Hopf dreams and diagonal harmonics

Nantel Bergeron, Cesar Ceballos, Vincent Pilaud*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 (Formula presented.) -trees related to (Formula presented.) -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.

Original languageEnglish
Pages (from-to)1546-1600
Number of pages55
JournalJournal of the London Mathematical Society
Issue number3
Publication statusPublished - Apr 2022

ASJC Scopus subject areas

  • Mathematics(all)

Fields of Expertise

  • Information, Communication & Computing


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