TITLE: Invariants of $\mathrm{GL}_n$ representations
QUESTION [10 upvotes]: $\DeclareMathOperator\GL{GL}$Let $V=\mathbb C^n$ be the natural representation of $\GL_n(\mathbb C)$ and let $W=\operatorname{Sym}^2(V)$ be the symmetric square representation. Let $W^k$ denote the direct sum of $k$-copies of $W$. A version of the first fundamental theorem gives a generating set for $\mathbb C[V^k \oplus (V^*)^k]^{\GL_n(\mathbb C)}$. Is a generating set for the ring of invariants $\mathbb C[W^k \oplus (W^*)^k]^{\GL_n(\mathbb C)}$ known?

REPLY [3 votes]: In classical terms, this is about joint invariants of $k$ quadratic forms $Q^{(1)}(x),\ldots,Q^{(k)}(x)$ in point coordinates $x=(x_1,\ldots,x_n)$ as well as $k$ quadratic forms $R^{(1)}(u),\ldots,R^{(k)}(u)$ in dual coordinates $u=(u_1,\ldots,u_n)$. By the FFT for $GL_n$, these invariants are linear combinations of graphs which tell you how to build invariant polynomials by contracting indices (along edges). Here the latter are made of two types of vertices of degree two (two incident edges), ones for the $Q$'s and ones for the $R$'s. Moreover all edges are $QR$, i.e., no $QQ$ or $RR$ edges. So basically these are disjoint collections of cycles of even length made by alternating $Q$'s and $R$'s.
For a proof of the relevant FFT see my two answers to
How to constructively/combinatorially prove Schur-Weyl duality?
Another description of these invariants involves introducing the $k^2$ matrices
$A^{(a,b)}$ with matrix elements
$$
A^{(a,b)}_{ij}=\sum_{\ell=1}^{n} Q_{i\ell}^{(a)} R_{\ell j}^{(b)}\ .
$$
Then the basic invariants are traces of arbitrary products or words made of $A$ matrices.
In the $k=1$ case, there is only one matrix $A$ and the generators are simply the traces of the powers $A,A^2,\ldots,A^n$ and they are algebraically independent because one can take $R$ to be the identity, and $Q$ some generic diagonal matrix and the result follows from the algebraic independence of the elementary symmetric functions. So one recovers the result in Mark's answer.
For general $k$ and $n$, finding a minimal set of (algebra) generators and a description of their relations, sounds to me like a hopeless task. Note that one may get some information via the relation to invariants under conjugation of $k^2$ matrices. This subject has been extensively investigated in work of Procesi, Rasmyslov, Formanek, etc.