Stable decompositions for some symmetric group characters arising in braid group cohomology

We prove that certain permutation characters for the symmetric group Σ n decompose in a manner that is independent of n for large n. This result is a key ingredient in the recent work of T. Church and B. Farb, who obtain a “representation stability” theorem for the character of Σ n acting on the cohomology H p ( P n , C ) of the pure braid group P n .