The necklace braid group $${\mathcal {NB}}_n$$ NB n is the motion group of the $$n+1$$ n + 1 component necklace link $$\mathcal {L}_n$$ L n in Euclidean $$\mathbb {R}^3$$ R 3 . Here $$\mathcal {L}_n$$ L n consists of n pairwise unlinked Euclidean circles each linked to an auxiliary circle. Partially motivated by physical considerations, we study representations of the necklace braid group $${\mathcal {NB}}_n$$ NB n , especially those obtained as extensions of representations of the braid group $$\mathcal {B}_n$$ B n and the loop braid group $${\mathcal {LB}}_n$$ LB n . We show that any irreducible $$\mathcal {B}_n$$ B n representation extends to $${\mathcal {NB}}_n$$ NB n in a standard way. We also find some non-standard extensions of several well-known $$\mathcal {B}_n$$ B n -representations such as the Burau and LKB representations. Moreover, we prove that any local representation of $$\mathcal {B}_n$$ B n (i.e., coming from a braided vector space) can be extended to $${\mathcal {NB}}_n$$ NB n , in contrast to the situation with $${\mathcal {LB}}_n$$ LB n . We also discuss some directions for future study from categorical and physical perspectives.

中文翻译：

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项链编织组的表示：拓扑和组合方法

项链编织群 $${\mathcal {NB}}_n$$ NB n 是 $$n+1$$ n + 1 分量项链链 $$\mathcal {L}_n$$ L n 的运动群欧几里得 $$\mathbb {R}^3$$ R 3 。这里 $$\mathcal {L}_n$$ L n 由 n 个成对未链接的欧几里得圆组成，每个欧几里得圆都链接到一个辅助圆。部分出于物理考虑，我们研究了项链编织群 $${\mathcal {NB}}_n$$ NB n 的表示，特别是那些作为编织群 $$\mathcal {B}_n$$ 表示的扩展获得的B n 和环编织群 $${\mathcal {LB}}_n$$ LB n 。我们证明任何不可约的 $$\mathcal {B}_n$$ B n 表示都以标准方式扩展到 $${\mathcal {NB}}_n$$ NB n。我们还发现了一些著名的 $$\mathcal {B}_n$$ B n -表示的一些非标准扩展，例如 Burau 和 LKB 表示。而且，我们证明 $$\mathcal {B}_n$$ B n 的任何局部表示（即来自编织向量空间）可以扩展到 $${\mathcal {NB}}_n$$ NB n ，相反到 $${\mathcal {LB}}_n$$ LB n 的情况。我们还从分类和物理角度讨论了未来研究的一些方向。