Summary: THE ASSOCIATIVE OPERAD AND THE WEAK ORDER ON THE
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 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.
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