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October 2022 Integral homology of real isotropic and odd orthogonal Grassmannians
Jordan Lambert, Lonardo Rabelo
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Osaka J. Math. 59(4): 853-880 (October 2022).


We obtain a combinatorial expression for the boundary map coefficients of real isotropic and odd orthogonal Grassmannians. It provides a natural generalization of the known formulas for Lagrangian and maximal isotropic Grassmannians. The results derive from the classification of Schubert cells into four types of covering pairs when identified with signed $k$-Grassmannian permutations. Our formulas show that the coefficients depend on the changed positions for each permutation pair type. We apply this to obtain an orientability criterion and compute the first and second homology groups for these Grassmannians. Furthermore, we exhibit an apparent symmetry of the boundary map coefficients.

Funding Statement

Supported by FAPESP grant number 13/10467-3 and 14/27042-8, and the Coordination for the Improvement of Higher Level Personnel - Capes.


We thank to David Anderson for helpful suggestions and valuable discussions on an earlier version. We thank an anonymous referee for comments on the manuscript.


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Jordan Lambert. Lonardo Rabelo. "Integral homology of real isotropic and odd orthogonal Grassmannians." Osaka J. Math. 59 (4) 853 - 880, October 2022.


Received: 31 August 2020; Revised: 5 August 2021; Published: October 2022
First available in Project Euclid: 4 October 2022

Primary: 05A05 , 05E16 , 14M15 , 57T15

Rights: Copyright © 2022 Osaka University and Osaka Metropolitan University, Departments of Mathematics


Vol.59 • No. 4 • October 2022
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