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?