Representation Stability for Cohomology of Configuration Spaces in ${\mathbb{R}}^d$

This paper studies representation stability in the sense of Church and Farb for representations of the symmetric group S n on the cohomology of the configuration space of n ordered points in R d. This cohomology is known to vanish outside of dimensions divisible by d − 1; it is shown here that the S n-representation on the i(d − 1)st coho-mology stabilizes sharply at n = 3i (resp. n = 3i + 1) when d is odd (resp. even). The result comes from analyzing S n-representations known to control the cohomology: the Whitney homology of set partition lattices for d even, and the higher Lie representations for d odd. A similar analysis shows that the homology of any rank-selected subposet in the partition lattice stabilizes by n ≥ 4i, where i is the maximum rank selected. Further properties of the Whitney homology and more refined stability statements for S n-isotypic components are also proven, including conjectures of J. Wiltshire-Gordon.