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](https://arxiv.org/pdf/1008.1368v3.pdf).



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> **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$$\chi\_{\mathcal{P}\_n^i} = P(X\_1, \ldots, X\_r) \text{ for all }n \ge 0.$$
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**Question.** Is it possible anybody could give me their intuition behind the proof of this result?



Church-Farb on the cohomology of pure braid groups and character polynomials, intuition behind proof of result?
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](https://arxiv.org/abs/1204.4533)" 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 Conf*n*(M) <- Conf*N*(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):



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> **Theorem (Church-Ellenberg-Farb).** If *V* is an FI#-module over **Q** with each *V**n* finite-dimensional, then there is a (unique) character polynomial $P(X\_1, \ldots, X\_r)$ in the cycle counting functions $X\_i$ so that$$\chi\_{V\_n} = P(X\_1, \ldots, X\_r) \text{ for all }n \ge 0.$$
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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):



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> **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 $$\chi\_{V\_n} = P(X\_1, \ldots, X\_r) \text{ for all }n \ge N\_0.$$
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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](https://arxiv.org/abs/1210.1854); 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 $n\geq N\_0=2k$.


