3.1. Generalized permutohedra

The class of generalized permutohedra include many classical polytopes: the usual permutohedron, the associahedron, hypersimplices, and many others (see [Reference PostnikovPo09]). Let us give several alternative ways to describe the class of generalized permutohedra. Let $(\mathbb {R}^n)^*$ denote the vector space of linear functions on $\mathbb {R}^n$ . For a face F of a polytope P, the normal cone $C_F \subset (\mathbb {R}^n)^*$ to F is given by

$$ \begin{align*}C_F = \{h \in (\mathbb{R}^n)^* \mid h(x) = \max\{h(y) \mid y \in P\} \text{ for }x \in F\}. \end{align*} $$

The normal fan ${\mathcal F}_P$ is the complete fan in $(\mathbb {R}^n)^*$ consisting of all cones $C_F$ as F varies over all the faces of P. We say that a complete fan ${\mathcal F}'$ is a coarsening of a complete fan ${\mathcal F}$ if the maximal cones of ${\mathcal F}'$ are unions of the maximal cones of ${\mathcal F}$ .

For a permutation $w\in S_n$ , let $v_w = -(w^{-1}(1),\dots ,w^{-1}(n))\in \mathbb {R}^n$ , and let $P_n := \mathrm {conv}(v_w\mid w\in S_n)$ be the standard permutohedron in $\mathbb {R}^n$ . For the following result, see [Reference Postnikov, Reiner and WilliamsPRW, Theorem 15.3] and [Reference Castillo and LiuCL].

For a polytope P, we define the support function $f_P: (\mathbb {R}^n)^* \to \mathbb {R}$ given by

$$ \begin{align*}f_P(h) = \max_{v \in P} h(v). \end{align*} $$

The function $f_P$ is a piecewise linear function on $(\mathbb {R}^n)^*$ whose maximal domains of linearity are exactly the top-dimensional cones of the normal fan of P.

The normal fan of the standard permutohedron $P_n$ is the braid fan (see Section 8.5), and by Theorem 3.2(2), a generalized permutohedron P is uniquely determined by the values $f_P(S):= f_P(h_S)$ , where $h_S(x_1,\ldots ,x_n)= \sum _{i \in S} x_i$ , and S varies over proper nonempty subsets of $[n]$ . The polytope P is then given by

(3.1) $$ \begin{align} x_1+\cdots + x_n = k = f_P([n]) \quad\text{and}\quad \sum_{i\in S} x_i \leq f_P(S). \end{align} $$

We write $f_P|_{2^{[n]}}$ for the function sending a subset $S \subset [n]$ to $f_P(S)$ , and, more generally, we will use notation, such as $f_P|_{\mathcal {S}}$ for a collection ${\mathcal {S}} \subseteq 2^{[n]}$ of subsets.

3.2. Alcoved polytopes

Let $h_1,\dots ,h_n$ be the basis of $(\mathbb {R}^n)^*$ , such that $h_i(x) = x_1 + x_2 + \cdots + x_i$ . Thus, $h_i:= h_{[1,i]}$ .

Definition 3.3 [Reference Lam and PostnikovLP07]

A polytope $P\subset \mathbb {R}^n$ is called an alcoved polytope if it is given by inequalities of the form

(3.2) $$ \begin{align} (h_i - h_j)(x) \leq a_{ij}, \qquad \text{for } i \neq j \text{ belonging to } [n] \end{align} $$

and the equation $x_1 + \cdots + x_n = k$ , for some realFootnote ¹ numbers $a_{ij}$ and k.

Alcoved polytopes are also called polytropes in [Reference Joswig and KulasJK]. We will always assume in (3.2) that the $a_{ij}$ have been chosen to be minimal, that is, they are values of the support function.

An alcoved polytope P is called generic if it is full-dimensional in H, and each equality $(h_i - h_j)(x) = a_{ij}$ defines a facet of P, for all $i,j\in [n]$ , $i \neq j$ .

Define the cyclic interval $[r,s]$ in $[n]$ as

$$ \begin{align*}[r,s]:=\left\{\begin{array}{ll} \{r,r+1,\dots,s\}&\text{if } 1\leq r\leq s \leq n,\\ \{r,r+1,\dots,n,1,2,\dots,s\}&\text{if } 1\leq s < r \leq n. \end{array} \right\} ,\end{align*} $$

and set $h_{[r,s]}:= \sum _{i \in [r,s]} x_i \in (\mathbb {R}^n)^*$ . Alcoved polytopes P are exactly all polytopes of the form

(3.3) $$ \begin{align} x_1+\cdots + x_n = k \quad\text{and}\quad \sum_{i\in [r,s]} x_i \leq f_{[r,s]} \end{align} $$

for cyclic intervals $[r,s]\subset [n]$ , where $f_{[r,s]}= f_P([r,s]):= f_P(h_{[r,s]})$ . Note that we have

(3.4) $$ \begin{align} a_{ij} = \begin{cases} f_{[j+1,i]} &\mbox{if } i> j \\ f_{[j+1,i]} -f_{[n]} = f_{[j+1,i]} -k& \mbox{if } i < j ,\end{cases} \end{align} $$

and it will be convenient to use both sets of parameters $f_{[r,s]}$ and $a_{ij}$ .

If $x =(x_1,x_2,\ldots ,x_n) \in \mathbb {R}^n$ , we shall use the shorthands $x_{[r,s]} := h_{[r,s]}(x)= \sum _{i \in [r,s]} x_i$ and $x_S:= h_S(x) = \sum _{i \in S} x_i$ .

