On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes
type: Item
instance of: scholarly article
title: On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes
zbMATH Open document ID: 0608.20010
DOI: 10.1016/0021-8693(86)90225-5
author: Louis Solomon
author: Gus I. Lehrer
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20B30
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 20F29
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 55Q10
zbMATH DE Number: 3983461
zbMATH Keywords: symmetric group
zbMATH Keywords: cohomology
zbMATH Keywords: graded algebra
zbMATH Keywords: finite Coxeter groups
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90225-5
OpenAlex ID: W1971735664
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description: scientific article; zbMATH DE number 3983461