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Mutation Sequences along Weaves and Amalgamation of Braid Varieties

  • Yuma Mizuno
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Abstract

Let p,q be positive braids with Demazure products u,v. The endpoint stratum Conf(p,q)u,v of the double Bott-Samelson cell splits as Conf(u,v) × X(p) × X(qop), a double Bruhat cell times two braid varieties. We prove that the stratum's cluster structure is given by a cluster localization of the one on Conf(p,q), and that the splitting map is a quasi-cluster isomorphism. This comes from a mutation sequence along a double Demazure weave, one mutation per trivalent vertex, ending at an amalgamation of an extension of the double word quiver with the weave quiver. In the half-decorated case, this proves the conjecture of Gorsky-Kim-Scroggin-Simental for the splicing map X(p) × X(ΔΔ) → X(pΔ), where Δ is a reduced positive braid for w0 and dem(p) = w0.

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Cite

BibTeXRIS
BibTeX
@misc{mizuno2026mutation,
  title = {{Mutation Sequences along Weaves and Amalgamation of Braid Varieties}},
  author = {Yuma Mizuno},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05833},
  eprint = {2609.05833},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05833},
}
RIS
TY  - UNPB
TI  - Mutation Sequences along Weaves and Amalgamation of Braid Varieties
AU  - Yuma Mizuno
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.05833
UR  - https://arxiv.org/abs/2609.05833
AB  - Let p,q be positive braids with Demazure products u,v. The endpoint stratum Conf(p,q)_u,v of the double Bott-Samelson cell splits as Conf(u,v) × X(p) × X(q^op), a double Bruhat cell times two braid varieties. We prove that the stratum's cluster structure is given by a cluster localization of the one on Conf(p,q), and that the splitting map is a quasi-cluster isomorphism. This comes from a mutation sequence along a double Demazure weave, one mutation per trivalent vertex, ending at an amalgamation of an extension of the double word quiver with the weave quiver. In the half-decorated case, this proves the conjecture of Gorsky-Kim-Scroggin-Simental for the splicing map X(p) × X(ΔΔ) → X(pΔ), where Δ is a reduced positive braid for w_0 and dem(p) = w_0.
ER  -