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 non-symmetric operad quotients of the associative operad. Moreover, the Hopf kernels of these quotient maps are non-symmetric suboperads of the associative operad.