3.3. Polypositroids

As for alcoved polytopes, polypositroids in $H_k$ and in $H_{\ell }$ are naturally in bijection. One of the main results of this paper is an explicit parametrization of all polypositroids.

We give some examples of polypositroids. For $a \in \mathbb {R}$ and P a polytope, the notation $aP$ denotes the polytope $\{ax \mid x \in P\}$ . For a cyclic interval $[r,s]$ , let

$$ \begin{align*}\Delta_{[r,s]} := \mathrm{conv}(e_r, e_{r+1},\ldots,e_s) \end{align*} $$

denote the corresponding coordinate simplex.

The cyclohedron is the Minkowski sum of simplices

(3.5) $$ \begin{align} P = \sum_{[r,s] \neq [n]} c_{r,s}\Delta_{[r,s]} + b \Delta{[n]} ,\end{align} $$

where each $c_{r,s}> 0$ and $b> 0$ . By [Reference PostnikovPo09, Theorem 7.4], the cyclohedron P is both an alcoved polytope and a generalized permutohedron, and in addition, P is simple. The $f_{[r,s]}$ from (3.3) are given by

(3.6) $$ \begin{align} f_{[r,s]}= b + \sum_{[t,u] \cap [r,s] \neq \emptyset} c_{t,u}, \end{align} $$

and we also set $k:= b+ \sum c_{r,s}$ so that $P \subset H_k$ .

If we set $c_{r,s} = 0$ whenever $r>s$ (i.e., whenever $[r,s]$ is not an honest interval), then we obtain the associahedron, which is also a polypositroid. Another possibility is to consider the polytope $P(b,c_{r,s})$ defined by (3.5), where we allow $b \in \mathbb {R}$ and $c_{r,s} \geq 0$ . When $b < 0$ , we use the Minkowski difference:

$$ \begin{align*}P - Q := \{ x \in \mathbb{R}^n \mid x + Q \subset P\}. \end{align*} $$

It is not hard to see that the deformed cyclohedron $P(b,c_{r,s})$ is either empty, or a polypositroid. When $b \geq 0$ , the polytope $P(b,c_{r,s})$ is a deformation of a cyclohedron: the normal fan of $P(b,c_{r,s})$ is a coarsening of that of the cyclohedron. However, when $b < 0$ , this may no longer be the case.

6.1. Coxeter necklaces

As before, we fix an affine plane $H=H_k:=\{x \in \mathbb {R}^n \mid x_1 + \cdots + x_n = k\}$ .

Definition 6.1. Let $\mathbf {v} = (v^{(1)},v^{(2)},\ldots ,v^{(n)})$ be a sequence of points in H. We say that $\mathbf {v}$ is a Coxeter necklace if, for each i, we have

(6.1) $$ \begin{align} v^{(i+1)} - v^{(i)} \mbox{ is nonnegative in all coordinates except the } i\mbox{-th coordinate.} \end{align} $$

Here, the superscript i is taken modulo n. Since $\mathbf {v} \subset H$ , (6.1) implies that the i-th coordinate of $v^{(i+1)} - v^{(i)}$ is nonpositive.

The sum of two Coxeter necklaces $\mathbf {v} \in H_k$ and $\mathbf {v}'\in H_{k'}$ is a Coxeter necklace $\mathbf {v}"\in H_{k+k'}$ . Recall that the polytope $Q(\mathbf {v})$ was defined in (5.2).

It will follow from Theorem 6.12 below that the image of this homomorphism of cones is ${\mathcal {C}}_{\mathrm {pol}}$ .

Proof. It suffices to show that $v^{(i)} \in Q(\mathbf {v})$ . We will show that $v^{(i)}_{[1,j]} \leq v^{(1)}_{[1,j]}$ ; the full set of inequalities follows by cyclic symmetry. Suppose $i \leq j+1$ . Then

$$ \begin{align*}(v^{(i)}-v^{(1)})_{[1,j]} = \sum_{k \in [2,i]} (v^{(k)}-v^{(k-1)})_{[1,j]} \leq 0 \end{align*} $$

using the definition (6.1) and the fact $k-1 \in [1,j]$ . Suppose $i> j +1$ . Then

$$ \begin{align*}(v^{(1)}-v^{(i)})_{[1,j]} = \sum_{k \in [i+1,1]} (v^{(k)}-v^{(k-1)})_{[1,j]} \geq 0 \end{align*} $$

using the definition (6.1) and the fact that $k-1 \notin [1,j]$ .

It follows from Lemma 6.4 that the map $\mathbf {v} \to Q(\mathbf {v})$ is injective.

Proof. For (1), we show that all but the first coordinate of $v_{c} - v_{\mathrm {id}}$ is nonnegative, and a similar argument shows that the entire sequence is a Coxeter necklace. We have $c = 23 \cdots 1$ in one-line notation. By Theorem 3.2(3),

$$ \begin{align*} v_{213 \cdots n} - v_{123\cdots n} &\in &&\mathbb{R}_{\geq 0}((-2,-1,-3,\ldots) - (-1,-2,-3,\ldots)) = \mathbb{R}_{\geq0}(e_2-e_1) \\ v_{2314 \cdots n} - v_{213\cdots n} &\in&& \mathbb{R}_{\geq 0}((-3,-1,-2,\ldots) - (-2,-1,-3,\ldots)) = \mathbb{R}_{\geq 0}(e_3-e_1) \\ \cdots \\ v_{23 \cdots (n-1) n 1}-v_{23 \cdots(n-1) 1 n} &\in &&\mathbb{R}_{\geq0}((-n,-1,-2,\ldots,1-n) - (1-n,-1,-2,\ldots,-n)) \\ &&& =\mathbb{R}_{\geq 0}(e_n-e_1). \end{align*} $$

It follows that all but the first coordinate of $v_{c} - v_{\mathrm {id}}$ is nonnegative.

