Cellular homology of real maximal isotropic Grassmannians
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.
scientific article; zbMATH DE number 6854767
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