Marked Order Polytope: Structure and Applications

• Marked order polytopes are convex sets formed by order‐preserving extensions on finite posets, generalizing Stanley’s order polytope with prescribed markings.
• They integrate combinatorial structures with geometric and algebraic properties, as evidenced by connections to Ehrhart theory, toric geometry, and polyhedral subdivisions.
• Their study unveils clear facet structures, bijections with marked chain polytopes, and links to representation theory via Gelfand–Tsetlin and related polytopes.

A marked order polytope is the convex set of order-preserving extensions of a prescribed marking on a distinguished subposet. For a finite poset $P$, a marked subset $A\subseteq P$ containing the extremal elements, and an order-preserving map $\lambda:A\to \mathbb{R}$, it can be realized either as

$$\mathcal{O}_{P,A}(\lambda)=\{\hat\lambda:P\to\mathbb{R}\ \text{order preserving and}\ \hat\lambda(a)=\lambda(a)\ \forall a\in A\}$$

or, after projecting away the marked coordinates, by the standard system of order inequalities together with lower and upper bounds induced by the marks. In this form it generalizes Stanley’s order polytope, while also encompassing Gelfand–Tsetlin polytopes and other representation-theoretic families, and interfacing with Ehrhart theory, marked chain and chain–order polytopes, flow polytopes, root polytopes, and toric geometry (Jochemko et al., 2012, Ardila et al., 2010, Rietsch et al., 2024).

1. Definitions and fundamental formulations

A marked poset consists of a finite poset $P$, a marked subset $A\subseteq P$ containing at least all minimal and maximal elements, and an order-preserving marking $\lambda\in L_A$, where

$$L_A := \{\lambda \in \mathbb{Z}^A \mid \lambda_a \le \lambda_b \text{ whenever } a \le b\}.$$

Writing coordinates only on the unmarked part $P\setminus A$, the marked order polytope is

$$O(P,A,\lambda)=\left\{ s \in \mathbb{R}^{P\setminus A} \,\middle|\, \begin{array}{l} s_x \le s_y \text{ for all } x<y \text{ in } P\setminus A,\ \lambda_a \le s_x \text{ for all } a\in A,\ x\in P\setminus A \text{ with } a\le x,\ s_x \le \lambda_b \text{ for all } x\in P\setminus A,\ b\in A \text{ with } x\le b \end{array} \right\}.$$

Equivalently, for any triple $A\subseteq P$0 with $A\subseteq P$1 and $A\subseteq P$2, one has $A\subseteq P$3 (Fourier, 2014).

The same object is often viewed in the full space $A\subseteq P$4 by fixing $A\subseteq P$5 for $A\subseteq P$6 and imposing $A\subseteq P$7 whenever $A\subseteq P$8. In this formulation, the marked order polyhedron is

$A\subseteq P$9

The term “marked order polytope” is used in the bounded case; if $\lambda:A\to \mathbb{R}$0 contains all minimal and maximal elements, boundedness is automatic, whereas $\lambda:A\to \mathbb{R}$1 yields the order cone $\lambda:A\to \mathbb{R}$2, which is unbounded (Pegel, 2016, Jochemko et al., 2012).

Classical order polytopes arise as the special case in which extremal marked elements are fixed to $\lambda:A\to \mathbb{R}$3 and $\lambda:A\to \mathbb{R}$4. This specialization recovers Stanley’s inequalities $\lambda:A\to \mathbb{R}$5 and $\lambda:A\to \mathbb{R}$6 for $\lambda:A\to \mathbb{R}$7 (Ardila et al., 2010, Pegel, 2016).

