Low-Dimensional Unitary Representations of B₃

1999/08/20 by Imre Tuba, Tuba, Imre
Mathematics · #16S34 (Secondary) #20C07 #20F36 #20H20 #81R10 (Primary) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT #msc:16S34 #msc:20C07 #msc:20F36 #msc:20H20 #msc:81R10

paper · pdf · doi:10.48550/arxiv.math/9908110

Added sections + some minor updates

arxiv created 1999/12/01 · arxiv updated 2009/11/30

Abstract

We characterize all simple unitarizable representations of the braid group B₃ on complex vector spaces of dimension d ≤ 5. In particular, we prove that if σ₁ and σ₂ denote the two generating twists of B₃, then a simple representation ρ:B₃ → \gl(V) (for dim V ≤ 5) is unitarizable if and only if the eigenvalues λ₁, λ₂, ..., λ_d of ρ(σ₁) are distinct, satisfy |λ_i|=1 and μ^(d)_1i > 0 for 2 ≤ i ≤ d, where the μ^(d)_1i are functions of the eigenvalues, explicitly described in this paper.

Related

• Representations of the braid group B₃ and of SL(2,Z)
• High-dimensional representations of the 3-component loop braid group
• Low-dimensional indecomposable representations of the braid group B₃
• Representations of the braid group B_n and the highest weight modules of U(sl_n-1) and U_q(sl_n-1)
• Irreducible representations of the braid group B₃ in dimension 6