On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes
Let \(S_ n\) be the symmetric group on \(\{\) 1,...,n\(\}\) acting on \({\mathbb{C}}^ n\) by permuting coordinates. Then \(S_ n\) acts on the manifold \(M_ n=\{(z_ 1,...,z_ n)\in {\mathbb{C}}^ n:\) \(z_ i\neq z_ j\) if \(i\neq j\}\) and on its cohomology \(H=H^*(M_ n,{\mathbb{C}})\). \textit{V. I. Arnol'd} computed H as a graded algebra [in Mat. Zametki 5, 227-231 (1969; Zbl 0277.55002); \(=\) Math. Notes 5, 138-140 (1969)]. (\textit{E. Brieskorn} generalized this to all finite Coxeter groups [in Sémin. Bourbaki, Lect. Notes Math. 317, 21-44 (1973; Zbl 0277.55003)].) The present paper shows that as \({\mathbb{C}}[S_ n]\)-module, for \(p=0,...,n- 1\), \(H^ p\) is isomorphic to a direct sum of induced modules \(Ind(Z(c),S_ n;\xi (c))\) where c runs over a set of representatives of the conjugacy classes of permutations which have n-p cycles and \(\xi\) (c) is an explicitly given one-dimensional representation of the centralizer Z(c) of c. The computation works directly from Arnol'd's presentation of H. The authors conjecture an analog for all finite Coxeter groups.
scientific article; zbMATH DE number 3983461