Order Structure on the Algebra of Permutations and of Planar Binary Trees

Abstract

Let X n be either the symmetric group on n letters, the set of planar binary n-trees or the set of vertices of the (n − 1)-dimensional cube. In each case there exists a graded associative product on ⊕ n≥0 K[X n]. We prove that it can be described explicitly by using the weak Bruhat order on S n, the left-to-right order on planar trees, the lexicographic order in the cube case.