A generalisation of the Burau representation and groups $G_{n}^{3}$ for classical braids

We consider a certain modification of the group $G^3_n$ which describes dynamics of point configurations, in particular braids, and define a representation of the modified $G^3_n$. The braid representation induced is powerful enough to detect the kernel of the Burau representation.

💡 Research Summary

The paper introduces a modified version of the previously defined groups G³ₙ, denoted ˆG³ₙ, and uses this modification to construct a new linear representation of the classical braid group Bₙ that is strictly stronger than the classical Burau representation.

The authors begin by recalling the Artin presentation of Bₙ with generators σ₁,…,σₙ₋₁ and the usual far‑commutativity and braid relations. The natural homomorphism ρ : Bₙ→Σₙ sending σᵢ to the transposition (i i+1) is introduced, and its kernel P Bₙ (the pure braid group) is identified.

Next, they define the group ˆG³ₙ. Its generators are symbols a_{ijk} indexed by three distinct integers i, j, k∈{1,…,n}. The defining relations are:

1. a_{kji}=a_{ijk}^{‑1} (inverse relation);
2. a_{ijk} commutes with a_{pqr} whenever the six indices i,j,k,p,q,r are all distinct;
3. a 4‑term relation a_{ijk}a_{ijl}a_{ikl}a_{jkl}=a_{jkl}a_{ikl}a_{ijl}a_{ijk} for any four distinct indices i,j,k,l.

These relations reflect the codimension‑1 events (three points becoming collinear) and codimension‑2 events (two such collinearities occurring simultaneously) in the configuration space of n points in the plane.

The symmetric group Σₙ acts on ˆG³ₙ by permuting the indices, which yields the semidirect product Σₙ⋉ˆG³ₙ. For each Artin generator σᵢ the authors define a map ϕₙ(σᵢ) = (ρ(σᵢ), Π_{m≠i,i+1} a_{m,i+1,i}), i.e. the permutation part is the usual transposition and the ˆG³ₙ‑part is the ordered product of all a_{m,i+1,i} with m ranging over the remaining strands. By checking the Artin relations directly (the “far‑commutativity” and the braid relation) they prove that ϕₙ extends to a homomorphism ϕₙ : Bₙ → Σₙ⋉ˆG³ₙ. Consequently, pure braids map into the subgroup {1}׈G³ₙ, which is naturally identified with ˆG³ₙ itself.

To obtain a linear representation, the authors consider the free module V = A⟨x_{ij} | i≠j, i,j∈{1,…,n}⟩ over the Laurent polynomial ring A = ℤ