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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!

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

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