Low-dimensional representations of the three component loop braid group
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© 2015 AIP Publishing LLC. Motivated by physical and topological applications, we study representations of the group LB3 of motions of 3 unlinked oriented circles in R3. Our point of view is to regard the three strand braid group B3 as a subgroup of LB3 and study the problem of extending B3 representations. We introduce the notion of a standard extension and characterize B3 representations admitting such an extension. In particular we show, using a classification result of Tuba and Wenzl [Pacific J. Math. 197, 491-510 (2001)], that every irreducible B3 representation of dimension at most 5 has a (standard) extension. We show that this result is sharp by exhibiting an irreducible 6-dimensional B3 representation that has no extensions (standard or otherwise). We obtain complete classifications of (1) irreducible 2-dimensional LB3 representations, (2) extensions of irreducible 3-dimensional B3 representations, and (3) irreducible LB3 representations whose restriction to B3 has abelian image.
author list (cited authors)
Bruillard, P., Chang, L., Hong, S., Plavnik, J. Y., Rowell, E. C., & Sun, M. Y.