Symmetric group actions on the cohomology of configurations in

  • Giacomo d'Antonio

    Universität Bremen, Germany
  • Giovanni Gaiffi

    Università di Pisa, Italy


In this paper we deal with the action of the symmetric group on the cohomology of the configuration space of points in . This topic has been studied by several authors and it is well-known that for even, while, for odd, .

On the cohomology algebra , there is, in addition to the natural -action, an extended action of ; this was shown for the case when is even by Mathieu, Robinson and Whitehouse and the second author using three different methods. For the case when 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 is even and show how to extend this proof to the case when is odd. We will also show how to locate among the homogeneous components of the graded algebra 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.

Cite this article

Giacomo d'Antonio, Giovanni Gaiffi, Symmetric group actions on the cohomology of configurations in . Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 21 (2010), no. 3, pp. 235–250

DOI 10.4171/RLM/569