Splicing braid varieties

TL;DR

The paper develops a comprehensive splicing framework for braid varieties and their double Bott-Samelson analogues, providing open charts that decompose complex spaces into products of simpler braid varieties. It formulates conjectures linking these splicing loci to single-cluster seeds and proves a key product decomposition for both braid varieties and double Bott-Samelson varieties in several cases, including open Richardson varieties. The work connects to cluster algebras, Demazure combinatorics, and geometric representation theory, and situates splicing within link-homology and Lusztig-type decompositions, offering a unified approach to cluster structures on these spaces. The results pave the way for cluster-theoretic descriptions of open loci in braid-type varieties and their relatives, with implications for positivity, monomial behavior of minors, and connections to Khovanov–Rozansky theories.

Abstract

For a positive braid $β\in \mathrm{Br}^{+}_{k}$, we consider the braid variety $X(β)$. We define a family of open sets $\mathcal{U}_{r, w}$ in $X(β)$, where $w \in S_k$ is a permutation and $r$ is a positive integer no greater than the length of $β$. For fixed $r$, the sets $\mathcal{U}_{r, w}$ form an open cover of $X(β)$. We conjecture that $\mathcal{U}_{r,w}$ is given by the nonvanishing of some cluster variables in a single cluster for the cluster structure on $\mathbb{C}[X(β)]$ and that $\mathcal{U}_{r,w}$ admits a cluster structure given by freezing these variables. Moreover, we show that $\mathcal{U}_{r, w}$ is always isomorphic to the product of two braid varieties, and we conjecture that this isomorphism is quasi-cluster. In some important special cases, we are able to prove our conjectures.

Figures (2)

• Figure 1: The left-to-right inductive weave for the braid $\beta = \sigma_2\sigma_1\sigma_3\sigma_2\sigma_2\sigma_3\sigma_1\sigma_2\sigma_2\sigma_1\sigma_3\sigma_2$. Taking $r_1 = 9$, the part of the weave above the dotted line is an inductive weave for $\beta^1$, while the part of the weave below the dotted line is an inductive weave for $\underline{w_0}\beta^2$, and the braid varieties $X(\underline{w_0}\beta^2)$ and $X(\beta^2\underline{w_0})$ are cluster quasi-isomorphic.
• Figure 2: The monomial $m_{r_1+6}$ equals $\left(u^{(1)}_{3}u^{(1)}_5u^{(1)}_1\right)^{-1}$.

Theorems & Definitions (91)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Conjecture 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• Example 1.9
• Lemma 2.1