The associative operad and the weak order on the symmetric groups

The associative operad is a certain algebraic structure on the sequence of group algebras of the symmetric groups. The weak order is a partial order on the symmetric group. There is a natural linear basis of each symmetric group algebra, related to the group basis by Mobius inversion for the weak order. We describe the operad structure on this second basis: the surprising result is that each operadic composition is a sum over an interval of the weak order. We deduce that the coradical filtration is an operad filtration. The Lie operad, a suboperad of the associative operad, sits in the first component of the filtration. As a corollary to our results, we derive a simple explicit expression for Dynkin's idempotent in terms of the second basis. There are combinatorial procedures for constructing a planar binary tree from a permutation, and a composition from a planar binary tree. These define set-theoretic quotients of each symmetric group algebra. We show that they are operad quotients of the associative operad. Moreover, the Hopf kernels of these quotient maps are suboperads of the associative operad.