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Consider a finite dimensional vector space $V$ and the general linear group $GL(V)$ acting on it. Both $GL(V)$ and the symmetric group $S_d$ act on the tensor product of $d$ copies of $V$, and by Weyl construction there corresponds to each partition of $d$ a representation of $GL(V)$. Taking $V=\mathbb{C}^n$ this in turn determines a representation of lie algebra $sl_n \mathbb{C}$. This is the approach adopted in Fulton and Harris.

My question is: What if we have an $sl_n\mathbb{C}$ structure on some space other than $\mathbb{C}^n$, do we have the same construction? To be precise, If we have such $sl_n$ representation $V$ then how can we decompose the tensor product $V^{\otimes d}$ into $sl_n$ irreps provided we have a $S_d$ irreducible decomposition.

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  • $\begingroup$ This is likely to get a better answer if you say what you mean by "the same construction." $\endgroup$
    – MTS
    Commented May 23, 2012 at 17:08
  • $\begingroup$ I mean if we have $V$ an $sl_2$ representation then how can we decompose the tensor product $V^{\otimes d}$ into $sl_2$ irreps provided we have a $S_d$ irreducible decomposition. $\endgroup$
    – George
    Commented May 24, 2012 at 5:10

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Depends what you mean by "the same", for example:

For which representations $W$ we can find various reps as summands in $W^{\otimes n}$? A good idea of course is to look at faithful self-dual $W$.

For which representations $W$ does the centraliser of $SL_n$ in $End(W^{\otimes n})$ admit a "nice" description? (This is how Schur--Weyl duality works, - you know that the centraliser is $\mathbb{C}S_n$, and that's where Young diagrams come from.) This is much less obvious.

Or you mean something entirely different from these questions?

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  • $\begingroup$ Thank u for the answer, will you please explain a little more on how to find various reps as summands in the tensor product or provide some reference. $\endgroup$
    – George
    Commented May 24, 2012 at 17:05
  • $\begingroup$ See my answer here: mathoverflow.net/questions/10126/… $\endgroup$
    – Vladimir Dotsenko
    Commented May 24, 2012 at 18:58

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