For (2), we note that all edges of P incident to $v_{c^{i-1}}$ are in the same direction as an edge of the cone $C_{i}$ . It follows that $P \subset v_{c^{i-1}} + C_i$ , so that $v_{c^{i-1}} = v^{(i)}(P)$ .

6.2. Balanced digraphs

Lemma 6.8. Suppose G is a balanced digraph. Then the quantity

$$ \begin{align*}S_i(G) := \sum_{j \in [n]} (m_{j,j} + m_{j+1,j} + \cdots + m_{i,j}) \end{align*} $$

does not depend on $i \in [n]$ (indices are taken modulo n in the sum $m_{j,j} + m_{j+1,j} + \cdots + m_{i,j}$ ).

For a balanced digraph G, we define $S(G) = S_i(G)$ to be the sum of Lemma 6.8.

Note that the space of balanced graphs forms a cone in a natural way: $G= \alpha G' + \beta G"$ has weights given by $m_{ij}= \alpha m^{\prime }_{ij} + \beta m^{\prime \prime }_{ij}$ .

Let $\mathbf {v} = (v^{(1)},v^{(2)},\ldots ,v^{(n)})$ be a balanced sequence of points in H. We define a weighted directed graph $G(\mathbf {v})$ by the formula

(6.3) $$ \begin{align} m_{ij} = \begin{cases} (v^{(i+1)}- v^{(i)})_j &\mbox{ if } j \neq i, \\ v^{(i+1)}_i & \mbox{if } j = i. \end{cases} \end{align} $$

Proof. We first check that $G(\mathbf {v})$ is a balanced graph. By definition of a balanced sequence, $m_{ij} \in \mathbb {R}_{\geq 0}$ . We have for each $i \in [n]$ ,

$$ \begin{align*} \sum_{j \in [n]} m_{ij} &=\sum_{j \in [n]} (v^{(i+1)} - v^{(i)})_j + v^{(i)}_i = v^{(i)}_i = \sum_{j \in [n]} (v^{(j+1)} - v^{(j)})_i + v^{(i)}_i = \sum_{j \in [n]} m_{ji}, \end{align*} $$

so $G(\mathbf {v})$ is balanced. Finally, using $v^{(i+1)}_j = m_{j,j}+ \cdots + m_{i,j}$ , one obtains $S(G) = \sum _j v^{(i+1)}_j = k$ .

Conversely, suppose we are given a balanced graph satisfying $S(G) = k$ . Then we define a sequence $\mathbf {v}$ by $v^{(i+1)}_j = m_{j,j}+ \cdots + m_{i,j}$ . It is easy to verify that $\mathbf {v}$ is a balanced sequence in H, and that this is inverse to $\mathbf {v} \mapsto G(\mathbf {v})$ .

The last statement follows immediately from the linearity of (6.3).

We write $\mathbf {v}(G)$ for the Coxeter necklace associated to a balanced digraph G.

Example 6.11. Let G be the balanced digraph on $[4]$ , such that $m_{ij} = 1$ for all $i \neq j$ . Then $S(G) = 6$ and $\mathbf {v}(G)$ is given by

$$ \begin{align*}v^{(1)} = (3,2,1,0), \qquad v^{(2)} = (0,3,2,1), \qquad v^{(3)} = (1,0,3,2), \qquad v^{(4)} = (2,1,0,3). \end{align*} $$

6.3. Parametrization

After Lemma 6.10, it suffices to show that the map $\mathbf {v} \to Q(\mathbf {v})$ is a bijection between Coxeter necklaces and polypositroids. We delay the proof of Theorem 6.12 to Section 6.5.

The following result follows easily from the definition of the cone of balanced digraphs.

The group $H_0=\mathbb {R}^{n-1}$ of translations preserves the affine hyperplane $H_k$ and acts on the cone ${\mathcal {C}}_{\mathrm {pol}}$ of polypositroids via the formula $z: (a_{ij}) \mapsto (a_{ij} + (h_i-h_j)(z))$ , where $z = (z_1,z_2,\ldots ,z_n) \in H_0$ . The lineality space of the cone ${\mathcal {C}}_{\mathrm {pol}}$ can be identified with $\mathbb {R}^{n-1}$ . Let ${\mathcal {C}}_{\mathrm {pol}}' := {\mathcal {C}}_{\mathrm { pol}}/\mathbb {R}^{n-1}$ denote the quotient cone, which may be identified with the set of polypositroids inside $H_0$ , modulo translations.

6.4. Face graphs

Let P be an alcoved polytope and F a face of P. We define the graph $T_F$ of P at F to be the directed graph on $[n]$ with a directed edge $(i \to j)$ whenever F lies on the hyperplane $(h_j- h_i)(x) \leq a_{ji}$ , or equivalently, on the hyperplane $x_{[i+1,j]} = f_{[i+1,j]}$ . We say that a digraph T on $[n]$ is noncrossing if, when the graph is drawn inside a circle with the vertices $[n]$ arranged in clockwise order, there are no intersections in the interior of the circle. We say that a digraph T on $[n]$ is alternating if no vertex has both incoming and outgoing edges.

Let us say that a Coxeter necklace $(v^{(1)},\ldots ,v^{(n)})$ is generic if every coordinate of $v^{(i+1)} - v^{(i)}$ is nonzero, for every i. This is equivalent to saying that all the nonloop edges of the graph $G(\mathbf {v})$ are nonzero.

Proof. Let F be a face, such that $T_F$ is either not noncrossing or not alternating.

