Framing Lattices and Flow Polytopes

TL;DR

The paper introduces framing lattices as a unifying lattice framework for the dual-graph structure of framing triangulations of flow polytopes, showing that the framing poset on maximal cliques forms an $ ext{HH}$-lattice with polygonal faces, enabling semidistributivity and congruence uniformity. Central to the theory are ccw and cw rotations of maximal cliques, the $C_{ ext{max}}/C_{ ext{min}}$ algorithms, and the core label order, which together generalize noncrossing partitions and connect join-irreducibles to ccw-extended paths and meet-irreducibles to cw-extended paths. The authors develop lattice congruences and quotients via $M$-moves, establishing that quotients correspond to framing lattices of modified graphs, and identify a distributive quotient $\mathscr{L}_{M(G)}$ independent of framing. The work then surveys a rich zoo of framing lattices, including Boolean, Tamari, Cambrian, Grassmann–Tamari, Grid-Tamari, cross-Tamari, and multipermutation lattices, all interpretable as framing lattices from coherent routes in framed graphs. This framework not only unifies known lattices but also offers a geometric interpretation through framing topes and core-label analysis, with open questions about broader quotients and computational aspects of the rotation graph. The results provide a versatile, combinatorial toolkit for linking flow-polytopal triangulations, noncrossing-partition theory, and a wide spectrum of lattice families.

Abstract

Flow polytopes of acyclic oriented graphs arise naturally in combinatorial optimization, and the study of their volumes and triangulations has revealed intriguing connections across combinatorics, geometry, algebra, and representation theory. In this work, we introduce the framing lattice associated with a framed graph, whose Hasse diagram is dual to a framed triangulation of the corresponding flow polytope. Framing lattices are remarkable in that they provide a unifying framework encompassing many classical and well-studied lattice structures, including the Boolean lattice, the Tamari lattice, and the weak order on permutations. They further subsume a broad array of examples such as all type-A Cambrian lattices, the Grassmann and grid-Tamari lattices, the alt-$ν$-Tamari and cross-Tamari lattices, the permutree lattices, and the $τ$-tilting posets of certain gentle algebras. We show, among several foundational structural properties, that the framing lattice is a semidistributive, congruence uniform, and polygonal lattice, with its polygons consisting of squares, pentagons, and hexagons. We study its connections to noncrossing partitions via Reading's core label orders, simple representations of its join and meet irreducible elements, and several of its lattice congruences and quotients induced by a graph operation called an M-move.

Figures (52)

• Figure 1: Four framed graphs and the Hasse diagrams of their framing lattices. The first is the Boolean lattice $\mathscr{B}_3$. The second is the lattice of multipermutations of $1^22^23$. The third is the $\varepsilon$-cambrian lattice with $\varepsilon = (-,-,+,-)$. The fourth is a cross-Tamari lattice of the cross-shaped grid shown below the right-most graph.
• Figure 2: Some popular exhibitions at the zoo of framing lattices.
• Figure 3: Some examples of the oruga graph and their flow polytopes.
• Figure 4: Two framings of the $G_2=\mathrm{oru}(2)$ graph and the framing triangulations of the corresponding flow polytope $\mathcal{F}_{G_2}$.
• Figure 5: The relation $\leq_{\mathscr{I}(v)}$ is a partial order on incoming paths to $v$ only if they all begin at the source.

Theorems & Definitions (144)

• Theorem A
• Theorem B
• Theorem C
• Example 1.1.1: The oruga graph and the cube
• Lemma 1.1.2
• proof
• Proposition 1.1.3: Danilov et al. DKK12
• Corollary 1.1.4
• Corollary 1.1.5
• Lemma 1.1.6