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I have been interested in representation theory lately in particular on that of Lie algebras. Now I have noticed that one way of building representations is to take tensor/exterior/symmetric powers. I have not studied multilinear algebra in depth before and I am looking for references on multilinear algebra relating to representation theory. I have had a look at Fulton and Harris and they do have an appendix section on this, but in the main text itself they seem to assume that the reader already has some prior knowledge of such material.

Can anyone recommend me any good references for methods of multilinear algebra in representation theory? In addition, what are common methods used to prove irreducibility of exterior/symmetric powers?


Edit: How much multilinear algebra is necessary in order to understand the ideas behind representation theory, such as symmetric and exterior powers of representations?

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It's a good exercise to prove irreducibility of exterior/symmetric powers (I assume by this you mean of a vector space $V$ as a $\text{GL}(V)$ or $\text{SL}(V)$ or etc. representation) by hand; I mean take a nonzero vector and explicitly show that you can get all other nonzero vectors by taking linear combinations. – Qiaochu Yuan Sep 20 '12 at 8:18
@QiaochuYuan Yes that is correct I am looking at $\bigwedge^k V$ as a $\mathfrak{g}$ - representation. – user38268 Sep 20 '12 at 9:50
The study of Lie algebras over $\mathbb{C}$ becomes easy once you have understood $\mathfrak{sl}_{2}$ and it might be a great idea to work out symmetric and tensor products of representation in this case by hand. Fulton Harris does that and if I remember correctly they have a beautiful (and very verbose) discussion on that topic. – s.b Sep 20 '12 at 18:20
up vote 5 down vote accepted

Lie Groups: An Approach through Invariants and Representations

Is probably what you want to look at.

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Thanks! for the link – s.b Sep 20 '12 at 18:13

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