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The associative operad and the weak order on the symmetric groups

Marcelo Aguiar and Muriel Livernet

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 isa 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 simpleexplicit expression for Dynkin's idempotent in terms of the second basis.

There are combinatorial procedures for constructing a planar binary tree from apermutation, and a composition from a planar binary tree. These define set-theoreticquotients of each symmetric group algebra. We show that they are non-symmetric operad quotients of theassociative operad. Moreover, the Hopf kernels of these quotient maps are non-symmetric suboperadsof the associative operad.

Journal of Homotopy and Related Structures, Vol. 2(2007), No. 1, pp. 57-84