Vertex Posets, Monotone Path Polytopes, and Chow Polynomials

Abstract: Let $P\subset\mathbb R<sup>n$ be a convex polytope and let $\ell$ be a linear functional which is nonconstant on every edge of $P$. The induced acyclic orientation determines positive and negative Białynicki-Birula type partitions of $P$ into unions of relative interiors of faces. Our first result establishes a duality: the positive partition is a stratification if and only if the negative one is a stratification. Our second result connects poset invariants with monotone path polytopes. Assuming the induced vertex relation admits the structure of a graded poset, we prove that the Chow polynomial of the resulting vertex poset agrees with the $h$-polynomial of a (dual) monotone path polytope.

• The paper demonstrates that BB-type partitions yield a dual stratification equivalence between O⁻ and O⁺ and introduces a graded vertex poset structure via the witness relation.
• It characterizes simplicity and the combinatorial structure of monotone path polytopes, showing that for simple polytopes, CH(P) is always simple and relates to products of simplices.
• The authors construct explicit poset kernels linking Chow polynomials with h-polynomials, offering new insights for toric varieties and algebraic invariants.

Vertex Posets, Monotone Path Polytopes, and Chow Polynomials

Introduction and Motivation

The paper "Vertex Posets, Monotone Path Polytopes, and Chow Polynomials" (2604.27515) investigates the interplay between combinatorics, polyhedral geometry, and algebraic geometry by studying poset structures induced by linear functionals on convex polytopes, monotone path polytopes (Chow quotients), and associated polynomial invariants. Its setting is that of a convex polytope $P\subset \mathbb R^n$ equipped with a generic linear functional $\ell$. The authors elucidate the structure of Biały-nicki-Birula-type partitions, monotone path polytopes, and their relation to the algebraic theory of Kazhdan–Lusztig–Stanley (KLS) polynomials and recent developments in the theory of Chow functions for posets.

The motivation is both combinatorial and geometric. In algebraic geometry, toric and related varieties with finite fixed points under $C^*$-action give rise, via moment maps, to polyhedral decompositions associated to the BB-partitions. The behavior of these decompositions (being stratifications or not) encodes subtle geometric and topological properties, such as simplicity of singularities in certain compactifications (notably in the wonderful models of matroids). Combinatorially, this points to connections among poset invariants, monotone path polytopes, and the geometry of convex polytopes.

BB-Partitions, Stratification, and Vertex Posets

Let $P$ be a convex polytope and $\ell$ a linear functional nonconstant on each edge. The $1$-skeleton of $P$ becomes an acyclically oriented graph by orienting each edge toward increasing $\ell$. For each vertex $v$, define:

• $O^-(v)$ as the union of relative interiors of faces where $\ell$0 attains its maximum at $\ell$1,
• $\ell$2 analogously for minima,
• $\ell$3 (resp. $\ell$4) as the closure of $\ell$5 (resp. $\ell$6).

The collection $\ell$7 (and similarly $\ell$8) partitions $\ell$9. BB-type partitions arise in the context of $C^*$0-actions on toric varieties: under the moment map, BB-attracting and repelling cells correspond precisely to $C^*$1 and $C^*$2, respectively.

The paper's first principal result is a duality theorem:

The partition $C^*$3 is a stratification if and only if the partition $C^*$4 is a stratification.

Here, stratification means that the closure of any stratum intersecting another is contained in the closure of the latter. This is shown, under genericity assumptions (notably, irreducibility: each $C^*$5 is a face), to be equivalent to several conditions: the so-called witness relation—for $C^*$6 vertices, there is a face on which $C^*$7 attains its minimum at $C^*$8 and maximum at $C^*$9—gives a graded poset structure if and only if stratification holds. The authors further give a complete catalogue of the implications and equivalences among various naturally induced vertex relations (chain order, Bruhat-type order, witness relation) and their interplay with the BB partitions.

