Séminaire Lotharingien de Combinatoire, 82B.19 (2019), 12 pp.

Nantel Bergeron, Cesar Ceballos, and Vincent Pilaud

Hopf dreams

Abstract. We introduce a Hopf algebra structure on a family of reduced pipe dreams with a natural surjection onto a commutative Hopf algebra of permutations. We then study three Hopf subalgebras of permutations whose preimages by the surjection yield three relevant Hopf subalgebras of pipe dreams. The first is the Loday-Ronco Hopf algebra on binary trees, the second is related to a special family of lattice walks on the quarter plane, and the third is a Hopf algebra on ν-trees related to ν-Tamari lattices. The latter motivates a new notion of Hopf chains in the Tamari lattice with applications in the theory of multivariate diagonal harmonics.


Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

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