  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.