Suppose $[r,s]$ and $[r',s']$ are cyclic intervals so that the corresponding directed edges $(r-1) \to s$ and $(r'-1) \to s'$ in $T_F$ are either crossing or form a directed path $(r-1) \to s = (r'-1) \to s'$ . In the crossing case, we may assume that $r < r' \leq s < s'< r$ (interpreted in a cyclic manner). We have the following equations and inequalities

(6.4) $$ \begin{align} x_{[r,s]} &= v^{(r)}_{[r,s]} \end{align} $$

(6.5) $$ \begin{align} x_{[r',s']}&= v^{(r')}_{[r',s']} \end{align} $$

(6.6) $$ \begin{align} x_{[r,s']} &\leq v^{(r)}_{[r,s']} \end{align} $$

(6.7) $$ \begin{align} x_{[r',s]} &\leq v^{(r')}_{[r',s]} \end{align} $$

for points $x = (x_1,x_2,\ldots ,x_n)$ on the face F. By (6.4) and (6.7), we have

$$ \begin{align*}x_{[r,r'-1]} = x_{[r,s]}- x_{[r',s]} \geq v^{(r)}_{[r,s]} - v^{(r')}_{[r',s]} ,\end{align*} $$

and by (6.5) and (6.6), we have

$$ \begin{align*}x_{[r,r'-1]} = x_{[r,s']} - x_{[r',s']} \leq v^{(r)}_{[r,s']} - v^{(r')}_{[r',s']}. \end{align*} $$

Equating the two expressions for $x_{[r,r'-1]}$ , we obtain

$$ \begin{align*}v^{(r)}_{[s+1,s']} \geq v^{(r')}_{[s+1,s']}. \end{align*} $$

This is impossible because $\mathbf {v}$ is generic balanced, implying that the coordinates in the positions $[s+1,s']$ of $v^{(r')}- v^{(r)}$ are all positive.

6.5. Proof of Theorem 6.12

Let $\mathbf {v} = (v^{(1)},\ldots ,v^{(n)})$ be a Coxeter necklace in H. We show that $Q(\mathbf {v})$ is a polypositroid. First, suppose that $\mathbf {v}$ is generic. Then by Lemmas 6.15 and 6.17, $Q(\mathbf {v})$ is a generalized permutohedron, and thus a polypositroid.

Now suppose that $\mathbf {v}$ is not generic, and let $G(\mathbf {v})$ be the balanced digraph under the bijection of Lemma 6.10. Let $G_{\varepsilon }$ be obtained from $G(\mathbf {v})$ by adding $\varepsilon>0$ to every nonloop edge. It is immediate that $G(\varepsilon )$ is again a balanced digraph, and we define the Coxeter necklace $\mathbf {v}_{\varepsilon }$ by $G(\mathbf {v}_{\varepsilon }) = G_{\varepsilon }$ . Then $\mathbf {v} = \lim _{\varepsilon \to 0} \mathbf {v}_{\varepsilon }$ is a limit of the generic balanced sequences $\mathbf {v}_{\varepsilon }$ . For sufficiently small but nonzero $\varepsilon $ , the combinatorial type of the polytope $Q(\mathbf {v}_{\varepsilon })$ corresponding to $\mathbf {v}_{\varepsilon }$ does not change. The alcoved polytope Q is thus a deformation of such a $Q(\mathbf {v}_{\varepsilon })$ , in the sense of moving facets. Since $Q(\mathbf {v}_{\varepsilon })$ is a generalized permutohedron, so is $Q(\mathbf {v})$ (see, for example [Reference Castillo and LiuCL]).

Now suppose P is a polypositroid with vertices $v_w$ . Since P is a generalized permutohedron, by Proposition 6.5 and Lemma 5.1, $\mathbf {v} = (v_{\mathrm {id}}, v_{c}, v_{c^{2}},\ldots ,v_{c^{n-1}})$ is balanced, and we have $\mathrm {env}(P) = Q(\mathbf {v})$ . But P is also alcoved, so we have $P = \mathrm {env}(P) = Q(\mathbf {v})$ . Thus, the map $\mathbf {v} \mapsto Q(\mathbf {v})$ is surjective. Finally, it follows from Lemma 6.4 that $\mathbf {v} \mapsto Q(\mathbf {v})$ is injective. This proves the equivalence of (1) and (2) in Theorem 6.12.

6.6. Proof of Theorem 4.8

The following result follows immediately from Proposition 6.5(1) and Theorem 6.12.

Recall that $\pi _{\mathcal {I}}$ denotes the composition of the restriction map $f_P \mapsto f_P|_{\mathcal {I}}$ with the transformation (3.4) from $f_{[r,s]}$ -coordinates to $a_{ij}$ coordinates. Let P be a generalized permutohedron and $f_P|_{2^{[n]}} \in {\mathcal {C}}_{\mathrm {sub}}$ be its support function. Then $\pi _{\mathcal {I}}(f_P)$ represents the alcoved polytope $\mathrm {env}(P)$ . By Proposition 6.18, we thus have $\pi _I({\mathcal {C}}_{\mathrm {sub}}) \subseteq {\mathcal {C}}_{\mathrm { pol}}$ . But if P is a polypositroid, then $\mathrm {env}(P) = P$ . It follows that $\pi _{\mathcal {I}}({\mathcal {C}}_{\mathrm {sub}}) = {\mathcal {C}}_{\mathrm {pol}}$ .

