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I am interested in a recommendation for a good book which discuses representation theory of GL(n)(say over field of complex numbers). I know only a basic representation theory. The question I am interested in are how looks decomposition of $GL(n)$ module $V\otimes W$, where $V$,$W$ irreps. I am interested in book or chapter in book which will not require too much preliminary.

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Are you interested in infinite-dimensional representations or only finite-dimensional representations? – B R Jan 11 '13 at 20:23
Made community wiki, I am interested only in finite dim reps. – Klim Efremenko Jan 12 '13 at 23:45
up vote 4 down vote accepted

As previous questions about books on representation theory and Lie theory indicate, there are a lot of them out there aimed at different parts of the subject. (So maybe community-wiki is indicated?) It's good to be clear at the outset that the problem of finite dimensional tensor product decomposition over $\mathbb{C}$ is essentially the same for general linear groups and for their Lie algebras. However, the group approach is sometimes more flexible.

The "classical" Schur-Weyl approach is exposed in many books, including classical texts by Boerner and others aimed at physicists. One mathematical source I'm fond of is the straightforward chapter in the Springer GTM 225 book Symmetry, representations, and invariants by Goodman and Wallach. Also, the symmetric group background needed is well covered in the first part of the earlier Springer GTM 129 Representation Theory by Fulton and Harris; but it may be alittle harder to extract from their book a focused account of Schur-Weyl duality.

The more "modern" thinking of Littelmann and others (which is combinatorial and aims at avoiding formulas with many cancellations) may not yet be as readily accessible in book form.

There are also some Russian alternatives which others will surely want to advocate for.

Once you get into the decomposition of arbitrary tensor products, rather than just tensor powers of the natural representation, life does get more complicated and you can't expect miracles from the textbook sources.

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I would second the recommendation for Fulton and Harris. It covers the basics of the representation theory of $GL(n)$ in a friendly and accessible way. – Chuck Hague Jan 14 '13 at 17:38

From the top of my head, I think those should answer the question nicely (isn't it a duplicate?) :

  • Goldfeld's "Automorphic forms and $L$-functions for the group $GL(n,\mathbb R)$"
  • Bump's "Automorphic forms and representations"
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It's easier to mention the keyword "Littlewood-Richardson coefficients" that gives the answer to your question than to come up with the best possible source explaining it. If you are only interested in the answer, I would suggest to read books on combinatorics, such as "Symmetric group" by Sagan, rather than trying to learn all the background from the representation theory justifying it.

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