Representation stability on the cohomology of complements of subspace arrangements (Q2316676)

This paper studies representation stability, in the sense of [\textit{T. Church} and \textit{B. Farb}, Adv. Math. 245, 250--314 (2013; Zbl 1300.20051)], of the cohomology groups of the complements of arrangements of linear subspaces of \(n\)-tuples in \(\mathbb{R}^n\) defined by sets of diagonal equalities and which are invariant under the action of the symmetric group \(S_n\) which permutes the coordinates. More precisely, let \(d\geq 2\) and \(n\) be natural numbers and let \(W_{\pi}^d\) denote the linear subspace of \(n\)-tuples \((w_1,\ldots,w_n)\) of points in \(\mathbb{R}^d\) such that \(w_i=w_j\), if \(i\) and \(j\) are in the same block in a set partition \(\pi\) of \(\{1,2,\ldots,n\}\). For an integer partition \(\lambda\), let \(\mathcal{A}_{\lambda}^d\) be the arrangement of subspaces \(W^d_{\pi}\) where \(\pi\) is of type \(\lambda\), and set \(\mathcal{A}_{\lambda}^d= \bigcup_{\lambda\in\Lambda} \mathcal{A}_{\lambda}^d\) for a finite set of integer partitions \(\Lambda\). The complements \(\mathcal{M}_{\Lambda}^d:= \mathbb{R}^{dn}\setminus \bigcup_{W\in\mathcal{A}_{\Lambda}^d} W\) come equipped with an \(S_n\)-action. Let \(\Lambda\) be a finite set of integer partitions of \(n_0\) not containing \((1^{n_0})\). For every \(n\geq n_0\), the author considers \(\Lambda^{(n)}\) as the set of integer partitions of \(n\) obtained from integer partitions in \(\Lambda\) by adding \(n-n_0\) parts of size \(1\). The main result of the paper, Theorem 1.1, shows that the sequence of reduced singular cohomology groups \(\left\{\tilde{H}^i(\mathcal{M}^d_{\Lambda^{(n)}};\mathbb{C})\right\}_n\) satisfies representation stability, giving an alternative proof of a special case of Theorem A in [\textit{N. Gadish}, Adv. Math. 322, 341--377 (2017; Zbl 1377.14012)] and of Theorem 4.15 in [\textit{D. Petersen}, Geom. Topol. 21, No. 4, 2527--2555 (2017; Zbl 1420.55027)]. Furthermore, Theorem 1.1 establishes new explicit stable ranges, and better stable bounds are obtained in Theorem 3.2 for the case of \(k\)-equal arrangements \(\mathcal{M}_{(k,1^{n-k})}^d\) for \(k\geq d+1\). In the particular case \(\Lambda=\{(2,1^{n-2})\}\), the complement \(\mathcal{M}^d_{\Lambda}\) is precisely the configuration space of \(n\) ordered points in \(\mathbb{R}^d\). Representation stability of the cohomology of this sequence of spaces was proved in [\textit{T. Church}, Invent. Math. 188, No. 2, 465--504 (2012; Zbl 1244.55012)] and optimal stable ranges were obtained in [\textit{P. Hersh} and \textit{V. Reiner}, Int. Math. Res. Not. 2017, No. 5, 1433--1486 (2017; Zbl 1404.20009)].