Hopf dreams and diagonal harmonics

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.

44 pages, 29 figures; Version 4: minor changes for final version

References in corpus (5)

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• Homology of torus knots
• Catalan numbers and Schubert polynomials for $w=1(n+1)...2$
• Compositional (km,kn)-Shuffle Conjectures
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Cited by in corpus (5)

• The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials
• A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions
• The Steep-Bounce Zeta Map in Parabolic Cataland
• Bijective link between Chapoton's new intervals and bipartite planar maps
• The $s$-weak order and $s$-permutahedra I: combinatorics and lattice structure