Representation Stability for Cohomology of Configuration Spaces in R Patricia Hersh and Victor Reiner

This paper studies representation stability in the sense of Church and Farb for representations of the symmetric group Sn on the cohomology of the configuration space of n ordered points in R. This cohomology is known to vanish outside of dimensions divisible by d− 1; it is shown here that the Sn-representation on the i(d− 1) cohomology stabilizes sharply at n = 3i (resp. n = 3i + 1) when d is odd (resp. even). The result comes from analyzing Sn-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 Sn-isotypic components are also proven, including conjectures of J. Wiltshire-Gordon.