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- W4301454175 abstract "In this paper, we exploit the combinatorics and geometry of triangulations of products of simplices to derive new results in the context of Catalan combinatorics of $nu$-Tamari lattices. In our framework, the main role of Catalan objects is played by $(I,overline{J})$-trees: bipartite trees associated to a pair $(I,overline{J})$ of finite index sets that stand in simple bijection with lattice paths weakly above a lattice path $nu=nu(I,overline{J})$. Such trees label the maximal simplices of a triangulation whose dual polyhedral complex gives a geometric realization of the $nu$-Tamari lattice introduced by Pr'evile-Ratelle and Viennot. In particular, we obtain geometric realizations of $m$-Tamari lattices as polyhedral subdivisions of associahedra induced by an arrangement of tropical hyperplanes, giving a positive answer to an open question of F.~Bergeron. The simplicial complex underlying our triangulation endows the $nu$-Tamari lattice with a full simplicial complex structure. It is a natural generalization of the classical simplicial associahedron, alternative to the rational associahedron of Armstrong, Rhoades and Williams, whose $h$-vector entries are given by a suitable generalization of the Narayana numbers. Our methods are amenable to cyclic symmetry, which we use to present type $B$ analogues of our constructions. Notably, we define a partial order that generalizes the type $B$ Tamari lattice, introduced independently by Thomas and Reading, along with corresponding geometric realizations." @default.
- W4301454175 created "2022-10-05" @default.
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- W4301454175 date "2016-11-29" @default.
- W4301454175 modified "2023-09-26" @default.
- W4301454175 title "Geometry of $nu$-Tamari lattices in types $A$ and $B$" @default.
- W4301454175 doi "https://doi.org/10.48550/arxiv.1611.09794" @default.
- W4301454175 hasPublicationYear "2016" @default.
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