454329
Decomposing representations of general linear group.
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In general understanding the composition of two Schur functors (called plethysm) is quite a difficult and subtle question. However in the special case of the symmetric and exterior squares we do have a nice combinatorial interpretation in terms of so called domino tableaux due to Carre and Leclerc. The relevant reference is here: https://link.springer.com/article/10.1023/A:1022475927626