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Here's a very special case for $\mathfrak{gl}_n$`$\mathfrak{gl}_n$ 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.
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Here's a very special case for $\mathfrak{gl}_n$ (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 $\lambda$).
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Here's a very special case for $\mathfrak{gl}n$ \mathfrak{gl}_n$ (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 ${\bf S}\lambda(V)$ S}_{\lambda}(V)$ denote the corresponding highest weight representation. Then $Sym^n(Sym^2 V) = \bigoplus_\lambda bigoplus_{\lambda} {\bf S}\lambda(V)$ 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, $Sym^n(\bigwedge^2 V) = \bigoplus\mu bigoplus_{\mu} {\bf S}_\mu(V)$ 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 $(\lambda_1 (\lambda_1 + r, \dots, \lambda_n + r)$ 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 $\lambda$).
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