Invariants in $S^n(S^k(\mathbb{C}^w)$

Question

Are the formulas for the multiplicity of $SL(w)$ invariants in $S^n(S^k(\mathbb{C}^w)$ known? This is a very classical topic. If no, in what ranges one can compute it (for certain paramaters fixed - e.g. it is obvious that this is zero for $n<w$)

Answer 1

There is an isomorphism $S^d(\mathbb{C}^n) \cong V_{n,d}$ where $V_{n,d}$ is the the vector $\mathbb{C}$-space of $n$-ary forms of degree $d.$

The number $\nu_{n,d}(k)$ of linearly independent homogeneous invariants of degree $k$ for $n$-ary form of degree $d$ is calculated by the formula: $$ \nu_{n,d}(k)=\sum_{s \in \mathcal{W}} (-1)^{|s|} c_{n,d}\bigl(k,(\rho-s(\rho))^*\bigr). $$

here $c_{n,d}(k,\mu) :=c_{n,d}(k,(\mu_1,\mu_2,\ldots,\mu_{n-1}))$ is the number of non-negative integer solutions of the system of equations $$ \left \{ \begin{array}{l} \omega_1(\alpha)-\omega_2(\alpha)=\mu_1, \\ \ldots \\ \omega_{n-2}(\alpha)-\omega_{n-1}(\alpha)=\mu_{n-2},\\ \omega_1(\alpha)+\omega_2(\alpha)+\cdots +2\,\omega_{n-1}(\alpha)=k\, d-\mu_{n-1}, \\ |\alpha|=k, \end{array} \right. $$ $\rho$ is half the sum of the positive roots of Lie algebra $\mathfrak{sl_{n}},$ $(-1)^{|s|}$ is the sign of the element $s \in \mathcal{W},$ $\mu^*$ the unique dominant weight on the orbit $\mathcal{W}(\mu)$ of the Weyl group $\mathcal{W}$ of the Lie algebra $\mathfrak{sl_{n}},$ and $\omega_i(\alpha)$ is some functions defined on a $n$-ary form.

Details you may find in the paper

L. Bedratyuk, Analogue of the Cayley-Sylvester formula and the Poincaré series for the algebra of invariants of n-ary form, Linear and Multilinear Algebra

Answer 2

You are asking about a composition of Schur functors, which is the same as a plethysm. Have you looked at pages 139-141 of Macdonald's "Symmetric functions and Hall polynomials"? He gives explicit formulas for the whole plethysm (i.e. not just the multiplicity of the invariant part) when $n=2$ and $n=3$, and an expression for the general case in terms of a kind of "generalized Kostka numbers".

Answer 3

There is an explicit formula in the paper of Leonid Bedratiuk, Analogue of Sylvester-Cayley formula for invariants of n-ary form.