Two basic model cases are especially transparent. For a chain $\lambda:A\to \mathbb{R}$8 with $\lambda:A\to \mathbb{R}$9,

$$\mathcal{O}_{P,A}(\lambda)=\{\hat\lambda:P\to\mathbb{R}\ \text{order preserving and}\ \hat\lambda(a)=\lambda(a)\ \forall a\in A\}$$0

so the marked order polytope is a simplex slice. For the diamond $$\mathcal{O}_{P,A}(\lambda)=\{\hat\lambda:P\to\mathbb{R}\ \text{order preserving and}\ \hat\lambda(a)=\lambda(a)\ \forall a\in A\}$$1 and $$\mathcal{O}_{P,A}(\lambda)=\{\hat\lambda:P\to\mathbb{R}\ \text{order preserving and}\ \hat\lambda(a)=\lambda(a)\ \forall a\in A\}$$2, one obtains

$$\mathcal{O}_{P,A}(\lambda)=\{\hat\lambda:P\to\mathbb{R}\ \text{order preserving and}\ \hat\lambda(a)=\lambda(a)\ \forall a\in A\}$$3

a rectangle (Fang et al., 2015).

2. Face structure, facets, and geometric criteria

The face structure of a marked order polyhedron admits a combinatorial description in terms of partitions of the underlying poset. For $$\mathcal{O}_{P,A}(\lambda)=\{\hat\lambda:P\to\mathbb{R}\ \text{order preserving and}\ \hat\lambda(a)=\lambda(a)\ \forall a\in A\}$$4, let $$\mathcal{O}_{P,A}(\lambda)=\{\hat\lambda:P\to\mathbb{R}\ \text{order preserving and}\ \hat\lambda(a)=\lambda(a)\ \forall a\in A\}$$5 be the partition obtained from the transitive closure of the relation $$\mathcal{O}_{P,A}(\lambda)=\{\hat\lambda:P\to\mathbb{R}\ \text{order preserving and}\ \hat\lambda(a)=\lambda(a)\ \forall a\in A\}$$6 when $$\mathcal{O}_{P,A}(\lambda)=\{\hat\lambda:P\to\mathbb{R}\ \text{order preserving and}\ \hat\lambda(a)=\lambda(a)\ \forall a\in A\}$$7 and $$\mathcal{O}_{P,A}(\lambda)=\{\hat\lambda:P\to\mathbb{R}\ \text{order preserving and}\ \hat\lambda(a)=\lambda(a)\ \forall a\in A\}$$8 are comparable and $$\mathcal{O}_{P,A}(\lambda)=\{\hat\lambda:P\to\mathbb{R}\ \text{order preserving and}\ \hat\lambda(a)=\lambda(a)\ \forall a\in A\}$$9. A block is free if it contains no marked element. If $P$0 is the minimal face containing $P$1, then

$P$2

where $P$3 denotes the free blocks (Pegel, 2016).

A partition $P$4 arises from a face if and only if it is connected, $P$5-compatible, and the induced marking on the quotient marked poset $P$6 is strict. Face inclusion is refinement of partitions, so the face lattice is identified with the poset of face partitions ordered by refinement (Pegel, 2016).

The recession cone is

$P$7

Hence $P$8 is bounded if and only if every $P$9 lies in some marked interval $A\subseteq P$0, equivalently, if for every $A\subseteq P$1 there exist $A\subseteq P$2 with $A\subseteq P$3 (Pegel, 2016).

Facet descriptions are especially clean after removing redundant cover inequalities. For regular marked posets, facets of $A\subseteq P$4 are in bijection with covering relations $A\subseteq P$5, and the facet-defining inequalities are exactly $A\subseteq P$6; the equations $A\subseteq P$7 cut out the affine hull (Pegel, 2016). In the strict regular case, the same facet-supporting hyperplanes can be written as

$A\subseteq P$8

together with

$A\subseteq P$9

(Stricker, 2024).

A sharp geometric criterion is available for 2-levelness. If $\lambda\in L_A$0 is strict and irredundant, then the following are equivalent: $\lambda\in L_A$1 is 2-level; each connected component of the Hasse diagram of $\lambda\in L_A$2 has exactly one maximal marked element and exactly one minimal marked element; and $\lambda\in L_A$3 is affinely isomorphic to an order polytope (Stricker, 2024). Thus marked order polytopes are not 2-level in general, even though ordinary order polytopes always are.

Disjoint unions behave multiplicatively: $\lambda\in L_A$4 and there is also a weighted Minkowski sum decomposition obtained from threshold $\lambda\in L_A$5–$\lambda\in L_A$6 markings on the marked set (Pegel, 2016, Fang et al., 2018).

