Lattices from graph associahedra and subalgebras of the Malvenuto-Reutenauer algebra

TL;DR

This paper compares two methods for constructing subalgebras of the Malvenuto-Reutenauer algebra, focusing on lattice quotients of the weak order and graph associahedra, and characterizes graphs related to these structures.

Contribution

It provides a detailed comparison of two constructions of subalgebras, linking lattice quotients of the weak order with graph associahedra within the Malvenuto-Reutenauer algebra.

Findings

Characterizes graphs whose maximal tubings form a lattice quotient of the weak order.

Establishes connections between lattice quotients and graph associahedra.

Provides criteria for when subalgebras include the Loday-Ronco algebra.

Abstract

The Malvenuto-Reutenauer algebra is a well-studied combinatorial Hopf algebra with a basis indexed by permutations. This algebra contains a wide variety of interesting sub Hopf algebras, in particular the Hopf algebra of plane binary trees introduced by Loday and Ronco. We compare two general constructions of subalgebras of the Malvenuto-Reutenauer algebra, both of which include the Loday-Ronco algebra. The first is a construction by Reading defined in terms of lattice quotients of the weak order, and the second is a construction by Ronco in terms of graph associahedra. To make this comparison, we consider a natural partial ordering on the maximal tubings of a graph and characterize those graphs for which this poset is a lattice quotient of the weak order.