Hopf dreams and diagonal harmonics

TL;DR

This paper constructs a new Hopf algebra structure on reduced pipe dreams, revealing connections to Catalan numbers, Tamari lattices, and diagonal harmonics, with implications for algebraic combinatorics.

Contribution

It introduces a novel Hopf algebra on pipe dreams, explores its subalgebras, and links these structures to Tamari lattices and diagonal harmonics, advancing combinatorial algebra theory.

Findings

Hopf algebra on reduced pipe dreams is free and cofree.

Contains subalgebras related to Catalan numbers and Tamari lattices.

Motivates new concepts like Hopf chains in Tamari lattices.

Abstract

This paper introduces a Hopf algebra structure on a family of reduced pipe dreams. We show that this Hopf algebra is free and cofree, and construct a surjection onto a commutative Hopf algebra of permutations. The pipe dream Hopf algebra contains Hopf subalgebras with interesting sets of generators and Hilbert series related to subsequences of Catalan numbers. Three other relevant Hopf subalgebras include the Loday-Ronco Hopf algebra on complete binary trees, a Hopf algebra related to a special family of lattice walks on the quarter plane, and a Hopf algebra on $\nu$-trees related to $\nu$-Tamari lattices. One of this Hopf subalgebras motivates a new notion of Hopf chains in the Tamari lattice, which are used to present applications and conjectures in the theory of multivariate diagonal harmonics.