**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](http://en.wikipedia.org/wiki/Amitsur%25E2%2580%2593Levitzki_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*](http://www.math.osu.edu/~kerler.2/VIGRE/InvResPres-Sp07/Cartier-IHES.pdf), 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 $2n$ ?



Alternating multilinear invariants of GL(n) on End (k^n)
Another proof of linear independence (which I should have typed in the first place - but there are too many spectral sequences on my mind these days): 


To prove linear independence of the forms you are considering, it is enough to prove that $\Omega\_1\wedge\Omega\_3\wedge\ldots\wedge\Omega\_{2n-1}\ne0$ (if there is a linear combination equal to zero, you can always multiply it by something so that one of the [lowest degree] terms in that combination becomes $\Omega\_1\wedge\Omega\_3\wedge\ldots\wedge\Omega\_{2n-1}$, and all others disappear). 


Now, let us note that in the case of a finite-dimensional algebra homology is dual to the cohomology, so it is sufficient to prove that the top homology of $gl\_n$ is non-zero. But it is very easy to check that $\wedge\_{i,j=1}^ne\_{ij}$ is closed in the Chevalley--Eilenberg complex, and there is nothing that may kill that homology class, so we are done.


Of course, this also uses cohomology, but in a much less heavy way :-)


