Introduction

The purpose of this work is to present an application of brick polyhedra of general subword complexes to produce geometric realizations of _ν _ν-associahedra.

There are several known connections between brick polytopes and generalizations of the associahedron. A main core for such connections is Knutson and Miller’s theory of subword complexes. Subword complexes are certain simplicial complexes motivated by the study of Gröbner geometry of Schubert varieties [1, 2]. One of the first connections between subword complexes and associahedra was discovered by Pilaud and Pocchiola in [3] using a slightly different terminology (of sorting networks), which was rediscovered using the subword complex terminology in [4]. A generalization for arbitrary finite Coxeter groups is due to Ceballos, Labbé and Stump in [5], who showed that c-cluster complexes arising in the theory of cluster algebras of finite type [6] can be obtained as well chosen subword complexes. The dual graph of the cluster complex, also known as the mutation graph for cluster algebras, is the edge graph of a well known polytope called the generalized associahedron [7, 8].

This last connection motivated the introduction of brick polytopes for spherical subword complexes by Pilaud and Stump [9], who generalized the notion of brick polytopes in type A by Pilaud and Santos in [10]. One of the main results in [9] provides a geometric realization of the generalized associahedron as the brick polytope of a spherical subword complex. Later on, Jahn and Stump presented a generalization of brick polyhedra for arbitrary subword complexes (not necessarily spherical) of finite type [11], who nicely connected them to the combinatorics and geometry of Bruhat intervals and Bruhat cones in Coxeter groups. Our work presents the first application of brick polyhedra to produce geometric realizations of _ν _ν-associahedra.

Fig. 1: Comparison of the νν-brick polyhedron and νν-associahedron for ν =ENEENν=ENEEN (top) and for ν =EENENν=EENEN (bottom)

Given a lattice path _ν _ν, consisting of finitely many north steps N and east steps E, the _ν _ν-associahedron [12] is a polytopal complex whose edge graph is the Hasse diagram of the _ν _ν-Tamari lattice introduced by Préville-Ratelle and Viennot in [13], and whose face poset is the poset of interior faces of the corresponding _ν _ν-Tamari complex [12] ordered by reverse inclusion. The case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =(NE)^n$$\end{document}ν=(NE)n recovers the classical associahedron, whose edge graph is the Hasse diagram of the classical Tamari lattice, which can be defined as the rotation poset of rooted binary trees and plays a fundamental role in many areas in mathematics, computer science and physics. The case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =(NE^m)^n$$\end{document}ν=(NEm)n recovers the m-Tamari lattices of Bergeron [14], whose interval enumeration has beautiful conjectural connections to the theory of trivariate diagonal harmonics in representation theory. Using techniques from tropical geometry, Ceballos, Sarmiento and Padrol produced the first geometric realizations of _ν _ν-associahedra [12], solving an open problem of Bergeron in this more general set up.

In this paper, we present a second geometric realization of the _ν _ν-associahedron as the complex of bounded faces of the brick polyhedron of a well chosen (non-spherical) subword complex. We also provide a suitable projection, in the special case where _ν _ν has non consecutive north steps, which provides a realization of the appropriate dimension, with a beautiful and elegant vertex-coordinate description. The brick polyhedron and the projection are illustrated for two examples in Fig. 1. Some 3-dimensional examples (resulting dimension after projecting) are illustrated in Figs. 21 and 22.

Brick polyhedra and _ν _ν-associahedra are central structures in their respective fields. The contributions in this paper do not only extend the notorious connection between brick polytopes related to type A cluster algebras and classical associahedra, but provides new insights for advances in both areas: Brick polyhedra: We provide the first known explicit combinatorial family of examples of brick polyhedra. The combinatorial understanding of _ν _ν-Tamari lattices and _ν _ν-associahedra is a powerful toolbox that provides intuition for a deeper understanding of the geometric structure of general brick polyhedra (see Sect. 5.1 and Proposition 6), motivating further questions for research in this area (see for instance Question 1).ν ν-associahedra and generalizations: Our description of _ν _ν-associahedra in terms of brick polyhedra opens new avenues of research in a much wider context, making this paper the starting point for various research directions. A very promising direction of research is related to the family of framing lattices arising from triangulations of flow polytopes [15]. This vast generalization captures many important lattices such as _ν _ν-Tamari lattices, type A Cambrian lattices, Grassmann- and grid Tamari lattices, the s-weak order and more, under the same umbrella. The connections in this paper motivate the introduction of framing polyhedra [16], which play the role of brick polyhedra for framing lattices, and relate Bruhat cones to normal cones of flow polytopes. We believe that framing polyhedra will be a fundamental tool in the study of framing lattices.