Low-dimensional unitary representations of 𝐵₃

Abstract

Abstract. We characterize all simple unitarizable representations of the braid group B 3 on complex vector spaces of dimension d ≤ 5. In particular, we prove that if σ 1 and σ 2 denote the two generating twists of B 3 , then a simple representation ρ : B 3 → GL(V ) (for dim V ≤ 5) is unitarizable if and only if the eigenvalues λ 1 , λ 2 , . . . , λ d of ρ(σ 1 ) are distinct, satisfy |λ i | = 1 and µ1i are functions of the eigenvalues, explicitly described in this paper.

Tri-bimaximal neutrino mixing, and the modular symmetry

“…At the leading order in 1/Λ and after the breaking of the flavour and electroweak symmetries, the mass terms from the Lagrangian (12) are…”

“…( 57) and, in the unidimensional case, there is a finite number of unitary representations. This is no-longer true when going to higher-dimensional representations [12]. We are particularly interested in the three-dimensional case which is discussed in detail in the appendix A.…”

The Lawrence-Krammer representation is a quantization of the symmetric square of the Burau representation

“…On the second step, we obtain the Humphry representation of B 3 in all dimension m + 1 as the m th symmetric power of the Burau representation ρ −1 3 , due to Theorem 2.5. Remarkably, that quantization and diagonal deformation (governed by the matrix Λ n = diag(λ r ) n r=0 , see Remark 2.2) of the Humphry representations (see (2.35)) include all irreducible representations of B 3 in dimension ≤ 5, due to results of Tuba and Wensel [28,29] and our equivalent description of their representations [1,17], see Theorem 2.7. Our approach using q-Pascal triangle allows us, for free, extend representation T 32 of B 3 for arbitrary dimension (see Theorem 2.6).…”

Hamiltonian monodromy via geometric quantization and theta functions

“…More details on the braid group and its relationship with the modular group P SL(2, Z) and, in particular, on related representations can be found e.g. in [23,32,33,6].…”