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2. Topology

    1. Quasi-polynomial behavior

          Let $V_n = \wedge^2 \mathbb{Q}^{n-1} = M_{n-2,1,1}$. Then $a_n := \langle \mathrm{H}^i(\mathrm{PConf}^n(\mathbb{C}), V_n\rangle_{S_n}$ is independent of $n$ eventually. More precisely,

      \begin{align*} a_n = \begin{cases} 2 i -2 &\mbox{if } i \equiv 0 \pmod{4} \\ 2 i -3 &\mbox{if } i \equiv 1 \pmod{4} \\ 2 i -2 &\mbox{if } i \equiv 2 \pmod{4} \\ 2 i -1 &\mbox{if } i \equiv 3 \pmod{4} \end{cases}. \end{align*}

      In fact, for any $\lambda$, $\lim_{n \to \infty }\langle \mathrm{H}^i(\mathrm{PConf}^n(\mathbb{C}), V(\lambda)\rangle_{S_n}$ is a quasi-polynomial in $i$ if $i \ge 2$.

      Add remark

      Problem 2.1.

      [W. Chen]
      1. Does the same hold if we replace $\mathbb{C}$ by a manifold $X$ $($for some class of manifolds$)$?
      2. For a general finitely generated $\mathrm{FI}$-algebra $A$, it is not true that $\lim_{n \to \infty } \langle A_n, V(\lambda)\rangle_{S_n}$ is a quasi-polynomial. Is there some hypothesis on $A$ that makes this true?
          P. Hersh: What’s the period of this quasi-polynomial? Can we interpret it as point counts on some rational polytope? W. Chen: It’s related to the size of $\lambda$.

      K. Casto: Is there a theory of quasi-character polynomials?

      J. Miller: Is there a mod $p$ version of this question? May be only when the manifold is open? More precisely, is $\lim_{n \to \infty } \dim \mathrm{H}^i(\mathrm{PConf}^n(\mathbb{C}), \mathbb{F}_p)$ a quasi-polynomial in $i$?

      Speyer: If $\lambda$ is the empty partition, then there is a multiplicative structure making it an algebra. Is it a finitely generated algebra? Maguire: We may have an answer for certain manifolds.

      J. Miller: For $X = \mathbf{P}^2_\mathbb{C}, \mathbf{P}^3_\mathbb{C}$ it holds when $\lambda$ is a single row (for $i \ge 12$ for $\mathbf{P}^2_\mathbb{C}$ and $i \ge 23$ for $\mathbf{P}^3_\mathbb{C}$).

      W. Chen: Another example with a similar behavior is $\mathrm{H}^i(\tilde{T}_n)$ where $\tilde{T}_n$ is the space of maximal tori in $\mathrm{GL}_n(\mathbb{C})$.

        • Naturally occurring non-trivial $\mathrm{FI}_2$ modules

              Let $X$ be a manifold with two specified boundary components. Claim: $V_n = \mathrm{H}_i(\mathrm{PConf}^n(X))$ has an $\mathrm{FI}_2$-module structure. In particular, if $X = S^1 \times I$ is the cylinder and $i=1$. Then $V$ has the following properties:
          1. There is an exact sequence $0 \to \tilde{M}(1) \to V \to \tilde{M}(2) \to 0$ where $\tilde{M}(1)$ and $\tilde{M}(2)$ are the pull-backs of $M(1)$ and $M(2)$ along the natural forgetful map $\mathrm{FI}_2 \to \mathrm{FI}$. (See [CEF] for $M$ notation.)
          2. $V$ has an $\mathrm{FI}$-module structure and $V$ is not in the image of the natural map $\mathrm{Mod}_{\mathrm{FI}} \to \mathrm{Mod}_{\mathrm{FI}_2}$ induced by the forgetful map $\mathrm{FI}_2 \to \mathrm{FI}$.


          In other words, $V$ is filtered by things that are pull-backs from $\mathrm{FI}$ but it is not itself a pull-back from $\mathrm{FI}$.

          Add remark

          Problem 2.2.

