Cellular homology of real maximal isotropic Grassmannians
type: Item
instance of: scholarly article
title: Cellular homology of real maximal isotropic Grassmannians
zbMATH Open document ID: 1414.57018
author: Lonardo Rabelo
publication date: 2018-03-29T00:00:00Z
review text: Over the complex numbers, the structure of the (co)homology rings of flag manifolds is easy to derive and the beauty of their structures both in combinatorics and algebra has been attracting many mathematicians. On the other hand, the (co)homology rings of real flag manifolds are highly nontrivial and the determination of their structure even for Grassmannians is still open. The difficulty, in contrast to the case of complex flag manifolds, is in the non-vanishing of the boundary map. \par The boundary map of cellular $ \mathbb{Z} $-homology was described by \textit{R. R. Kocherlakota} [Adv. Math. 110, No. 1, 1--46 (1995; Zbl 0832.22020)] and \textit{L. Casian} and \textit{Y. Kodama} [Invent. Math. 165, No. 1, 163--208 (2006; Zbl 1118.22008)] in completely different ways. However, both results involve serious calculations and it is not easy to extract an explicit formula for the boundary operators. \par In the paper under review, the author determines boundary maps of the cellular $ \mathbb{Z} $-homology of real maximal isotropic Grassmannians of type C, B, D explicitly (Theorem 6.5). The description is quite concrete based on the combinatorics of strict partitions, also used in [\textit{T. Ikeda} and \textit{H. Naruse}, Trans. Am. Math. Soc. 361, No. 10, 5193--5221 (2009; Zbl 1229.05287)], and the precise structure of Weyl groups. \par As an application, the orientability of these isotropic Grassmannians is obtained. \par Reviewer's remark: the statements of Corollary 4.2 and Proposition 6.4 contain misprints.
reviewed by: Kyo Nishiyama
Mathematics Subject Classification ID: 57T15
Mathematics Subject Classification ID: 53D05
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 51M35
zbMATH DE Number: 6854767
zbMATH Keywords: real flag manifolds
zbMATH Keywords: cellular homology
zbMATH Keywords: isotropic Grassmannians
zbMATH Keywords: strict partitions
zbMATH Keywords: orientability
MaRDI profile type: Publication
full work available at URL: advgeom-2016-0014
OpenAlex ID: W2473514931
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DOI: 10.1515/ADVGEOM-2016-0014
published in: advg
description: scientific article; zbMATH DE number 6854767