Symmetric group actions on the cohomology of configurations in $R^d$

TL;DR

This paper explores how symmetric groups act on the cohomology of configuration spaces in Euclidean space, revealing extended symmetries and providing elementary algebraic constructions for these actions.

Contribution

It introduces an elementary algebraic approach to understanding extended symmetric group actions on cohomology of configuration spaces, especially for odd dimensions.

Findings

Extended $S_{n+1}$-action on cohomology for even $d$

Elementary algebraic construction for odd $d$

Interplay between standard and extended actions

Abstract

In this paper we deal with the action of the symmetric group on the cohomology of the configuration space $C_n(d)$ of $n$ points in $\mathbb{R}^d$. This topic has been studied by several authors (see the introduction). On the cohomology algebra $H^*(C_n(d); \mathbb{C})$ there is, in addition to the natural $S_n$-action, an extended action of $S_{n+1}$; this was first shown for the case when $d$ is even by Mathieu, Robinson and Whitehouse and the second author. For the case when $d$ is odd it was shown by Mathieu (anyway we will give an elementary algebraic construction of the extended action for this case). The purpose of this article is to present some results that can be obtained, in an elementary way, exploiting the interplay between the extended action and the standard action.