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Hi,

I am faced with a finite-dimensional representation $V$ of $\mathfrak{gl}_{2n}$, whose character I know. I know how to use this character to determine the irreducibles for $\mathfrak{gl}_{2n}$ which appear in $V$, using the Weyl character formula.

I want to understand, instead, the restriction of $V$ to $\mathfrak{sp}_{2n}$, and in particular I want to use what I know above to determine dimension of the space of highest weight vectors, equivalently the number of summands when $V$ is expressed as a sum of irreducible $\mathfrak{sp}_{2n}$-modules.

In theory, I know at least one approach, which would be to write the $W=S_{2n}$-symmetric polynomials I get as characters for $V$ relative to $\mathfrak{gl}_{2n}$ instead in some suitable basis of $W'=(S_n\times S_n)\rtimes S_2$-symmetric polynomials, and to compare the corresponding Weyl character formulas. However, I wonder if there is some textbook or article where this sort of thing is worked through as an example, or at least explained more completely than I have sketched above.

So my question is really a reference request, ideally which would lay out a reasonable basis for W'-symmetric functions, corresponding to characters of $\mathfrak{sp}_{2n}$, and ideally would use this machinery in some example to determine how some $V_\lambda$ for $\mathfrak{gl}_{2n}$ splits up into irreducibles for $\mathfrak{sp}_{2n}$. If no reference can be produced, but someone knows how to work through some non-trivial examples, it would be great.

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Depending on how explicitly you know the decomposition of V into $\mathfrak{gl}_{2n}$ irreducibles, a definitely old-school approach would be the tensor calculus: the $\mathfrak{sp}_{2n}$-invariants are generated by the symplectic form $\omega$, so it's a question of 'subtracting the $\omega$-traces' from the irreducible tensors. This quickly gets messy, though, for complicated irreps. –  José Figueroa-O'Farrill Dec 30 '10 at 16:39

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Hmmm... a few hours after posting this, I found a pretty satisfactory answer in Fulton and Harris, Representation Theory, (25.39) p. 427 of my copy. I'll leave the question open in case anyone already started to think about it, or has any other helpful replies, but for now I think I'm satisfied.

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