3. Enumeration: arithmetic, reciprocity, Ehrhart theory, and face numbers

For integral markings, the lattice-point enumerator

$\lambda\in L_A$7

counts integer-valued order-preserving extensions of $\lambda\in L_A$8. This function is piecewise polynomial over the order cone

$\lambda\in L_A$9

with chambers determined by the relative order of the marked values. On each chamber, $$L_A := \{\lambda \in \mathbb{Z}^A \mid \lambda_a \le \lambda_b \text{ whenever } a \le b\}.$$0 agrees with a polynomial whose degree is $$L_A := \{\lambda \in \mathbb{Z}^A \mid \lambda_a \le \lambda_b \text{ whenever } a \le b\}.$$1 (Jochemko et al., 2012, Jochemko et al., 9 Apr 2026).

The chamberwise structure is explicit. For a compatible chain of order ideals $$L_A := \{\lambda \in \mathbb{Z}^A \mid \lambda_a \le \lambda_b \text{ whenever } a \le b\}.$$2, the corresponding cell is a product of simplices, and its lattice-point count factors as

$$L_A := \{\lambda \in \mathbb{Z}^A \mid \lambda_a \le \lambda_b \text{ whenever } a \le b\}.$$3

where the $$L_A := \{\lambda \in \mathbb{Z}^A \mid \lambda_a \le \lambda_b \text{ whenever } a \le b\}.$$4 record the numbers of elements between consecutive marked layers (Jochemko et al., 2012). Ehrhart–Macdonald reciprocity then yields

$$L_A := \{\lambda \in \mathbb{Z}^A \mid \lambda_a \le \lambda_b \text{ whenever } a \le b\}.$$5

as the number of strict order-preserving extensions of $$L_A := \{\lambda \in \mathbb{Z}^A \mid \lambda_a \le \lambda_b \text{ whenever } a \le b\}.$$6 (Jochemko et al., 2012).

A more explicit Ehrhart formula for $$L_A := \{\lambda \in \mathbb{Z}^A \mid \lambda_a \le \lambda_b \text{ whenever } a \le b\}.$$7 is available in terms of linear extensions. For strict and irredundant $$L_A := \{\lambda \in \mathbb{Z}^A \mid \lambda_a \le \lambda_b \text{ whenever } a \le b\}.$$8,

$$L_A := \{\lambda \in \mathbb{Z}^A \mid \lambda_a \le \lambda_b \text{ whenever } a \le b\}.$$9

where $P\setminus A$0 is the number of unmarked elements on the interval, $P\setminus A$1 is the number of descents on that interval, and $P\setminus A$2 (Stricker, 2024). The same paper shows that this is also the Ehrhart polynomial of the associated marked chain and marked chain–order polytopes.

A recent refinement is multivariate Ehrhart positivity. If a family of posets is closed under ideals and filters and every order polynomial in the family has nonnegative linear term, then on each chamber the counting function $P\setminus A$3 is a polynomial in the differences

$P\setminus A$4

with nonnegative coefficients. This gives Ehrhart positivity for marked order polytopes of skew shapes and proves conjectures on skew Gelfand–Tsetlin polytopes and $P\setminus A$5-generalized Pitman–Stanley polytopes (Jochemko et al., 9 Apr 2026).

Face numbers can also be recovered cohomologically. Given any polyhedral subdivision $P\setminus A$6 of a convex polytope $P\setminus A$7, one constructs a cochain complex over $P\setminus A$8,

$P\setminus A$9

whose cohomology dimensions satisfy

$$O(P,A,\lambda)=\left\{ s \in \mathbb{R}^{P\setminus A} \,\middle|\, \begin{array}{l} s_x \le s_y \text{ for all } x<y \text{ in } P\setminus A,\ \lambda_a \le s_x \text{ for all } a\in A,\ x\in P\setminus A \text{ with } a\le x,\ s_x \le \lambda_b \text{ for all } x\in P\setminus A,\ b\in A \text{ with } x\le b \end{array} \right\}.$$0

