Braids and Permutations

E. Artin described all irreducible representations of the braid group B k to the symmetric group S(k). We strengthen some of his results and, moreover, exhibit a complete picture of homomorphisms ψ : B k → S(n) for n ≤ 2k. We show that the image Im ψ of ψ is cyclic whenever either (*) n < k = 4 or (* *) ψ is irreducible and 6 < k < n < 2k. For k > 6 there exist, up to conjugation, exactly 3 irreducible representations B k → S(2k) with non-cyclic images but they all are imprimitive. We use these results to prove that for n < k = 4 the image of any braid homomorphism ϕ : B k → B n is cyclic, whereas any endomorphism ϕ of B k with non-cyclic image preserves the pure braid group P B k. We prove also that for k > 4 the intersection P B k ∩ B ′ k of P B k with the commutator subgroup B ′ k = [B k , B k ] is a completely characteristic subgroup of B ′ k. Some results of this paper were obtained and the draft version [Lin96b] was written during my stay at the Max-Planck-Institut für Mathematik in Bonn in 1996. I am deeply grateful to MPI for hospitality.