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

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 $R^d$. This topic has been studied by several authors and it is well-known that for $d$ even, $H^{\ast }(C_n(d);C)≌2{\mathrm{Ind}}_{S_2}^{S_n}1$ while, for $d$ odd, $H^{\ast }(C_n(d);C)≌C{S}_n$.

On the cohomology algebra $H^{\ast }(C_n(d);C)$, there is, in addition to the natural $S_n$-action, an extended action of $S_{n_{+}1}$; this was shown for the case when $d$ is even by Mathieu, Robinson and Whitehouse and the second author using three different methods. 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. Among these we will recall a quick proof for the formula cited above for the case when $d$ is even and show how to extend this proof to the case when $d$ is odd. We will also show how to locate among the homogeneous components of the graded algebra $H^{\ast }(C_n(d);C)$ the copies of the standard, sign and standard tensor sign representations and we will give explicit formulas for both the extended and the canonical actions on the low-degree cohomology modules.