Numerically, this analysis yields extensible descriptions of when the induced poset relations are graded, when the stratifications restrict well to all faces, and when the intersection structure can be characterized in terms of face incidence.

Monotone Path Polytopes and Their Simplicity

A core secondary object in the paper is the monotone path polytope $P$0 (the fiber polytope of $P$1 with respect to $P$2). Monotone path polytopes generalize well-studied examples such as the permutohedron and arise as moment polytopes of Chow quotients by the $P$3-action. The faces of $P$4 correspond to chains of faces of $P$5 (with certain compatibility conditions), which simplifies dramatically under the stratification (irreducibility and poset) assumptions.

The paper presents several strong results:

• When the stratification condition holds, the faces of $P$6 are described purely by monotone chains of faces indexed by the BB-type poset, and the poset structure is explicit.
• Simplicity of $P$7 is characterized entirely in terms of combinatorial data from $P$8: for every directed edge, the required number of triangle faces (adjacency) yields a necessary and sufficient criterion.
• For simple polytopes $P$9 where the BB stratification holds, $\ell$0 is always simple (i.e., its dual is simplicial), and the polytope $\ell$1 is necessarily a product of simplices.
• The paper also analyzes operations (products, pyramids) that preserve these stratification and simplicity properties.

Numerical implications are detailed for faces, vertices, and edges of monotone path polytopes, and the possible failures of stratification are tightly constrained—e.g., polygons with $\ell$2 sides never admit stratification for any generic $\ell$3.

Poset Invariants: KLS Functions and Chow Polynomials

A significant theoretical advance is the explicit construction of poset kernels for the vertex poset induced by $\ell$4 in the stratified regime and the identification of their associated KLS functions and Chow polynomials. Specifically:

• Under simplicity, the kernel for the poset is given by sums over faces (dimensions) between vertex intervals, matching the structure of the incidence algebra with polynomial weights.
• A major result is that the Chow polynomial of the vertex poset coincides with the $\ell$5-polynomial of the dual monotone path polytope, explicitly relating combinatorial topology of paths with algebraic-geometric invariants.

These polynomials exhibit positivity, palindromicity, and, in the Cohen–Macaulay case (e.g., products of simplices), $\ell$6-positivity. The kernel constructed sometimes coincides with the characteristic kernel (i.e., the Möbius function) but not always, and explicit examples in the paper demonstrate this dichotomy with detailed computations.

Implications and Future Directions

The paper’s results provide conceptual and computational bridges among several fields:

• Algebraic geometry: the results give combinatorial interpretations of intersection homology invariants in toric and related varieties, particularly via Chow quotients and BB-stratifications.
• Combinatorics and polyhedral theory: the explicit connections among poset kernels, $\ell$7-polynomials, and monotone path structures yield new tools to analyze polytopes with rich face lattices.
• Representation theory: the poset structure, together with the KLS polynomial theory, relates to Kazhdan–Lusztig theory and generalizations in type $\ell$8 and beyond.

Open directions include a systematic description of polytopes for which the poset kernel matches the characteristic kernel, new classes of polytopes realizing specified combinatorial invariants, and further connections to BB decompositions beyond the toric setting (e.g., for Grassmannians and flag varieties), as highlighted by parallels to ongoing work in log resolutions and wonderful models for matroids.

There are also algorithmic and computational implications: the explicit descriptions provide efficient criteria for checking stratification, computing $\ell$9- and Chow polynomials, and classifying simple polytopes via local data (e.g., triangle/quadrilateral faces).

Conclusion

This work establishes fundamental equivalences relating stratification of BB-type partitions of convex polytopes, monotone path polytopes, and the combinatorics of graded vertex posets. It advances the theory by constructing explicit poset kernels and showing, in the simple and stratified cases, that the Chow polynomial of the vertex poset equals the $1$0-polynomial of the dual monotone path polytope. These results deepen the connections among stratified polyhedral decompositions, algebraic–geometric quotients, and algebraic combinatorics, opening pathways for further advances in all these directions.