Skip to content
DraconiticResearch
Search Sources API
201,818 studies31 sourcesUpdated 5 min ago
PreprintCC BY 4.0arXiv

Double Bruhat cells in symmetric spaces

  • Huanchen Bao
  • Jinfeng Song
Open at sourcePDFDOICite

Abstract

This paper studies the closures of the double Bruhat cells of symmetric spacess in its equivariant embeddings. We call them large double Bruhat varieties. Our aim is to establish the double Bruhat stratification as a natural setting for extending the geometry of Richardson varieties to symmetric spaces. For the wonderful compactification of the associated adjoint symmetric space, we show their scheme-theoretic intersections with the unique closed orbit are reduced unions of projected Richardson varieties. This connection with Richardson geometry on partial flag varieties supports large double Bruhat varieties as a natural class of subvarieties in symmetric-space embeddings. In positive characteristic, we prove that every equivariant embedding admits a Frobenius splitting compatible with all large double Bruhat varieties and all boundary divisors. These varieties are strongly F-regular, and globally F-regular when the ambient embedding is projective. In characteristic zero, they are of strongly F-regular type, and of globally F-regular type in the projective case; in particular, they are normal, Cohen-Macaulay, and have rational singularities.

Related

  1. On cluster deep loci in double Bruhat cells
    Hanwen Quan
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  2. Double-arc rearrangement and the symmetric isoperimetric problem in Gauss space
    Marco Barchiesi, Filippo Cagnetti, Vesa Julin
    arXivPreprintarXiv licenceFull textAnalysis & PDEsProbabilityarXiv
  3. ISOMETRIC EMBEDDING OF NONCOMMUTATIVE L_p -SPACES INTO NONCOMMUTATIVE SYMMETRIC SPACES
    Jinghao Huang, Marius Junge, Fedor Sukochev +1
    HALPreprintPublisher termsFull textAnalysis & PDEsHAL
  4. Locally symmetric hypersurfaces in globally symmetric spaces
    Jihun Kim, Yuri Nikolayevsky, Ruy Tojeiro
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  5. Shifted wave equation on noncompact symmetric spaces
    Yulia Kuznetsova, Zhipeng Song
    HALPreprintLicence not statedFull textGeneral mathematicsHAL

Cite

BibTeXRIS
BibTeX
@misc{bao2026double,
  title = {{Double Bruhat cells in symmetric spaces}},
  author = {Huanchen Bao and Jinfeng Song},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16362},
  eprint = {2610.16362},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16362},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Double Bruhat cells in symmetric spaces
AU  - Huanchen Bao
AU  - Jinfeng Song
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16362
UR  - https://arxiv.org/abs/2610.16362
AB  - This paper studies the closures of the double Bruhat cells of symmetric spacess in its equivariant embeddings. We call them large double Bruhat varieties. Our aim is to establish the double Bruhat stratification as a natural setting for extending the geometry of Richardson varieties to symmetric spaces. For the wonderful compactification of the associated adjoint symmetric space, we show their scheme-theoretic intersections with the unique closed orbit are reduced unions of projected Richardson varieties. This connection with Richardson geometry on partial flag varieties supports large double Bruhat varieties as a natural class of subvarieties in symmetric-space embeddings. In positive characteristic, we prove that every equivariant embedding admits a Frobenius splitting compatible with all large double Bruhat varieties and all boundary divisors. These varieties are strongly F-regular, and globally F-regular when the ambient embedding is projective. In characteristic zero, they are of strongly F-regular type, and of globally F-regular type in the projective case; in particular, they are normal, Cohen-Macaulay, and have rational singularities.
ER  -