Let $V = (\mathbb{R}^n, g)$, where $g$ is the Euclidean inner product on $V$. Denote by $G$ the orthogonal group $O(V) = O(n)$ and by $\mathfrak{g}$ the Lie algebra of $G$.


Let $W \subset \Lambda^2V^\* \odot \Lambda^2V^\*$ be the subset satisfying the algebraic Bianchi identity. More precisely, let $R(v\_1,v\_2,v\_3,v\_4)$ denote an element of $\Lambda^2V^\* \odot \Lambda^2V^\*$. Thus $R$ is skew-symmetric in $v\_1$ and $v\_2$ and it is also skew-symmetric in $v\_3$ and $v\_4$. Moreover


$$ R(v\_3,v\_4,v\_1,v\_2) = R(v\_1,v\_2,v\_3,v\_4). $$


Then $R \in W$ if and only if, in addition to the conditions above, $R$ also satisfies the following identity (known as the algebraic Bianchi identity):


$$ R(v\_1,v\_2,v\_3,v\_4) + R(v\_2,v\_3,v\_1,v\_4) + R(v\_3,v\_1,v\_2,v\_4) = 0. $$


Now my question can be formulated. What is an explicit description of the ring $\mathbb{C}^G[W]$ of $G$-invariant polynomials on $W$ (with $W$ being the space of algebraic curvature tensors, if I may call it so) and $G$ acting on $W$ by restricting its natural action on $\Lambda^2V^\* \odot \Lambda^2V^\*$.


Also, if one fixes a degree $d > 0$, what is an explicit description of the space of $G$-invariant homogeneous polynomials in $W$ of degree $d$?


I was thinking at first about the Chern-Weil homomorphism, but I think this only gives a proper subspace of $G$-invariant polynomials on $W$, and not all of them (I am not a 100% sure). This has probably been studied in the literature. I don't have access to MathSciNet anymore though (due to some budget cuts by my University).


Edit 1: I notice some overlap with the post [Invariant polynomials in curvature tensor vs. characteristic classes](https://mathoverflow.net/questions/323540/invariant-polynomials-in-curvature-tensor-vs-characteristic-classes), but the posts are sufficiently different.



What are all invariant polynomials on the space of algebraic curvature tensors?
I am not sure that this has a "nice" answer. Your question can be reformulated as follows. Let $\mathcal{A}\_n$ be the space of algebraic curvature tensors on $\mathbb{R}^n$. A homogenous polynomial $P$ on $\mathcal{A}\_n$ is the same as an element of $S^k\mathcal{A}\_n$, the $k$-th symmetric tensor power of $\mathcal{A}\_n$. Now if $H\_k$ is the space of homogeneous polynomials of degree $k$ on $\mathcal{A}\_n$, then $H\_k \subset S^k \mathcal{A}\_n$ is a subrepresentation of $G$.


In other words, a recipe to obtain an answer to your question for specific $k$, $n$, is the following. Decompose the $G$-representation $S^k \mathcal{A}\_n$ into irreducible $G$-representations and count the number of trivial representations among those. This can be done for low $k$, $n$ using software such as [LiE](http://wwwmathlabo.univ-poitiers.fr/%7Emaavl/LiE/form.html).


Note that as a $G$-representation, the space $\mathcal{A}\_n$ splits into the direct sum of three irreducible representations:
$$ \mathcal{A}\_n = \mathbb{R} \oplus S^2\_0(\mathbb{R}^n) \oplus \mathcal{W},$$
where $\mathcal{W}$ is the space of Weyl curvature tensors (i.e. those curvature tensors that are additionally entirely trace-free).


A quick check on LiE shows that there are plenty such polynomials: For example, looking for polynomials that depend on the Weyl part alone and $n$ large, there is one of degree 2 and four of degree 3. I doubt there is a good general answer.


