Representation stability for cohomology of configuration spaces in 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 cohomology 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\geq 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.