Skip to content
DraconiticResearch
Search Sources API
186,631 studies31 sourcesUpdated 58 min ago
PreprintCC BY 4.0arXiv

On cluster deep loci in double Bruhat cells

  • Hanwen Quan
Open at sourcePDFDOICite

Abstract

We study deep loci for the reduced-word Berenstein-Fomin-Zelevinsky cluster atlas on type-A double Bruhat cells. As defined in CGSS24, for a seed collection on a cluster variety X, the deep locus is the complement of the union of all corresponding cluster tori. We focus on double Bruhat cells G(u,v) for G = SL(n). Our main results are several theorems describing the structure of these deep loci, including their interaction with the Poisson structure, the relationships between the deep loci of different cells, and a tracking formula that allows us to compute the reduced-word deep locus of a cell from a lower-dimensional one. The paper also includes explicit low-rank calculations for double Bruhat cells in SL(3) and Borel double Bruhat cells in SL(4). Finally, we discuss the relationship between these results and some general properties of deep loci.

Related

  1. Double Bruhat cells in symmetric spaces
    Huanchen Bao, Jinfeng Song
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  2. Cluster structures from Legendrian double twist knots: a comparative approach
    James Hughes, Jiajie Ma
    arXivPreprintCC BY-NC-SA 4.0Full textGeometry & topologyarXiv
  3. Deep Points of Cluster Algebras II: Punctured Marked Surfaces
    James Beyer
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  4. Double cluster swapping for spin models: Pfaffian relations and sharpness
    Diederik van Engelenburg, Lorca Heeney, Marcin Lis
    arXivPreprintCC BY 4.0Full textMathematical physicsProbabilityarXiv
  5. Double Gravity-Assist Rendezvous Trajectory to Halley's Comet Using Deep-Space Low Thrust
    Roberto Flores, Alessandro Beolchi, Chiara Pozzi +4
    arXivPreprintCC BY 4.0Full textInstrumentation & methodsPlanetary science & exoplanetsarXiv

Cite

BibTeXRIS
BibTeX
@misc{quan2026cluster,
  title = {{On cluster deep loci in double Bruhat cells}},
  author = {Hanwen Quan},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27870},
  eprint = {2608.27870},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27870},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On cluster deep loci in double Bruhat cells
AU  - Hanwen Quan
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.27870
UR  - https://arxiv.org/abs/2608.27870
AB  - We study deep loci for the reduced-word Berenstein-Fomin-Zelevinsky cluster atlas on type-A double Bruhat cells. As defined in CGSS24, for a seed collection on a cluster variety X, the deep locus is the complement of the union of all corresponding cluster tori. We focus on double Bruhat cells G(u,v) for G = SL(n). Our main results are several theorems describing the structure of these deep loci, including their interaction with the Poisson structure, the relationships between the deep loci of different cells, and a tracking formula that allows us to compute the reduced-word deep locus of a cell from a lower-dimensional one. The paper also includes explicit low-rank calculations for double Bruhat cells in SL(3) and Borel double Bruhat cells in SL(4). Finally, we discuss the relationship between these results and some general properties of deep loci.
ER  -