TITLE: Symmetric tensor products of irreducible representations
QUESTION [8 upvotes]: I wonder if there is a way to compute the symmetric tensor power of irreducible representations for classical Lie algebras: $\mathfrak{so}(n)$, $\mathfrak{sp}(n)$, $\mathfrak{sl}(n)$. 
The question is motivated by reading
"Introduction to Quantum Groups and Crystal Bases" by Hong, J. and Kang, S.-J.
The book provides an algorithm for computing the tensor product of any two irreducible representations for classical Lie algebras. Could it be generalized to symmetric parts of tensor products? Any references are very much appreciated!

REPLY [4 votes]: 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.