For a marked order polytope, the relevant subdivision is the cubosimplicial subdivision indexed by $$O(P,A,\lambda)=\left\{ s \in \mathbb{R}^{P\setminus A} \,\middle|\, \begin{array}{l} s_x \le s_y \text{ for all } x<y \text{ in } P\setminus A,\ \lambda_a \le s_x \text{ for all } a\in A,\ x\in P\setminus A \text{ with } a\le x,\ s_x \le \lambda_b \text{ for all } x\in P\setminus A,\ b\in A \text{ with } x\le b \end{array} \right\}.$$1-admissible chains of order ideals; in this setting the complex admits a purely combinatorial description, yielding a direct computation of the $$O(P,A,\lambda)=\left\{ s \in \mathbb{R}^{P\setminus A} \,\middle|\, \begin{array}{l} s_x \le s_y \text{ for all } x<y \text{ in } P\setminus A,\ \lambda_a \le s_x \text{ for all } a\in A,\ x\in P\setminus A \text{ with } a\le x,\ s_x \le \lambda_b \text{ for all } x\in P\setminus A,\ b\in A \text{ with } x\le b \end{array} \right\}.$$2-vector (Melikhova, 18 Jul 2025).

4. Marked chain polytopes, chain–order interpolations, and equivalence questions

The marked chain polytope of $$O(P,A,\lambda)=\left\{ s \in \mathbb{R}^{P\setminus A} \,\middle|\, \begin{array}{l} s_x \le s_y \text{ for all } x<y \text{ in } P\setminus A,\ \lambda_a \le s_x \text{ for all } a\in A,\ x\in P\setminus A \text{ with } a\le x,\ s_x \le \lambda_b \text{ for all } x\in P\setminus A,\ b\in A \text{ with } x\le b \end{array} \right\}.$$3 replaces order inequalities by nonnegativity and chain-sum constraints: $$O(P,A,\lambda)=\left\{ s \in \mathbb{R}^{P\setminus A} \,\middle|\, \begin{array}{l} s_x \le s_y \text{ for all } x<y \text{ in } P\setminus A,\ \lambda_a \le s_x \text{ for all } a\in A,\ x\in P\setminus A \text{ with } a\le x,\ s_x \le \lambda_b \text{ for all } x\in P\setminus A,\ b\in A \text{ with } x\le b \end{array} \right\}.$$4 Ardila–Bliem–Salazar constructed a piecewise-affine bijection

$$O(P,A,\lambda)=\left\{ s \in \mathbb{R}^{P\setminus A} \,\middle|\, \begin{array}{l} s_x \le s_y \text{ for all } x<y \text{ in } P\setminus A,\ \lambda_a \le s_x \text{ for all } a\in A,\ x\in P\setminus A \text{ with } a\le x,\ s_x \le \lambda_b \text{ for all } x\in P\setminus A,\ b\in A \text{ with } x\le b \end{array} \right\}.$$5

with coordinate formula

$$O(P,A,\lambda)=\left\{ s \in \mathbb{R}^{P\setminus A} \,\middle|\, \begin{array}{l} s_x \le s_y \text{ for all } x<y \text{ in } P\setminus A,\ \lambda_a \le s_x \text{ for all } a\in A,\ x\in P\setminus A \text{ with } a\le x,\ s_x \le \lambda_b \text{ for all } x\in P\setminus A,\ b\in A \text{ with } x\le b \end{array} \right\}.$$6

and an explicit inverse defined recursively by maxima along lower covers. For every $$O(P,A,\lambda)=\left\{ s \in \mathbb{R}^{P\setminus A} \,\middle|\, \begin{array}{l} s_x \le s_y \text{ for all } x<y \text{ in } P\setminus A,\ \lambda_a \le s_x \text{ for all } a\in A,\ x\in P\setminus A \text{ with } a\le x,\ s_x \le \lambda_b \text{ for all } x\in P\setminus A,\ b\in A \text{ with } x\le b \end{array} \right\}.$$7, these maps induce bijections on the $$O(P,A,\lambda)=\left\{ s \in \mathbb{R}^{P\setminus A} \,\middle|\, \begin{array}{l} s_x \le s_y \text{ for all } x<y \text{ in } P\setminus A,\ \lambda_a \le s_x \text{ for all } a\in A,\ x\in P\setminus A \text{ with } a\le x,\ s_x \le \lambda_b \text{ for all } x\in P\setminus A,\ b\in A \text{ with } x\le b \end{array} \right\}.$$8-th rational lattice, so marked order and marked chain polytopes always have the same Ehrhart polynomial (Ardila et al., 2010).

