The decomposition of the exterior of the symmetric square over Lie algebra sl(3)

Question

I am studying the representation theory of finite dimensional modules over the simple Lie algebra $\operatorname{sl}(3)$. I know some basics facts about the decomposition of some construction of modules (into irreducibles) e.g. decomposition of tensors of irreducible modules and a fundamental module.

But I have no ideal about the decomposition of exterior algebra of symmetric square. I really want to know and to learn more about how they decompose into a direct sum of irreducible representations.

Notations： Let $V$ be a representation over $\operatorname{sl}(3)\wedge(V)$ denoted the exterior algebra of $V$. $\operatorname{Sym}^2(\mathfrak{C}^3)$ is the symmetric square of the standard representation $\mathfrak{C}^3$.

My Question： How do I decompose $\wedge(\operatorname{Sym}^2(\mathfrak{C}^3)$) into irreducible modules?

Any suggestion are deeply appreciated. Thanks very much！

Answer 1

See Fulton, Harris,Representation Theory,A First Course, Lectures 13,14