Low-dimensional representations of the three component loop braid group

TL;DR

This paper investigates low-dimensional representations of the loop braid group $ ext{LB}_3$, focusing on extending braid group $ ext{B}_3$ representations and classifying irreducible representations of $ ext{LB}_3$.

Contribution

It introduces the concept of standard extensions of $ ext{B}_3$ representations and classifies low-dimensional irreducible $ ext{LB}_3$ representations, highlighting the limits of extendability.

Findings

Every irreducible $ ext{B}_3$ representation of dimension ≤ 5 admits a standard extension.

A 6-dimensional irreducible $ ext{B}_3$ representation has no extension.

Complete classifications of certain $ ext{LB}_3$ representations are provided.

Abstract

Motivated by physical and topological applications, we study representations of the group $\mathcal{LB}_3$ of motions of $3$ unlinked oriented circles in $\mathbb{R}^3$. Our point of view is to regard the three strand braid group $\mathcal{B}_3$ as a subgroup of $\mathcal{LB}_3$ and study the problem of extending $\mathcal{B}_3$ representations. We introduce the notion of a \emph{standard extension} and characterize $\mathcal{B}_3$ representations admiting such an extension. In particular we show, using a classification result of Tuba and Wenzl, that every irreducible $\mathcal{B}_3$ representation of dimension at most $5$ has a (standard) extension. We show that this result is sharp by exhibiting an irreducible $6$-dimensional $\mathcal{B}_3$ representation that has no extensions (standard or otherwise). We obtain complete classifications of (1) irreducible $2$-dimensional $\mathcal{LB}_3$ representations (2) extensions of irreducible $3$-dimensional $\mathcal{B}_3$ representations and (3) irreducible $\mathcal{LB}_3$ representations whose restriction to $\mathcal{B}_3$ has abelian image.