Subgroups of braid groups generated by Birman-Ko-Lee generators

TL;DR

This work introduces Young subgroups B_Q of the braid group B_n, generated by Birman-Ko-Lee generators a_{ij} within blocks of a partition Q, and provides an intrinsic description via Hurwitz action on tuples over a free group. It proves that B_Q equals the stabilizer of a structured tuple t in $F_m^n$ and gives a precise orbit description of the Hurwitz action on these tuples, along with a practical membership test and a constructive algorithm to express elements of B_Q in terms of the generators. The authors develop arc diagrams as a diagrammatic toolkit to translate reduction sequences and conjugate sequences into planar objects, analyze their behavior under half twists, and establish compatibility with the braid group action. The results have potential applications to mutations of exceptional collections and related braid-action problems, offering concrete criteria and algorithms for working with subgroups generated by BKL generators.

Abstract

We define a Young subgroup of the braid group as a subgroup generated by an arbitrary subset of the Birman-Ko-Lee generators. We give an intrinsic description of such subgroups which yields, in particular, an easy criterion to decide membership. We also give an algorithm to write an element of a Young subgroup as a product of the generators. Our methods are based on analyzing the Hurwitz action on tuples over free groups via a diagrammatic approach.

Figures (20)

• Figure 1.1: The Birman-Ko-Lee generators $a_{ij}$
• Figure 4.1:
• Figure 4.2: Mutations of reduction sequences
• Figure 4.3:
• Figure 4.4:

Theorems & Definitions (41)

• Remark 1.1
• Theorem 1.2: Theorem \ref{['th:mainth']}\ref{['it:mainth2']}
• Example 1.3
• Theorem 1.4: Theorem \ref{['th:mainth']}\ref{['it:mainth1']}
• Lemma 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Lemma 2.4
• Remark 2.5