8.1. Normal fans to generic simple alcoved polytopes

Recall that we say that an alcoved polytope is generic if every inequality $(h_i-h_j)(x) \leq a_{ij}$ determines a facet. The f-vector $(f_0,f_1,\ldots ,f_{d})$ of a d-dimensional polytope P is given by

$$ \begin{align*}f_i := \#\{i\text{-dimensional faces of } P\}. \end{align*} $$

Theorem 8.1. The f-vectors of any two generic simple alcoved polytopes $P\subset \mathbb {R}^n$ are the same. The face numbers are given by

(8.1) $$ \begin{align} f_{i}= \binom{n-1}{i} \binom{2n-i-2}{n-1} \qquad \mbox{ for } i=0,1,\dots,n-1. \end{align} $$

The root polytope R is the convex hull of the vectors $\{ h_i - h_j \mid i, j \in \{1,2,\ldots ,n\}\}$ , which we think of as lying inside $(\mathbb {R}^n)^*/h_n$ . If T is a directed tree on $[n]$ (or, more generally, a directed graph on $[n]$ ), we let $\Delta _T \subset (\mathbb {R}^n)^*$ denote the convex hull of the points $\{h_i - h_j \mid j \to i \; \text {is an edge of } T\} \cup \{0\}$ . A local triangulation of R is a triangulation, such that every simplex is one of the $\Delta _T$ .

Let P be a generic simple alcoved polytope. The normal fan ${\mathcal F}_P$ of P lies in $(\mathbb {R}^n)^*/h_n$ . The condition that P is generic implies that every root $h_i - h_j \neq 0$ is an edge of ${\mathcal F}_P$ . The condition that P is simple implies that each maximal cone $C_v$ of P is spanned by $(n-1)$ roots. Thus, the collection of maximal cones of ${\mathcal F}_P$ induces a local triangulation of P. Let $C_T := \mathrm {span}_{\geq 0} \Delta _T$ be the cone spanned by $\Delta _T$ .

8.2. Matching ensembles

Let P be a generic simple alcoved polytope. Recall that in Section 6.4, we have defined the graph $T_F$ for any face F of P. When $F=v$ is a vertex of P, the graph $T_v$ is a tree that we call a vertex tree. Let ${\mathcal {T}}(P)$ denote the set of vertex trees of P. The data of ${\mathcal {T}}(P)$ are equivalent to the knowledge of the normal fan ${\mathcal F}_P$ . Thus, the fan ${\mathcal F}_P$ is complete, and the maximal cones $C_v$ of ${\mathcal F}_P$ are indexed by vertices v, such that $C_v$ is the positive span of the vectors $h_i - h_j$ for $j \to i$ an edge of $T_v$ .

The first part of the following result is similar to [Reference PostnikovPo09, Lemma 13.2].

Suppose T is an alternating tree $[n]$ . A matching of $(I,J)$ in T is a collection of edges of T which form a matching of I with J, such that vertices in I are sources, and the vertices in J are sinks. Say that two directed alternating trees $T,T'$ on $[n]$ are compatible if there do not exist disjoint subsets $I, J \subset [n]$ of the same cardinality, such that both T and $T'$ contain matchings of $(I,J)$ , and these matchings disagree.

Proof. Suppose T and $T'$ are not compatible. Let $I, J \subset [n]$ be disjoint, such that T (respectively, $T'$ ) contains a matching M (respectively, $M'$ ) from I to J, such that $M \neq M'$ . We assume that I and J are chosen to be minimal, so that $M \cap M' = \emptyset $ . Let

$$ \begin{align*}x=\sum_{j \in J} h_j - \sum_{i \in I} h_i = \sum_{(i \to j) \in M} h_j - h_i = \sum_{(i \to j) \in M'} h_j - h_i. \end{align*} $$

Clearly, $x \in C_T \cap C_{T'}$ . The minimal face of $C_T$ containing x is $C_M$ . The minimal face of $C_{T'}$ containing x is $C_{M'}$ . Since $M \neq M'$ , we conclude that $C_T \cap C_{T'}$ is not a common face.

Conversely, suppose that T and $T'$ are compatible. Let $F = T \cap T'$ be the intersection, a directed forest on $[n]$ . Define a partial order $\prec $ on the connected components (denoted A) of F, by letting $A \prec A'$ if there is a (necessarily unique) sequence $A=A_0, A_1, \ldots ,A_{\ell } = A'$ of distinct components of F, such that T has a (unique) directed edge $f_i$ joining $A_i$ to $A_{i+1}$ for $i \in [0,\ell -1]$ . Similarly, define $\prec '$ using $T'$ . We claim that $A \prec A'$ if and only if $A' \prec ' A$ . Assuming otherwise, the sequence of components from A to $A'$ for T and from $A'$ to A for $T'$ can be assumed to be distinct except for A and $A'$ . Using the directed edges $f_i \in T$ from A to $A'$ and $g_i \in T'$ from $A'$ to A, together with some of the edges in F, one obtains an alternating cycle of even length, such that (picking an orientation) the clockwise edges belong to T, and the counterclockwise edges belong to $T'$ . This immediately contradicts the compatibility of T and $T'$ .

Now, let $f:[n] \to \mathbb {R}$ be a function with the following properties: it is constant with value $f(A)$ on the components A of F, and such that $f(A) < f(A')$ if and only if $A \prec A'$ if and only if $A' \prec ' A$ . Assume that $f(n) = 0$ . Then f extends to a linear function $\phi _f: (\mathbb {R}^n)^*/h_n \mapsto \mathbb {R}$ , by setting $h_i \mapsto f(i)$ . It follows by construction that $\phi _f(C_T) \geq 0$ , $\phi _f(C_{T'}) \leq 0$ , and $\phi _f(C_F) = 0$ . It follows that $C_T \cap C_{T'} = C_F$ is a common face of both cones.