Equal Ehrhart polynomials do not imply combinatorial or unimodular equivalence. For regular marked posets, the number of facets of $$O(P,A,\lambda)=\left\{ s \in \mathbb{R}^{P\setminus A} \,\middle|\, \begin{array}{l} s_x \le s_y \text{ for all } x<y \text{ in } P\setminus A,\ \lambda_a \le s_x \text{ for all } a\in A,\ x\in P\setminus A \text{ with } a\le x,\ s_x \le \lambda_b \text{ for all } x\in P\setminus A,\ b\in A \text{ with } x\le b \end{array} \right\}.$$9 is $A\subseteq P$00, while the number of facets of $A\subseteq P$01 is

$A\subseteq P$02

where $A\subseteq P$03 counts saturated chains between marked elements. The key obstruction is the star relation, namely a subposet with a central element $A\subseteq P$04 and

$A\subseteq P$05

with incomparable lower and upper pairs. For a regular marked poset, the following are equivalent: $A\subseteq P$06 and $A\subseteq P$07 are unimodularly equivalent; they have the same $A\subseteq P$08-vector; they have the same number of facets; and $A\subseteq P$09 has no star relation (Fourier, 2014).

Marked chain–order polytopes interpolate between the two extremes by partitioning the unmarked set as $A\subseteq P$10. The extremal choices recover the marked order polytope ($A\subseteq P$11) and the marked chain polytope ($A\subseteq P$12). For all admissible decompositions, the polytopes are Ehrhart equivalent, and every marked chain–order polytope is a normal lattice polytope (Fang et al., 2015).

There is also a continuous family parametrized by the hypercube $A\subseteq P$13, with universal transfer maps

$A\subseteq P$14

These are mutually inverse, and $A\subseteq P$15 restricts to a piecewise-linear bijection from the marked order polyhedron to the $A\subseteq P$16-deformed marked poset polyhedron. At the vertices $A\subseteq P$17, one recovers all marked chain–order polytopes; their combinatorial type is constant on the relative interior of each face of the hypercube, and for generic $A\subseteq P$18 the vertices are exactly the vertices in the tropical subdivision associated to the marked poset (Fang et al., 2017).

5. Rank markings, root polytopes, flow polytopes, and toric geometry

For ranked marked posets, marked order polytopes acquire a polarity description. Let $A\subseteq P$19 be a starred poset, let $A\subseteq P$20 be the rank function, and define the rank marking by $A\subseteq P$21. The corresponding marked order polytope $A\subseteq P$22 contains a unique interior lattice point $A\subseteq P$23 with

$A\subseteq P$24

and the translated polytope $A\subseteq P$25 is reflexive (Rietsch et al., 2024).

The same ranked poset determines a starred quiver $A\subseteq P$26, whose arrows encode vectors $A\subseteq P$27, $A\subseteq P$28, and $A\subseteq P$29. The resulting root polytope is

$A\subseteq P$30

With the polarity convention

$A\subseteq P$31

Theorem 4.10 identifies the translated rank-marked order polytope as the polar dual of the root polytope: $A\subseteq P$32 Equivalently, the inequalities of the translated marked order polytope are in bijection with cover relations, and the vertices of the dual are precisely the root vectors $A\subseteq P$33 (Rietsch et al., 2024).

In the planar setting, this picture interfaces with flow polytopes. If $A\subseteq P$34 is a connected plane acyclic quiver and $A\subseteq P$35 is its dual starred quiver, then

$A\subseteq P$36

so the root polytope of the dual quiver is integrally equivalent to the polar dual of the flow polytope (Rietsch et al., 2024). A related direct statement holds for marked order polytopes: if $A\subseteq P$37 is strongly planar and satisfies the left-boundary marking condition, then $A\subseteq P$38 is integrally equivalent to a flow polytope $A\subseteq P$39 (Liu et al., 2019).

