On the signature of homogeneous spaces
type: Item
instance of: scholarly article
title: On the signature of homogeneous spaces
zbMATH Open document ID: 0764.53040
DOI: 10.1007/BF00181301
author: Peter Slodowy
published in: Geometriae Dedicata
publication date: 1992-09-27T00:00:00Z
review text: The author calculates the signatures of real and quaternionic Grassmannians and all homogeneous spaces of compact exceptional Lie groups. Let \(G\), \(H\) be compact connected Lie groups such that \(H\subset G\) and \(\text{rank}(G)=\text{rank}(H)\). Relative to a common maximal torus \(T\subset H\subset G\) one denotes by \(\Sigma\) resp. \(\Sigma'\) the root systems of \(T\) in \(G\) resp. \(H\), and by \(W\) resp. \(W'\) the corresponding Weyl groups. Let \(\Psi=\Sigma^ +\backslash(\Sigma')^ +\), where ``\(+\)'' denotes the corresponding sets of positive roots. The calculation of the signature is based on the general formula \[ \text{sign}(G/H)={1\over| W'|}\sum_{w\in W}(-1)^{\mu(w)}, \] where \(\mu(w)\) counts the number of complementary roots \(\gamma\in \Psi\) made negative by \(w^{-1}\). The numbers \(\mu(w)\) are calculated separately for each case of \(\Psi\) by combinatorial methods. The paper is an appendix to \textit{F. Hirzebruch} and \textit{P. Slodowy} [ibid. 35, No. 1-3, 309-343 (1990; Zbl 0712.57010)]. The latter contains basic definitions, motivations, the above formula for the signature, calculations of \(\text{sign}(G/H)\) for other symmetric spaces and the main equality \(\phi(X)=\text{sign}(X)\) for all connected oriented homogeneous spaces admitting a spin structure \((\phi(X)\) denotes the normalized elliptic genus of a homogeneous space \(X\)).
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 57R15
zbMATH DE Number: 64292
zbMATH Keywords: quaternionic Grassmannians
zbMATH Keywords: compact exceptional Lie groups
zbMATH Keywords: root systems
zbMATH Keywords: Weyl groups
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00181301
OpenAlex ID: W2044149763
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description: scientific article; zbMATH DE number 64292