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 Möbius 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. 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 symmetric group on n letters. In particular this entails structure maps, for each n,m ≥ 1 and 1 ≤ i ≤ n, Asn ⊗ Asm ◦i −→ Asn+m−1 satisfying certain axioms (Section 1). The Lie operad Lie is a symmetric suboperad of As. The space As∗ := ⊕ n≥1 kSn carries the structure of a (non-unital) graded Hopf algebra, first defined by Malvenuto and Reutenauer [15], and studied recently in a number of works, including [1, 7, 14]. It is known that Lie∗ := ⊕ n≥1 Lien sits inside the subspace of As∗ consisting of primitive elements for this Hopf algebra. This led us to consider whether this subspace is itself a suboperad of As. In the process to answering this question, we found a number of interesting results linking the non-symmetric operad structure of As to the combinatorics of the symmetric groups, and in particular to a partial order on Sn known as the left weak Bruhat order (or weak order, for simplicity). Date: October 17, 2006. 2000 Mathematics Subject Classification. Primary 18D50, 06A11; Secondary 06A07, 16W30.