These identifications have toric consequences. Let $A\subseteq P$40 be the face fan of $A\subseteq P$41. If $A\subseteq P$42 is strongly-connected, then $A\subseteq P$43 is a projective Gorenstein Fano toric variety with at most terminal singularities. When $A\subseteq P$44 comes from a ranked poset $A\subseteq P$45, the face fan $A\subseteq P$46 refines the normal fan $A\subseteq P$47 of the order polytope, and if $A\subseteq P$48 is graded then

$A\subseteq P$49

Consequently, $A\subseteq P$50 is a small partial desingularization; moreover, there exists a refinement $A\subseteq P$51 such that the toric morphism to $A\subseteq P$52 is small and crepant, and $A\subseteq P$53 is smooth. In particular, the Hibi toric variety $A\subseteq P$54 has a small resolution of singularities for any ranked poset $A\subseteq P$55 (Rietsch et al., 2024).

The same framework supports mirror-symmetry constructions. For strongly-connected starred quivers, the Laurent polynomial superpotential $A\subseteq P$56 has Newton polytope $A\subseteq P$57, and in the poset case the superpotential polytope equals $A\subseteq P$58 for the canonical choice of weights (Rietsch et al., 2024).

6. Representation-theoretic realizations and distinguished families

Marked order polytopes arise naturally in highest-weight representation theory. In type $A\subseteq P$59, the Gelfand–Tsetlin polytope $A\subseteq P$60 is exactly a marked order polytope: $A\subseteq P$61 for a marked poset built from the Gelfand–Tsetlin pattern. The companion marked chain polytope is the Feigin–Fourier–Littelmann polytope: $A\subseteq P$62 The transfer map $A\subseteq P$63 therefore gives a direct combinatorial bijection between the lattice points of the two families in all dilations, explaining the equality of the corresponding basis cardinalities (Ardila et al., 2010).

The same marked-order viewpoint extends beyond type $A\subseteq P$64. Generalized Gelfand–Tsetlin patterns for $A\subseteq P$65 and $A\subseteq P$66 lie in the family of marked order polytopes, while the generalized Gelfand–Tsetlin polytopes for type $A\subseteq P$67 do not (Ardila et al., 2010). In the strongly planar setting, skew Gelfand–Tsetlin polytopes also appear as marked order polytopes and are integrally equivalent to flow polytopes (Liu et al., 2019).

For the type $A\subseteq P$68 flag variety $A\subseteq P$69, every marked chain–order polytope of the Gelfand–Tsetlin poset is realized as a Newton–Okounkov body, up to integral translation. The two extremes recover the classical Gelfand–Tsetlin and Feigin–Fourier–Littelmann–Vinberg polytopes: $A\subseteq P$70

$A\subseteq P$71

The associated value semigroups are finitely generated and saturated, so the flag variety admits flat degenerations to the irreducible normal projective toric varieties determined by these polytopes (Fujita, 2021).

An explicit Gröbner-theoretic realization is also available. For every marked chain–order polytope of the Gelfand–Tsetlin poset, the associated toric variety is realized as a sagbi degeneration of the flag variety. This construction generalizes the classical Gelfand–Tsetlin degeneration and the weighted PBW degeneration, and it extends standard monomial theories and PBW monomial bases to arbitrary marked chain–order decompositions (Makhlin, 2022). In the marked order case, it recovers the Gelfand–Tsetlin polytope, the usual semistandard-tableau realization, and the corresponding monomial bases.

These realizations place marked order polytopes at the intersection of combinatorics, polyhedral geometry, and representation theory. On the combinatorial side they encode order-preserving extensions, chamber decompositions, and face partitions; on the geometric side they participate in polarity, reflexivity, toric desingularization, and Newton–Okounkov theory; and on the representation-theoretic side they recover Gelfand–Tsetlin and related pattern polytopes together with their toric degenerations (Ardila et al., 2010, Fujita, 2021, Makhlin, 2022).