My advisor and I are working through a paper on partition functions, and we got to the following passage:
Fix $n \in \mathbb N$ and let $W := ((\mathbb R^n)^{\otimes 3})^{C_3}$, where the $C_3$ means that we are looking at the subspace that is invariant under cyclic actions. Let $\mathcal O(W)$ denote the algebra of polynomials on $W$. For any $q \in \mathcal O(W)$, let $dq$ denote its derivative in $\mathcal O(W) \otimes W^*$.
We have been completely unable to track down any books or other sources on this algebra of polynomials. The paper does not have any references in this paragraph, and none of the references define these polynomials. I'm now wondering:
1) Are there any books or articles that explain what this algebra of polynomials is? The section on multilinear algebra in our library contained no books explaining it, and googling doesn't seem to turn up anything relevant.
2) Could anyone tell me what an element in this algebra looks like?
3) How is the derivative of this kind of polynomial defined? If it was just a formal derivative, shouldn't it be in $\mathcal O(W)$, not $\mathcal O(W) \otimes W^*$?
The algebra of polynomials on a vector space $W$ (of dimension $N$, say) is just a coordinate-free way of saying "polynomials in $N$ variables". So, if you like, a polynomial on $W$ is just a function $q:W\to \mathbb{R}$ such that if you pick a linear isomorphism $f:\mathbb{R}^N\to W$, the composition $qf:\mathbb{R}^N\to\mathbb{R}$ is a polynomial function in $N$ variables. Note that it is easy to show that this condition is in fact independent of the linear isomorphism $f$ chosen. Given a polynomial $q\in\mathcal{O}(W)$ and a vector $w\in W$, you can take the directional derivative $\partial_w q$, which is another polynomial on $W$. In this way, the (total) derivative of $q$ is a linear map $W\to\mathcal{O}(W)$ sending $w$ to $\partial_w q$, or equivalently an element of $\mathcal{O}(W)\otimes W^*$.
More abstractly, $\mathcal{O}(W)$ can be defined as the symmetric algebra $\operatorname{Sym}(W^*)$ on the dual of $W$. Here an element of $W^*$ is thought of as representing a homogeneous linear polynomial on $W$, and $\operatorname{Sym}(W^*)$ is the free commutative $\mathbb{R}$-algebra generated by the vector space $W^*$. That is, arbitrary polynomials are obtained formally as linear combinations of products of homogeneous linear polynomials. This definition has the advantage of being completely coordinate-free and obviously functorial, and also working over any field or even any commutative ring (the definition in the previous paragraph fails over finite fields, since two different polynomials can represent the same function).
I don't know of any particular reference for this, but it has a Wikipedia page you might find helpful.