A matching field on $[n]$ is a collection ${\mathcal {E}} = \{M_{I,J}\}$ of matchings, one for each pair $(I,J)$ of disjoint subsets of $[n]$ of equal size, such that for each $I' \subset I$ and $J' \subset J$ , where $I'$ is matched to $J'$ in $M_{I,J}$ , we have $M_{I',J'}$ is the restriction of $M_{I,J}$ to $(I',J')$ . If $M_{I',J'}$ is a restriction of $M_{I,J}$ , we shall say that $M_{I,J}$ contains $M_{I',J'}$ .

We shall call a matching field noncrossing, if every matching $M_{I,J}$ is noncrossing when drawn on the circle.

Proof. Let $k = |I| = |J|$ . Consider the vector

$$ \begin{align*}x = \left(1+ \frac{1}{k}\right)\sum_{i \in I} h_i + \sum_{j \in J \cup\{j'\}} h_j. \end{align*} $$

Then x lies in the cone $C_T$ for some $T \in {\mathcal {T}}(P)$ . Let $\tilde T = T|_{I \cup J \cup \{j'\}}$ denote the induced subgraph on $I \cup J \cup \{j'\}$ . It is not difficult to see that $\tilde T$ must be connected, and thus itself a tree. Let $A_1,A_2,\ldots ,A_r \subset I \cup J$ be the (vertex sets of the) connected components of the forest $\tilde T \setminus \{j'\}$ obtained by removing $j'$ . Looking at the coefficients of $h_t$ , $t \in A_s$ in x, we deduce that $|A_s \cap I| = |A_s \cap J|$ for each $s = 1,2,\ldots ,r$ . It follows that the matching $M_{I,J} \in {\mathcal {E}}(P)$ restricts to a matching on $(A_s \cap I, A_s \cap J)$ for each $s = 1,2,\ldots ,r$ .

Now let $i \in (A_1 \cap I)$ be the vertex in $A_1$ connected to $j'$ and let $(i,j) \in M_{I,J}$ be the edge of $M_{I,J}$ incident to i. Then T also contains the matching $M_{I,J} \setminus \{(i,j)\} \cup \{(i,j')\}$ , and, by Theorem 8.4, so does ${\mathcal {E}}(P)$ . This establishes condition (1) of the linkage axiom for ${\mathcal {E}}(P)$ . Condition (2) is similar.

The following examples support Conjecture 8.7.

Example 8.10. Let $n = 4$ . Let P be a generic simple polypositroid. We use Theorem 8.4 to understand the possible choices for ${\mathcal {T}}(P)$ . By the noncrossing condition, a matching field ${\mathcal {E}}(P)$ is uniquely determined except for matchings on $(I,J) = (\{1,3\},\{2,4\})$ and $(I,J) = (\{2,4\},\{1,3\})$ , each of which there are two choices of matchings, giving four possibilities for ${\mathcal {E}}(P)$ , all of which satisfy the linkage axiom. By an explicit calculation, for example, by computing whether putative vertices $v_T$ lie inside P, we find that the matching $M_{\{1,3\},\{2,4\}}$ (respectively, $M_{\{2,4\},\{1,3\}}$ ) depends on the sign of $a_{41}+a_{23}- a_{43}-a_{21}$ (respectively, $a_{12}+a_{34} - a_{14}-a_{32}$ ) (if $a_{41}+a_{23}- a_{43}-a_{21} = 0$ , then P is not simple). Suppose that P arises from the balanced graph G via Theorem 6.12. Then we have

(8.2) $$ \begin{align}\begin{aligned} a_{41}+a_{23}- a_{43}-a_{21} &= m_{43}+m_{13} - m_{24}-m_{34} \\a_{12}+a_{34} - a_{14}-a_{32}&= m_{14}+m_{24} - m_{31}-m_{41} \end{aligned}\end{align} $$

(using (6.2), the right hand side (RHS) can be written in a number of equivalent ways). It is easy to construct generic balanced G, such that the RHS has any of the four possible ordered pairs of signs. For example, a balanced directed cycle $(1 \to 4 \to 3 \to 1)$ (respectively, $(4 \to 3 \to 2 \to 4)$ ) makes the first (respectively, second) quantity in (8.2) positive and the second (respectively, first) 0. It follows that there are exactly four normal fans of generic simple polypositroids for $n = 4$ .

We consider the cyclohedron defined in (3.5).

For example, let $n = 4$ . Then the cyclohedron has $20$ vertices, and each vertex tree is a rotation of one of the trees in Figure 1.

Figure 1

Noncrossing, decreasing, alternating trees on $\{1,2,3,4\}$ .

8.3. Proof of Theorem 4.9

Proof. Suppose P is a generic simple polypositroid. By the uniqueness part of Theorem 8.4 (or directly from the proof), there is a vertex tree $T_v$ of P, which contains a noncrossing matching of $(\{k,l\},\{i,j\})$ . Since $i,j,k,l$ are in cyclic order, the matching must match i with l and match j with k. We thus have

$$ \begin{align*} a_{il} + a_{jk} &= (h_{i}-h_{l})(v) +(h_j - h_k)(v) \\ &= (h_i - h_k)(v) + (h_j - h_l)(v) \\ &\leq a_{ik} + a_{jl}. \end{align*} $$

Equality cannot occur, for otherwise, $T_v$ will contain a cycle. Conversely, if P is a generic simple alcoved polytope and $a_{ik} + a_{jl}> a_{il} + a_{jk}$ holds for any four indices $i,j,k,l$ in cyclic order, then we deduce that ${\mathcal {E}}(P)$ is noncrossing, so ${\mathcal {T}}(P)$ consists of noncrossing trees, and, by Lemma 6.15, we conclude that P is a polypositroid.

