A canonical realization of the alt $ν$-associahedron

Abstract: Given a lattice path $\nu$, the alt $\nu$-Tamari lattice is a partial order recently introduced by Ceballos and Chenevi`ere, which generalizes the $\nu$-Tamari lattice and the $\nu$-Dyck lattice. All these posets are defined on the set of lattice paths that lie weakly above $\nu$, and posses a rich combinatorial structure. In this paper, we study the geometric structure of these posets. We show that their Hasse diagram is the edge graph of a polytopal complex induced by a tropical hyperplane arrangement, which we call the alt $\nu$-associahedron. This generalizes the realization of $\nu$-associahedra by Ceballos, Padrol and Sarmiento. Our approach leads to an elegant construction, in terms of areas below lattice paths, which we call the canonical realization. Surprisingly, in the case of the classical associahedron, our canonical realization magically recovers Loday's ubiquitous realization, via a simple affine transformation.