          [J. Ellenberg]
          1. What can we say about $\mathrm{H}_i(\mathrm{PConf}(X))$ generally along these lines?
          2. Can we calculate $\mathrm{Ext}(M,N)$ for $\mathrm{FI}_2$-modules?
              A. Snowden: One can calculate these Ext groups in the category of $\mathrm{FI}_2$-modules using Koszul resolutions.

          [CEF] Thomas Church, Jordan Ellenberg, Benson Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015), no. 9, 1833–1910

            • Cohomology of $\mathrm{PConf}^n$


              Add remark

              Problem 2.3.

              [P. Hersh] Find the Specht module decomposition for $\mathrm{H}^i(\mathrm{PConf}^n \mathbb{C})$, and its $\mathrm{FI}$-module generators. It might involve a derangement recurrence. Recurrence is going to involve both $i$ and $n$. $($This question might relate to W. Chen’s question.$)$
                  J. Ellenberg: Compare with the theorem that states: \[ \bigoplus_{i \ge 0} \mathrm{H}^i(\mathrm{PConf}^n \mathbb{C})\otimes \mathrm{sgn}^i = \mathbb{Q}[S_n] = \bigoplus_{i \ge 0} \mathrm{H}^i(\mathrm{PConf}^n \mathbb{R}^d) \] when $d$ is odd.

                • $S_{\infty}$ structure and configuration spaces


                  Add remark

                  Problem 2.4.

                  [P. Patzt] Compute $S_{\infty}$ representations of configuration spaces directly.

                    • Highly acyclic complexes

                          Working over a field $\mathbf{k}$, define $\mathrm{PBC}_{\bullet}^n$ with \[ \mathrm{PBC}_{p}^n = \{(v_1, \ldots, v_p, C) \colon v_i \mbox{ are } \mathbf{k}\mbox{-linearly independent, } C \mbox{ a complement of span of } v_i \} . \] Then $\mathrm{PBC}^n_{\bullet}$ is a semi-simplicial $\mathbf{VIC}(\mathbf{k})$-set. It is highly connected ($(n-3)/2$-connected). The corresponding augmented chain complex \[ \mathbb{Z}[\mathrm{PBC}_{\bullet}] \xrightarrow{\epsilon} \mathbb{Z} \to 0 \] is acyclic (exact). The boundary maps are the signed sums of all ways to take a vector and put it in the complement. This all holds if we replace the field $\mathbf{k}$ by a general ring $R$, and in this case we have $\mathbb{Z}[\mathrm{PBC}_{p}] = \mathbb{Z}\mathrm{GL}_n(R) \otimes_{\mathbb{Z}\mathrm{GL}_{n - (p+1)}(R)} \mathbb{Z} $. Generalizing this (and changing the coefficient ring from $\mathbb{Z}$ to $R$) we get the following question:

                      Add remark

                      Problem 2.5.

                      [P. Patzt] Is $R\mathrm{GL}_n(R) \otimes_{R\mathrm{GL}_{n - (\bullet+1)}(R)} R^{n - (\bullet+1)} \to R^n \to 0$ highly acyclic?
                          If true for $R =\mathbb{Z}$, it would prove that $\mathrm{H}_2(\mathrm{IA}_n)$ is presented in finite degree.

                        • Schur weyl duals


                          Add remark

                          Problem 2.6.

                          [A. Snowden] Are there direct constructions for the Schur–Weyl dual of $\mathrm{FI}$-modules like $\mathrm{H}^{\bullet}(\mathrm{PConf}^n{X}, \mathbb{Q})$? E.g., can one construct a $\mathbf{GL}_{\infty}$-space and a $\mathbf{GL}_{\infty}$-equivariant sheaf whose cohomology is the Schur–Weyl dual?
                              V. Reiner: for $X=\mathbb{R}^d$ with $d$ odd, perhaps get tensor algebra.

                          P. Tosteson: “space version” of Schur functors: given $S_n$-space $X$ and space $Y$ can form $S_X(Y)=(Y^n \times X)/S_n$. “Polynomial functor” of spaces in sense of Goodwillie.

                          T. Church: $\mathrm{FI}$-algebras in spaces lead to models for infinite loop spaces.

                              Cite this as: AimPL: Representation stability, available at http://aimpl.org/repnstability.