Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
    
This is likely to get a better answer if you say what you mean by "the same construction." –  MTS May 23 '12 at 17:08
    
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. –  George May 24 '12 at 5:10

1 Answer 1

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?

share|improve this answer
    
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. –  George May 24 '12 at 17:05
    
See my answer here: mathoverflow.net/questions/10126/… –  Vladimir Dotsenko May 24 '12 at 18:58

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.