Representations of Braid Groups via Conjugation Actions on Congruence Subgroups

where α and β are nonzero complex numbers. If α, β lie in some subfield F of C, then the representations are defined over F (in fact, over Z[α±1, β±1]). The starting point for ρn(α) is the reduced Burau representation βn : Bn → GLn−1(Z[t, t −1]). For each i ≥ 1, set K(α) = {A ∈ SLn−1(C[t, t ]) : A ≡ I mod (t− α)}. The sequence {K(α)}i≥1 is a central series inK(α) = K 1(α) (i.e., Ki+j(α) ⊇ [K(α),K(α)]). Moreover, the graded quotients satisfy K(α)/K(α) ∼= sln−1(C). The conjugation action of GLn−1(C[t, t −1]) on K(α) induces a homomorphism fn(α) : GLn−1(C[t, t ]) −→ Aut(K(α)/K(α)) ∼= GLn(n−2)(C) (note that sln−1(C) is a vector space of dimension (n− 1) 2 − 1 = n(n− 2)). We define ρn(α) = fn(α) ◦ βn. The kernel of ρn(1) is easily described: it is the subgroup Pn of pure braids. The map ρn(1) is thus a representation of the symmetric group Σn.