Now, the inequalities $a_{ik} + a_{jl}> a_{il} + a_{jk}$ define an open subcone C of ${\mathcal {C}}_{\mathrm { alc}}$ , each point of which represents a generic alcoved polytope. It follows from Proposition 8.11 that the cyclohedron is a generic simple polypositroid and thus C is nonempty. An open dense subset of $C' \subseteq C$ corresponds to generic alcoved polytopes that are simple. Applying Proposition 8.13, we see that these polytopes are generic simple polypositroids. The closure of $C'$ is the closed cone cut out by (4.3). The corresponding limits of generic simple polypositroids are nonempty (possibly not generic, possibly not simple) polypositroids, finishing the proof of the Theorem 4.9.

8.4. Duality for alternating trees

A noncrossing tree T on $[n]$ is called circular-alternating if for each vertex v, the edges incident to v alternate between incoming and outgoing as they are read in order when $T_v$ is drawn on a circle. Thus, for example, if v is incident to $u_1,u_2,u_3,u_4$ with $u_1<v<u_2<u_3<u_4$ , then the edges $(v,u_1)$ , $(v,u_4)$ , $(v,u_3)$ , $(v,u_2)$ alternate in direction.

Let ${\mathcal {T}}_{\mathrm {alt}}$ denote the set of alternating, noncrossing trees, and let ${\mathcal {T}}_{\mathrm {cir}}$ denote the set of noncrossing, circular-alternating trees. For $T \in {\mathcal {T}}_{\mathrm {alt}}$ , let $T'$ be obtained from T as follows: place the numbers $1',1,2',2,\ldots ,n',n$ in clockwise order around a circle, and draw T using the numbers $1,2,\ldots ,n$ . Then $T'$ is the unique tree on the numbers $1',2',\ldots ,n'$ , such that each directed edge $c' \to d'$ of $T'$ intersects a unique directed edge $a \to b$ of T, and furthermore, $a, d', b, c'$ are in clockwise order.

For a tree T, let $C^{\prime }_T$ denote the cone in $\mathbb {R}^n$ spanned by $e_i - e_j$ for each directed edge $j \to i$ in T.

Proof. We have

$$ \begin{align*}(h_b - h_a)(e_{d}-e_{c}) = \begin{cases} 0 & \mbox{if } \overline{ab} \mbox{ and } \overline{d'a'} \mbox{ are noncrossing,} \\ 1 & \mbox{if } a,d',b,c' \mbox{ are in clockwise order,} \\ -1 & \mbox{if } a,d',b,c' \mbox{ are in anticlockwise order.} \end{cases} \end{align*} $$

The result then follows from the definition of $\varphi $ .

Example 8.16. A noncrossing alternating tree $T \in {\mathcal {T}}_{\mathrm {alt}}$ and the corresponding noncrossing, circular-alternating tree ${\mathcal {T}}'=\varphi (T)$ is given in Figure 2. One can check that the cones $C_T = \mathrm {span}_{\geq 0}(h_3-h_4, h_2-h_4,h_2-h_6,h_2-h_1,h_5-h_6)$ and $C^{\prime }_{T'}=\mathrm { span}_{\geq 0}(e_3-e_4, e_5-e_3,e_1-e_5,e_2-e_1,e_5-e_6)$ are dual, in agreement with Proposition 8.15.

Figure 2

The bijection of Proposition 8.14: when T is the graph consisting of the solid arrows, $T' = \varphi (T)$ is the graph consisting of the dashed arrows.

Let T be a directed tree on $[n]$ . An edge is directed away from n if it is directed away from n as part of some path connected to n. Define two statistics on T by

$$ \begin{align*} \mathrm{up}(T) &= \#\{ \text{edges directed away from } n\} \\ \mathrm{des}(T) &= \#\{ \text{edges } i \to j \text{ with } i> j\}. \end{align*} $$

The edges counted by $\mathrm {des}(T)$ are called descent edges.

Let P be a generic simple polypositroid. Let ${\mathcal {T}}(P) = \{T_v\}$ be the collection of its vertex trees: by Lemma 8.2, ${\mathcal {T}}(P) \subset {\mathcal {T}}_{\mathrm {alt}}$ .

We now describe the 1-skeleton of P in terms of the trees $T_v$ . Suppose $E = (v,v')$ is an edge of P. The forest $T_E$ has two components and one has $T_v = T_E \cup \{e\}$ and $T_{v'} = T_E \cup \{e'\}$ for distinct directed edges $e,e'$ . The graph $T_E \cup \{e,e'\}$ has a unique (nondirected) cycle containing both e and $e'$ .

For a vertex v of a generalized permutohedron P, define the tree $T^{\prime }_v$ as follows (cf. [Reference Postnikov, Reiner and WilliamsPRW]): $T^{\prime }_v$ has a directed edge $j \to i$ if there is an edge incident with v which goes in the direction $\mathbb {R}_{\geq 0}(e_i - e_j)$ . When P is a polypositroid, we thus have two directed trees $T_v$ and $T^{\prime }_v$ on $[n]$ for each vertex $v \in P$ .

Let $h_P(t)$ denote the h-polynomial of a simple d-dimensional polytope P. It is given by the equality $\sum _{i=0}^d f_i(P) t^i = h_P(t+1)$ .

