 
Summary: THE ASSOCIATIVE OPERAD AND THE WEAK ORDER ON THE
SYMMETRIC GROUPS
MARCELO AGUIAR AND MURIEL LIVERNET
Abstract. 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 M¨obius 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 permu
tation, and a composition from a planar binary tree. These define settheoretic quotients of
each symmetric group algebra. We show that they are nonsymmetric operad quotients of the
associative operad. Moreover, the Hopf kernels of these quotient maps are nonsymmetric
suboperads of the associative operad.
Introduction
One of the simplest symmetric operads is the associative operad As. This is an algebraic
structure carried by the sequence of vector spaces Asn = kSn, n 1, where Sn is the
