A geometric and combinatorial exploration of Hochschild lattices

TL;DR

This paper investigates Hochschild lattices, revealing their geometric structures, shellability, and combinatorial properties, including enumeration and polynomial computations, thus deepening understanding of their mathematical nature.

Contribution

It provides the first geometric realizations of Hochschild lattices, proves their EL-shellability, and explores their combinatorial and enumerative properties.

Findings

Hochschild lattices admit cubic realizations.

They are EL-shellable and constructible by interval doubling.

The paper enumerates k-chains and computes degree polynomials.

Abstract

Hochschild lattices are specific intervals in the dexter meet-semilattices recently introduced by Chapoton. A natural geometric realization of these lattices leads to some cell complexes introduced by Saneblidze, called the Hochschild polytopes. We obtain several geometrical properties of the Hochschild lattices, namely we give cubic realizations, establish that these lattices are EL-shellable, and show that they are constructible by interval doubling. We also prove several combinatorial properties as the enumeration of their $k$-chains and compute their degree polynomials.