Homomorphisms of commutator subgroups of braid groups

We give a complete classification of homomorphisms from the commutator subgroup of the braid group on n n strands to the braid group on n n strands when n n is at least 7. In particular, we show that each non-trivial homomorphism extends to an automorphism of the braid group on n n strands. This answers four questions of Vladimir Lin. Our main new tool is the theory of totally symmetric sets.