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

Abstract

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 ≥ 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.2010 Mathematics Subject Classification. 55R80, 05E45, 05E05, 20C30, 06A07.

Polynomial Factorization Statistics and Point Configurations in ℝ3

“…If λ = (1 m 1 2 m 2 · · · ) is a partition, let a λ := j≥1 a m j j . Hersh and Reiner [11,Thm. 2.17] state the following identity of formal power series d≥0 λ⊢d…”

“…(2) To get the formula for ν sf (λ) we start with another formal power series identity from [11,Thm. 2.17].…”

The equivariant Kazhdan–Lusztig polynomial of a matroid

“…The polynomial K n (t) is well understood, going back to Lehrer and Solomon [LS86]. An explicit formula for ch K n (t) appears in [HR,Theorem 2.7]. We cannot give an explicit formula for Q n (t) in the same way that we did for uniform matroids, but we will decribe the recursion that can be used to compute it and calculate some examples for small n.…”

Polynomial Splitting Measures and Cohomology of the Pure Braid Group

“…We show in Proposition 6.2 that the representations A k n are isomorphic to others appearing in the literature known to exhibit representation stability. Hersh and Reiner [15,Corollary 5.4] determine the precise rate of stabilization of these representations, yielding the following result. Theorem 1.5 (Representation stability for A k n ).…”