Permutation Flows I: Triangulations of Flow Polytopes (Research Announcement)

TL;DR

This work introduces permutation flows, a unifying combinatorial framework that lifts integer flows on framed graphs to encode unimodular triangulations of flow polytopes. It builds a rich dictionary among cliques, vines, groves, and permutation flows, and develops the permutation flow triangulation that extends the DKK construction, enabling a geometric proof of the Lidskii volume formula and linking the h*-polynomial to the G-Eulerian polynomial. The paper also establishes a lattice structure for the weak order on permutation flows and constructs an extensive framework of augmented-framed graphs, shuffles, and associated combinatorial objects that index simplices in the triangulation. Overall, it forges deep connections between polyhedral geometry, combinatorics of classical objects (permutations, Catalan structures, parking functions), and representation-theoretic aspects of flow polytopes, with Part II promising further developments.

Abstract

We introduce a new broadly unifying family of combinatorial objects, which we call permutation flows, associated to an acyclic directed graph $G$ together with a framing $F$. This new family is combinatorially rich and contains as special cases various families of combinatorial objects that are frequently studied in the literature, as is the case of permutations, circular permutations, multipermutations, Stirling permutations, Catalan objects and their generalizations. When permutation flows are decorated with compatible shuffles, they also include the combinatorics of parking functions and their generalizations. This model is geometrically rich. We show that permutation flow shuffles define a family of unimodular triangulations of the flow polytope $F_G(a)$ on $G$ with an integer balanced netflow vector a where only the last entry is negative. As an application we provide a new proof of the Lidskii volume formula of Baldoni and Vergne for this family of polytopes and a reformulation of the same formula where every term is explained by the nature of the combinatorial objects involved. Permutation flow triangulations extend the Danilov, Karzanov, and Koshevoy triangulations that were defined for the case where a=e_0-e_n. We provide a formula for the h^*-polynomial of the flow polytope as the descent enumerating polynomial of permutation flows. The model comes with an order structure induced by intuitive operators on permutation flows which we call the weak order. This order includes as special cases the weak order on permutations, the Tamari lattice, order ideals in Young's lattice, and their generalizations, among others. It was conjectured in 2020 by the three authors, together with Benedetti, Harris, and Morales, that this poset is in general a lattice. This conjecture has been recently established with independent proofs by Bell and Ceballos, and by Berggren and Serhiyenko.

Figures (40)

• Figure 1: The Pitman-Stanley polytope $\mathcal{F}_{\mathrm{PS}_3}(1,1,1,-3)$ and its integer flows.
• Figure 2: A framed graph $(G,F)$. The framing of the incoming and outgoing edges at each vertex is ordered pictorially from bottom to top.
• Figure 3: $t$ is a direct split of $e$ at $v$.
• Figure 4: Two permutation flows on the framed graph $(G,F)$ of Figure \ref{['fig:framed_graph']}. Note that occurrences of $x_0$ and $t_j$ are replaced by their indices for legibility.
• Figure 5:

Theorems & Definitions (156)

• Theorem 1.1
• Theorem 1.2: Reformulated generalized Lidskii formula
• Theorem 1.3
• Theorem 1.5
• Theorem 1.6
• Conjecture 1.7: Conjecture 7.6GonzalezHanusaMoralesYip2023
• Theorem 2.1: Generalized Lidskii formula BaldoniVergne2008MeszarosMorales2019
• Remark 2.2
• Theorem 2.3: Hille2003 Theorem 3.2
• Corollary 2.4: Zangwill1968Hille2003