2005/11/29 by Marcelo Aguiar, Aguiar, Marcelo, Muriel Livernet +1 · 1 citation
Mathematics · #06A11 #18D50 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #math.CO #math.QA #msc:06A11 #msc:18D50
paper · pdf · doi:10.48550/arxiv.math/0511698
Precision concerning the use of non-symmetric and symmetric operad
openalex publication_date 2005/11/29 · arxiv created 2006/10/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 operad quotients of the associative operad. Moreover, the Hopf kernels of these quotient maps are suboperads of the associative operad.