rt.representation theory - Symmetric tensor products of irreducible representations

Here's a very special case for $mathfrak{gl}_n$ in characteristic 0 (which I have found useful in my work). Let $V$ be the vector representation, and for a partition $lambda$ with at most $n$ parts, let ${bf S}_{lambda}(V)$ denote the corresponding highest weight representation. Then $Sym^n(Sym^2 V) = bigoplus_{lambda} {bf S}_{lambda}(V)$ where the direct sum is over all partitions $lambda$ of size $2n$ with at most $n$ parts such that each part of $lambda$ is even. Similarly, $Sym^n(bigwedge^2 V) = bigoplus_{mu} {bf S}_{mu}(V)$ where the direct sum is over all partitions $mu$ of $2n$ with at most $n$ parts such that each part of the conjugate partition $mu'$ is even. If you want the corresponding result for $mathfrak{sl}_n$ we just introduce the equivalence relation $(lambda_1, dots, lambda_n) equiv (lambda_1 + r, dots, lambda_n + r)$ where $r$ is an arbitrary integer.

One reference for this is Proposition 2.3.8 of Weyman's book Cohomology of Vector Bundles and Syzygies (note that $L_lambda E$ in that book means a highest weight representation with highest weight $lambda'$ and not $lambda$).

Another reference is Example I.8.6 of Macdonald's Symmetric Functions and Hall Polynomials, second edition, which proves the corresponding character formulas.