Hochschild lattices and shuffle lattices

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In his study of a Hochschild complex arising in connection with the free loop fibration, S. Saneblidze defined the freehedron, a certain polytope constructed via a truncation process from the hypercube. It was recently conjectured by F. Chapoton and proven by C. Combe that a certain orientation of the 1-skeleton of the freehedron carries a lattice structure. The resulting lattice was dubbed the Hochschild lattice and it is interval constructable and extremal. These properties allow for the definition of three associated structures: the Galois graph, the canonical join complex and the core label order. In this article, we study and characterize these structures. We exhibit an isomorphism from the core label order of the Hochschild lattice to a particular shuffle lattice of C. Greene. We also uncover an enumerative connection between the core label order of the Hochschild lattice, a certain order extension of its poset of irreducibles and the freehedron. These connections nicely parallel the situation surrounding the better-known Tamari lattices, noncrossing partition lattices and associahedra.


Original languageEnglish
Article number103521
Number of pages31
JournalEuropean Journal of Combinatorics
Publication statusPublished - Jun 2022

External IDs

unpaywall 10.1016/j.ejc.2022.103521
WOS 000784071300006


DFG Classification of Subject Areas according to Review Boards


  • Complexes, Model

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