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

By Jean-Louis Loday, Maria O. Ronco. 2001

141 citations recorded by the source corpus.

Let Xn 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≥0K[Xn]. We prove that it can be described explicitly by using the weak Bruhat order on Sn, the left-to-right order on planar trees, the lexicographic order in the cube case.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5c645693173fe38f51012853ca153b9cc4d25a68

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

On quadri-algebras (2014, background): “It is also suspected that the sub-quadri-algebra of FQSym generated by the permutation (12) is free.”
Sets with two associative operations (2003, methodology): “One observes that (see [5]) u ¢ v (resp.”
Realization of the Stasheff polytope (2002, background): “for instance [6]), and then forgetting the levels, one gets a well-defined map ψ : Sn → Yn.”
Combinatorial Hopf algebras and generalized Dehn–Sommerville relations (2003, methodology): “The Hopf algebra YSym has linear bases indexed by planar binary trees and was introduced by Loday and Ronco [LR98, LR02].”
Algebraic and combinatorial structures on pairs of twin binary trees (2012, methodology): “…we describe its product as an interval of this order Moreover, we prove that this Hopf algebra is free as an algebra by constructing two multiplicative bases, and introduce two operators on pairs of twin binary trees, analogous to the operators over and under of Loday–Ronco on binary trees [LR02].”
Comparison of Hopf algebras on trees (2003, background): “[ 11 ], we get an associative (graded) algebra.”
Combinatorial Hopf algebras and K-homology of Grassmanians (2007, unknown): “We extend the formula by linearity and continuity to give a multiplication ∗ : mMR⊗mMR → mMR. Define the standardization operator st : N∗ → Sm ∞ by sending a word w to the unique u ∈ Sm∞ of the same length (if it exists) such that wi ≤ wj if and only if ui ≤ uj for each 1 ≤ i, j ≤ (w).”