Higher conjugation cohomology in commutative Hopf algebras

Let A be a graded, commutative Hopf algebra. We study an action of the symmetric group Sigma (n) on the tensor product of n - 1 copies of A; this action was introduced by the second author in [8] and is relevant to the study of commutativity conditions on ring spectra in stable homotopy theory [6]. We show that for a certain class of Hopf algebras the cohomology ring H*(Sigma (n); A(circle timesn-1)) is independent of the coproduct provided n and (n - 2)! are invertible in the ground ring. With the simplest coproduct structure, the group action becomes particularly tractable and we discuss the implications this has for computations.