Exercise from Etingof's notes on Representation Theory I am reading through these notes of Etingof on Representation theory and I am stuck with one exercise (it's problem 4.69 in the notes). The problem is the following.
Consider the space $X=Mat_n(\mathbb{C})^k$ of $k$-tuples of $n\times n$ complex matrices. It has natural action of $GL_n(\mathbb{C})$, which is by simultaneous conjugation. This action induces the action on the space $k[X]$ of polynomial functions on $X$. 

Question: prove that the algebra $k[X]^{GL_n}$ of invariant polynomials is generated by the functions of the form $(A_1,\dots,A_k)\mapsto trace(w(A_1,\dots,A_k))$, where $w$ is just some word on $k$ letters.

There is even a hint. Namely, to embed the space of polynomials with fixed degree $d_i$ in the variable $A_i$ into $End(\mathbb{C}^n)^{\otimes \sum d_i}$ and use Schur-Weyl duality. 
I do not quite understand how to use this hint. Schur-Weyl duality tells me how to decomope a representation of the form $V^{\otimes r}$ as $GL(V)\times S_r$-module. I guess in my case $V$ should be $End(\mathbb{C}^n)$. But in my case the group is $GL_n$, not $GL_{n^2}$. Am I missing something? Can you, please, help me with this exercise?
Thank you very much for your help!

I have found this book by Kraft and Procesi. They describe this problem on page 21, and give the proof in the section 4.7.
