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?

  • $\begingroup$ I corrected the theorem to n >= 0 (rather than n >= 1). $\endgroup$ – Tom Church Dec 5 '16 at 4:31
  • $\begingroup$ Are you sure this is really a theorem of Church-Farb? I cannot find it in [CF] anywhere -- can you tell me where you found it? I think it might be due to Church-Ellenberg-Farb instead. $\endgroup$ – Tom Church Dec 12 '16 at 16:42

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 $n\geq N_0=2k$.

  • 2
    $\begingroup$ One more thing about the $FI$-module approach that (at least in my opinion) adds to the intuition behind this result. If $n \ge d$ the irreducible $S_n$ representations that occur in $FI$-modules of weight $d$ (over $\mathbb{Q}$ let's say) are exactly those that contain a $S_{n-d}$-invariant vector. So in some sense this theorem says that $H^k(Conf_n(M),\mathbb{Q})$ is spanned by cohomology classes realized by at most $2k$ of the points moving around one another while the rest of the points are fixed, and once $n$ is large enough we just see the "same classes" with more fixed points. $\endgroup$ – Nate Dec 5 '16 at 17:14
  • $\begingroup$ @TomChurch Thanks for the wonderful answer! (From the man himself, no less.) I still have two questions. 1. If it's possible to not invoke the language of FI-modules, what is your intuition behind the proof of my mentioned result? 2. For what $i$ is the computation for the character polynomial $\chi_{\mathcal{P}_n^i}$ known? $\endgroup$ – user102036 Dec 12 '16 at 15:52
  • $\begingroup$ @user102036: Here is one answer: for any $S_d$-representation $W_d$, the family of $S_n$-representations $V_n:=\text{Ind}^{S_n}_{S_d\times S_{n-d}} W_d\boxtimes \mathbb{Q}$ has such a character polynomial $P$ with $\chi_{V_n}=P(\cdots)$ for all $n\geq 0$. So you can prove it by showing the cohomology decomposes as a sum of such families [with finitely many contributing to each dimension]. We deduced this from Orlik-Solomon's work on hyperplane complements, which implies the cohomology decomposes as a sum over "Young subgroups" (explained in the proof of Thm 4.1 in [CF] on p41-43 on arXiv). $\endgroup$ – Tom Church Dec 12 '16 at 16:39
  • $\begingroup$ HOWEVER I am not sure that we actually did ever prove this result without FI-modules: I looked back at [CF] and I cannot find any results on character polynomials. I think this might actually be a theorem of [CEF] (i.e. it was misattributed in your question). If so, the answer in my previous comment still makes sense, but it was not the way things actually went historically. $\endgroup$ – Tom Church Dec 12 '16 at 16:41
  • $\begingroup$ As for your second question, I computed $\chi_{\mathcal{P}^2_n}=2\binom{X_1}{3}+3\binom{X_1}{4}+\binom{X_1}{2}X_2-\binom{X_2}{2}-X_3-X_4$ (this is in the intro to [CEF]), and you should be able to compute $\chi_{\mathcal{P}^1_n}$ yourself. I don't know any explicit computations beyond this, but @JohnWiltshireGordon might. $\endgroup$ – Tom Church Dec 12 '16 at 16:54

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