Grassmannian Realizations of Classical Pseudo-Riemannian Symmetric Spaces

Abstract

We establish a unified fixed-point Grassmannian framework for the classical pseudo-Riemannian symmetric spaces from Berger's classification. For a classical group G=mathopAut(V,b), we identify G with the open graph-like locus inside an ordinary or maximal-isotropic Grassmannian G̅. Natural involutions of G̅ are constructed via classical forms, compatible scalar structures, and subspace configurations. We explicitly describe the full fixed loci of these involutions in terms of classical Grassmannians. The corresponding graph-like fixed loci (or their signature strata) are identified, respectively, with the moduli spaces of classical forms, compatible scalar structures, and subspace configurations; these moduli spaces coincide with classical pseudo-Riemannian symmetric spaces. In summary, the 54 classical series are organized into four families: 10 group-type models, 7 classical-form Grassmannians, 20 scalar-structure Grassmannians, and 17 subspace-configuration Grassmannians.

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BibTeX RIS

BibTeX

@misc{chen2026grassmannian,
  title = {{Grassmannian Realizations of Classical Pseudo-Riemannian Symmetric Spaces}},
  author = {Yunxia Chen and Naichung Conan Leung},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36724},
  eprint = {2610.36724},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36724},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Grassmannian Realizations of Classical Pseudo-Riemannian Symmetric Spaces
AU  - Yunxia Chen
AU  - Naichung Conan Leung
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.36724
UR  - https://arxiv.org/abs/2610.36724
AB  - We establish a unified fixed-point Grassmannian framework for the classical pseudo-Riemannian symmetric spaces from Berger's classification. For a classical group G=mathopAut(V,b), we identify G with the open graph-like locus inside an ordinary or maximal-isotropic Grassmannian G̅. Natural involutions of G̅ are constructed via classical forms, compatible scalar structures, and subspace configurations. We explicitly describe the full fixed loci of these involutions in terms of classical Grassmannians. The corresponding graph-like fixed loci (or their signature strata) are identified, respectively, with the moduli spaces of classical forms, compatible scalar structures, and subspace configurations; these moduli spaces coincide with classical pseudo-Riemannian symmetric spaces. In summary, the 54 classical series are organized into four families: 10 group-type models, 7 classical-form Grassmannians, 20 scalar-structure Grassmannians, and 17 subspace-configuration Grassmannians.
ER  -

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