# Restriction from $\mathfrak{gl}_{2n}$ to $\mathfrak{sp}_{2n}$

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