Lattice congruences, fans and Hopf algebras

Abstract

We give a unified explanation of the geometric and algebraic properties of two well-known maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak order on a Coxeter group we construct a complete fan of convex cones with strong properties relative to the corresponding lattice quotient of the weak order. We show that if a family of lattice congruences on the symmetric groups satisfies certain compatibility conditions then the family defines a sub Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations. Such a sub Hopf algebra has a basis which is described by a type of pattern avoidance. Applying these results, we build the Malvenuto-Reutenauer algebra as the limit of an infinite sequence of smaller algebras, where the second algebra in the sequence is the Hopf algebra of non-commutative symmetric functions. We also associate both a fan and a Hopf algebra to a set of permutations which appears to be equinumerous with the Baxter permutations.

arXiv comment34 pages, 1 figure. Version 2: Very belatedly updating the arXiv version to agree with the last pre-publication version

References in corpus (4)

• Lattice congruences of the weak order
• An analogue of the plactic monoid for binary search trees
• The Order Dimension of the Poset of Regions in a Hyperplane Arrangement
• The Algebra of Binary Search Trees

Cited by in corpus (44)

• Clusters, Coxeter-sortable elements and noncrossing partitions
• Cambrian Lattices
• Hopf Algebras in Combinatorics
• Cambrian fans
• Sortable elements and Cambrian lattices
• Lattice congruences of the weak order
• Algebraic and combinatorial structures on pairs of twin binary trees
• Cambrian Hopf Algebras
• Sortable elements in infinite Coxeter groups
• The Hopf algebra of diagonal rectangulations
• Lattice theory of torsion classes: Beyond $\tau$-tilting theory
• Combinatorial frameworks for cluster algebras
• Non-kissing complexes and tau-tilting for gentle algebras
• Noncrossing partitions and the shard intersection order
• Generic rectangulations
• Brick polytopes, lattice quotients, and Hopf algebras
• Quotientopes
• Polyhedral models for generalized associahedra via Coxeter elements
• The facial weak order and its lattice quotients
• Removahedral congruences versus permutree congruences
• Celebrating Loday's Associahedron
• Crosscut-simplicial Lattices
• Hopf algebras on decorated noncrossing arc diagrams
• Enumerative properties of Grid-Associahedra
• Shard polytopes
• A lattice of combinatorial Hopf algebras, Application to binary trees with multiplicities
• Acyclic reorientation lattices and their lattice quotients
• Combinatorial Realization of the Hopf Algebra of Sashes
• Linear compactness and combinatorial bialgebras
• Permutahedra and generalized associahedra
• Rectangulotopes
• The category of a partitioned fan
• Noncrossing arc diagrams and canonical join representations
• Hochschild polytopes
• Algebraic and combinatorial structures on Baxter permutations
• Operads in algebraic combinatorics
• Coxeter-biCatalan combinatorics
• Lattice homomorphisms between weak orders
• The Primitive Eulerian polynomial
• Biclosed sets in real hyperplane arrangements
• Geometric realizations of the $s$-weak order and its lattice quotients
• The Hopf algebra of generic rectangulations
• Coarsening polyhedral complexes
• Separating trees and simple congruences of the weak order