**Introduction.** Let $k$ be a field of characteristic $0$, and let $n\in\mathbb N$. Let $V=k^n$. The group $\mathrm{GL}_n\left(k\right)=\mathrm{GL} V$ acts on $\mathrm{End} V$ by conjugation, and thus also on the space of $s$-multilinear forms $\left(\mathrm{End} V\right)^s\to k$ for each $s\in\mathbb N$.

For every $p\in\mathbb N$ and every $p$-multilinear form $f:\left(\mathrm{End} V\right)^p \to k$, we define the *antisymmetrization* of $f$ to be the $p$-multilinear form

$g:\left(\mathrm{End} V\right)^p \to k,$

$\left(A_1,A_2,...,A_p\right) \mapsto \dfrac{1}{p!}\sum\limits_{\sigma\in S_p}\left(-1\right)^{\sigma} f\left(A_{\sigma\left(1\right)},A_{\sigma\left(2\right)},...,A_{\sigma\left(p\right)}\right)$.

This $g$ is an antisymmetric $p$-multilinear form.

For any integers $p\geq 0$ and $q\geq 0$, any antisymmetric $p$-multilinear form $\alpha:\left(\mathrm{End} V\right)^p\to k$ and any antisymmetric $q$-multilinear form $\beta:\left(\mathrm{End} V\right)^q\to k$, we can define an antisymmetric $p+q$-multilinear form $\alpha\wedge\beta:\left(\mathrm{End} V\right)^{p+q}\to k$ as the antisymmetrization of the form

$\left(\mathrm{End} V\right)^{p+q}\to k,$

$\left(A_1,A_2,...,A_p,B_1,B_2,...,B_q\right)\mapsto \alpha\left(A_1,A_2,...,A_p\right)\beta\left(B_1,B_2,...,B_q\right)$.

(We could also define it using shuffle products, but that's not important in characteristic $0$.)

For any $p\in\mathbb N$, let $\Omega_p : \left(\mathrm{End} V\right)^p \to k$ be the antisymmetrization of the form

$\left(\mathrm{End} V\right)^p \to k,$

$\left(A_1,A_2,...,A_p\right)\mapsto \mathrm{Tr}\left(A_1A_2...A_p\right)$.

Then, it is known that the ring of antisymmetric $\mathrm{GL}_n\left(k\right)$-invariant multilinear forms on $\mathrm{End}V$ (with multiplication being given by $\wedge$) is generated by the $\Omega_p$ for $p\in\mathbb N$ (this follows from the First Fundamental Theorem for $\mathrm{GL}_n\left(k\right)$, which actually gives all multilinear invariants rather than just the antisymmetric ones). It is also easy to see that $\Omega_p=0$ for all even $p$, and the Amitsur-Levitzki theorem yields that $\Omega_p=0$ for all $p\geq 2n$.

Thus, the family $\left(\Omega_{p_1}\wedge\Omega_{p_2}\wedge ...\wedge\Omega_{p_r}\right)$ (indexed by all strictly increasing sequences $\left(p_1,p_2,...,p_r\right)$ of odd positive integers smaller than $2n$) generates the vector space of all antisymmetric $\mathrm{GL}_n\left(k\right)$-invariant multilinear forms on $\mathrm{End}V$.

**Question.** How to prove that this family is a basis of this space?

**Context.** This is quoted as a consequence of (not further specified) invariant theory in Pierre Cartier's *A primer of Hopf algebras*, page 9, §2.1. I am suspecting Cartier wants to involve some kind of Second Fundamental Theorem, but I don't know it well enough. Maybe there is a slick proof in the same vein as one shows that Amitsur-Levitzki does not hold in smaller degrees than $2n$ ?