Corollary 8.20. Suppose P is a generic simple polypositroid. The h-polynomial of P is

$$ \begin{align*}h_P(t) = \sum_{T^{\prime}_v \in {\mathcal{T}}(P)} t^{\mathrm{des}(T^{\prime}_v)} = \sum_{T_v \in {\mathcal{T}}(P)} t^{\mathrm{up}(T_v)}. \end{align*} $$

8.5. Coarsenings of braid arrangements

Recall that the braid arrangement ${\mathcal B}_n \subset (\mathbb {R}^n)^*/h_n$ is the central arrangement which is the union of all hyperplanes of the form $y_i - y_j = 0$ , $i \neq j \in [n]$ , where $y = (y_1,y_2,\ldots ,y_n)$ is identified with the linear function $y_1x_1 + y_2x_2 + \cdots + y_n x_n \in (\mathbb {R}^n)^*$ . We shall consider the hyperplane arrangement ${\mathcal B}_n$ as a complete fan, the braid fan. The maximal cones of ${\mathcal B}_n$ are indexed by $w \in S_n$ :

$$ \begin{align*}C_w = \{y = (y_1,y_2,\ldots,y_n) \in (\mathbb{R}^n)^* \mid y_{w(1)} \leq y_{w(2)} \leq \cdots \leq y_{w(n)}\}. \end{align*} $$

and the rays of ${\mathcal B}_n$ are the $h_S= \sum _{i \in S} x_i \in (\mathbb {R}^n)^*/h_n$ for $S \in 2^{[n]}-\{\emptyset ,[n]\}$ .

The normal fan to the permutohedron $P_n \subset H$ is the braid fan ${\mathcal B}_n$ . More generally, any generalized permutohedron P that is sufficiently generic has ${\mathcal B}_n$ as its normal fan (see [Reference Postnikov, Reiner and WilliamsPRW, Proposition 3.2]).

Now, let P be a generic simple polypositroid and ${\mathcal F} = {\mathcal F}(P) \subset (\mathbb {R}^n)^*/h_n$ be its normal fan. The rays of ${\mathcal F}$ are the $h_{[r,s]}$ , $[r,s] \in \mathcal {I}$ . While there are many possibilities for ${\mathcal F}$ , as we saw in Theorem 8.1, all such ${\mathcal F}$ have the same f-vector. By definition, P is a generalized permutohedron, so ${\mathcal F}$ is a coarsening of the fan ${\mathcal B}_n$ . In particular, each maximal cone $C_T$ , $T \in {\mathcal {T}}(P)$ in ${\mathcal F}$ is a union of a number of the cones $C_w$ .

Example 8.24. Let $n =3$ . In this case, the normal fan of a generic simple polypositroid is the braid arrangement. There are six trees in ${\mathcal {T}}_{\mathrm {alt}}$ :

and six in ${\mathcal {T}}_{\mathrm {cir}}$ :

Since the underlying graph of every $T' \in {\mathcal {T}}_{\mathrm {cir}}$ is a path, there is a unique $w = w_{T'} \in S_n$ satisfying the condition of Proposition 8.22. This gives a bijection between ${\mathcal {T}}_{\mathrm {alt}}$ and $S_n$ , identifying $C_T$ , $T \in {\mathcal {T}}_{\mathrm {alt}}$ , and $C_w$ , $w \in S_n$ .

Example 8.25. Let $n = 4$ . We have $|{\mathcal {T}}_{\mathrm {alt}}|=24$ , consisting of $8$ trees that are stars and 16 trees that are paths. For a generic simple polypositroid P, we have $|{\mathcal {T}}(P)| = f_0(P) = \binom {6}{3} = 20$ by Theorem 8.1. There are 16 trees in $T \in {\mathcal {T}}_{\mathrm {alt}}$ , such that $\varphi (T)$ is a directed path. By Corollary 8.23, these trees belong to ${\mathcal {T}}(P)$ , for any P.

There are eight trees in ${\mathcal {T}}_{\mathrm {alt}}$ , such that $\varphi (T)$ is a (circular-alternating) star. These eight trees are the four cyclic rotations of the following two trees

$$ \begin{align*}(2 \to 1 \leftarrow 4 \to3)\qquad \text{and} \qquad (3 \to 4 \leftarrow 1 \to 2). \end{align*} $$

For each of these trees, $C_T$ is a union of two of the cones $C_w$ . For example, take $T = (2 \to 1 \leftarrow 4 \to 3)$ with dual tree $T' = (2,4 \to 3 \to 1)$ . According to Proposition 8.22, we have $C_{2 \to 1 \leftarrow 4 \to 3} = C_{2431} \cup C_{4231}$ . Similarly, we obtain

$$ \begin{align*}C_{4\to 3 \leftarrow 2 \to 1} = C_{4213} \cup C_{2413}, \qquad C_{4\to 1 \leftarrow 2 \to 3} = C_{2413} \cup C_{2431}, \qquad C_{2\to 3 \leftarrow 4 \to 1} = C_{4213} \cup C_{4231}. \end{align*} $$

Thus, we have

$$ \begin{align*}C_{2 \to 1 \leftarrow 4 \to3} \cup C_{4\to 3 \leftarrow 2 \to 1} = C_{4\to 1 \leftarrow 2 \to 3} \cup C_{2\to 3 \leftarrow 4 \to 1}. \end{align*} $$

Each ${\mathcal {T}}(P)$ contains either both $(2 \to 1 \leftarrow 4 \to 3)$ and $(4\to 3 \leftarrow 2 \to 1)$ or both $(4\to 1 \leftarrow 2 \to 3)$ and $(2\to 3 \leftarrow 4 \to 1)$ . Switching between these two choices corresponds to switching the matching $M_{\{2.4\},\{1,3\}}$ in ${\mathcal {E}}(P)$ .