Church-Farb on the cohomology of pure braid groups and character polynomials, intuition behind proof of result? Fix $n \ge 2$. Let $V_n$ be the $\binom{n}{2}$-dimensional vector space (over $\mathbb{C}$) generated by a set of vectors $\{w_{ij} : 1 \le i < j \le n\}$. Let $\bigwedge^* V_n$ be the exterior algebra on $V_n$. Note that $\bigwedge^* V_n$ is graded as an algebra (e.g. as a vector space) by $\bigwedge^* V_n = \bigoplus_{i = 0}^\infty \bigwedge^i V_n$, where of course $\bigwedge^i V_n = 0$ for $i > n$. Now let $\mathcal{P}_n$ be the quotient of the algebra $\bigwedge^* V_n$ by the ideal generated by all elements $R_{jkl}$, where $j$, $k$, $l$ are distinct and$$R_{jkl} := w_{jk} \wedge w_{kl} + w_{kl} \wedge w_{lj} + w_{lj} \wedge w_{jk}.$$Thus $\mathcal{P}_n$ is just $\bigwedge^* V_n$ where in addition we have the "Jacobi identity" $R_{jkl} = 0$ on all triples of generators. Note that the grading on $\bigwedge^* V_n$ induces a grading$$\mathcal{P}_n = \bigoplus_{i = 0}^n \mathcal{P}_n^i.$$It is a famous theorem of Arnol'd from 1969 that the algebra $\mathcal{P}_n$ is isomorphic as an algebraic to the cohomology of the pure braid group on $n$ strands, or equivalent the space of $n$-tuples of distinct points in the plane, with the $i$th cohomology group isomorphic to $\mathcal{P}_n^i$.
The symmetric group $S_n$ acts on $\mathcal{P}_n^1 = \wedge^1 V_n = V_n$ via$$\sigma \cdot w_{ij} = w_{\sigma(i) \sigma(j)}.$$This action induces an action of $S_n$ on each $\mathcal{P}_n^i$.
Consider the following result of Church and Farb, established in their paper "Representation theory and homological stability" here.

Theorem (Church-Farb). For each $i$ there is a (unique) character polynomial $P(X_1, \ldots, X_r)$ in the cycle counting functions $X_i$ so that$$\chi_{\mathcal{P}_n^i} = P(X_1, \ldots, X_r) \text{ for all }n \ge 0.$$

Question. Is it possible anybody could give me their intuition behind the proof of this result?
 A: This turns out to be a completely general phenomenon for configuration spaces on any open manifold, though we did not know this at the time; it came in the later paper "FI-modules and stability for representations of symmetric groups" by Church, Ellenberg, Farb (all references are to that paper).
Consider the following two categories:
FI = the category of finite sets and injections
FI# = the category of finite sets and "partially-defined injections" (Def 4.1.1)
The category FI arises naturally here because it parametrizes configuration spaces: having fixed a space M, associated to the finite set S is the configuration space ConfS(M) := Inj(S,M), and an injection S -> T defines a restriction ConfS(M) <- ConfT(M). Therefore the cohomology groups Hk(ConfS(M)) define an FI-module, meaning a functor from FI to abelian groups.
The category FI# is a slightly larger one that captures some additional structure that exists on (the cohomology of) configuration spaces of open manifolds. The maps ConfS(M) <- ConfT(M) "forget" points (they would go from Confn(M) <- ConfN(M) with n <= N), but on an open manifold you can also add points near the boundary, or "at infinity". This is only well-defined up to homotopy, so you don't quite get an FI#-space; but on cohomology this does extend the FI-module structure on Hk(ConfS(M)) to an FI#-module. (Prop 6.4.2)
The result you state comes from a general structure theorem for FI#-modules (Theorem 4.1.5). This theorem is very general (it holds over Z, without any finiteness conditions) but for your purposes we can use this corollary, Theorem 4.1.7(vii):

Theorem (Church-Ellenberg-Farb). If V is an FI#-module over Q with each Vn finite-dimensional, then there is a (unique) character polynomial $P(X_1, \ldots, X_r)$ in the cycle counting functions $X_i$ so that$$\chi_{V_n} = P(X_1, \ldots, X_r) \text{ for all }n \ge 0.$$

In particular, the result you asked about holds for the cohomology of configurations on any open manifold.
There is also a similar theorem for FI-modules, as long as they are finitely generated (Theorem 3.3.4):

Theorem (Church-Ellenberg-Farb). If V is a finitely generated FI-module over Q, then there is a (unique) character polynomial $P(X_1, \ldots, X_r)$ in the cycle counting functions $X_i$ and some $N_0\in \mathbb{N}$ so that $$\chi_{V_n} = P(X_1, \ldots, X_r) \text{ for all }n \ge N_0.$$

This applies to configurations on any manifold, closed or open (Theorem 6.2.1). I emphasize that the existence of this $N_0$ comes from a Noetherian theorem, which in general makes it completely nonconstructive. (The Noetherian theorem was  proved by Church-Ellenberg-Farb-Nagpal; however for this application you only need it over Q, where it had been proved earlier by Snowden and Church-Ellenberg-Farb, which is all you need here.) However in the special case of configuration spaces, we could do some tricks to bound it; in particular when M has dim > 2, we find (Theorem 1.8) that the character polynomial for Hk(Confn(M)) has degree at most k and holds for $n\geq N_0=2k$.
