Symmetric Alcoved Polytopes

Generalized alcoved polytopes are polytopes whose facet normals are roots in a given root system. We call a set of points in an alcoved polytope a generating set if there does not exist a strictly smaller alcoved polytope containing it. The type $A$ alcoved polytopes are precisely the tropical polytopes that are also convex in the usual sense. In this case the tropical generators form a generating set. We show that for any root system other than $F_4$, every alcoved polytope invariant under the natural Weyl group action has a generating set of cardinality equal to the Coxeter number of the root system.


Introduction
In this paper we investigate alcoved polytopes which are symmetric under the action of the Weyl group from the viewpoint of tropical convexity. Replacing addition by tropical addition and multiplication by tropical multiplication, one can define tropically convex subsets of finite dimensional real vector spaces. Develin and Sturmfels showed that tropical polytopes have a natural polyhedral cell decomposition such that the cells are tropically convex and classically convex at the same time [De-St]. Such sets are named polytropes by Joswig and Kulas [Jo-Ku]. Every polytrope can be written as an intersection of halfspaces given by root hyperplanes for a root system of type A [De-St].
This paper grew out of an attempt to generalize notions of tropical convexity to other types of root systems. The polytropes, which are building blocks of tropically convex sets, have a characterization that natural generalizes to other root systems: they are alcoved polytopes of type A studied by Lam and Postnikov [La-Po]. We consider them in a slightly more general situation than in [La-Po], since we also allow non-integral coefficients. Any d-dimensional alcoved polytope of type A is the tropical convex hull of d + 1 vertices [Jo-Ku]. We are interested in generating sets for alcoved polytopes associated to arbitrary root systems. Here a set of points in an alcoved polytope P is called a generating set if P is the smallest alcoved polytope containing it.
Our main result states that every alcoved polytope for an irreducible and reduced root system Φ not of type F 4 that is symmetric under the action of the Weyl group can be generated by h vertices, where h is the Coxeter number of Φ. We also discuss the case F 4 , where a symmetric alcoved polytope may need up to 24 generators while the Coxeter number is 12.
In the A n -case however, every alcoved polytope has a generating set of size n + 1 which is the Coexter number, regardless of symmetry. We give a short proof of this fact. The symmetry assumption cannot be dropped in general, since there exists a non-symmetric alcoved polytope of type D 4 requiring eight generators, whereas the Coxeter number of D 4 is six.

Tropical convexity and alcoved polytopes
We begin by recalling some facts on tropical convexity. Let (R, ⊕, ) be the min-plus tropical semiring. The space R n together with componentwise addition ⊕ is a semimodule under the semiring (R, ⊕, ), if we put a (x 1 , . . . , x n ) = (a + x 1 , . . . , a + x n ) for a ∈ R and (x 1 , . . . , x n ) ∈ R n .
. A subset S of R n is called tropically convex if for all x, y in S and for all µ, ν ∈ R the element (µ x) ⊕ (ν y) is also contained in S. The tropical convex hull of a subset V in R n is the smallest tropically convex set containing V .
Note that any tropically convex subset of R n containing x also contains x + (r, . . . , r) = r x for all r ∈ R. Therefore we project tropically convex subsets of R n to R n /R(1, . . . , 1) and call their image in R n /R(1, . . . , 1) also tropically convex.
In Theorem 15] it is shown that the tropical convex hull of a finite subset of R n /R(1, . . . , 1) admits a polyhedral cell decomposition. These bounded cells are at the same time tropically convex and convex in the ordinary sense. Such a set is also called a polytrope; see [Jo-Ku].
Tropical convexity is closely related to root systems of type A in the following way. Let U be the subspace is an isomorphism of real vector spaces. Hence the restriction of the canonical scalar product on R n to U induces a scalar product , on R n /R(1, . . . , 1). Consider the elements a ij = e i − e j in R n /R(1, . . . , 1), where e i denotes the canonical unit vectors. Then Φ = {a ij : i = j} is a root system of type A n−1 in R n /R(1, . . . , 1).
Lemma 2.2. For any polytrope P in R n /R(1, . . . , 1) there are non-negative real constants c ij such that Proof. By [De-St, Lemma 10 and Proposition 18], polytropes are precisely the bounded intersections of half-spaces of the form {x = (x 1 , . . . , x n ) ∈ R n /R(1, . . . , 1) : x i − x j ≤ c ij } for i = j and c ij ∈ R, which implies our claim.
Hence polytropes look like alcoved polytopes of type A, as defined in [La-Po]. To be precise, let V be a real vector space endowed with a scalar product , , and let Φ be any irreducible and reduced root system in V . By t we denote the type of the corresponding Dynkin diagram, i.e. t is equal to A n for n ≥ 1, B n , C n for n ≥ 2, D n for n ≥ 4 or to E 6 , E 7 , E 8 , F 4 , or G 2 . Moreover, let W be the Weyl group of V , i.e. W is the finite group of orthogonal endomorphisms of V generated by the reflections at the hyperplanes H a = {x ∈ V : a, x = 0} perpendicular to the roots. Definition 2.3. A generalized alcoved polytope of type t is a bounded intersection of root halfspaces of the form where Ψ is a subset of a root system Φ of type t and c a ∈ R.
Thus a generalized alcoved polytope is a bounded subset of V defined as an intersection of root halfspaces. Alternatively, we can describe it as a polytope whose set of facet normal vectors is contained in Φ. If the right hand side values c a are all integers, a generalized alcoved polytope we defined is an alcoved polytope from [La-Po]. Payne [Pa] showed that alcoved polytopes have a Koszul property. For simplicity we will refer to "generalized alcoved polytopes" simply as "alcoved polytopes".
Some examples of alcoved polytopes include associahedra (see [Jo-Ku] and references therein), cubes, cross polytopes, and the 24-cell as a degenerate F 4 alcoved polytope.

Generating sets of alcoved polytopes
We will define generating sets of alcoved polytopes and prove results about their cardinality for type A. Besides, we discuss an interesting example for type D 4 .
Definition 3.1. Let Φ be a root system of type t in V , and let P be an alcoved polytope of type t in V . A subset Σ ⊂ V is called a generating set of P if P is the smallest alcoved polytope in V containing Σ.
Let P be an alcoved polytope for the root system Φ of type A n−1 as in Lemma 2.2. Since alcoved polytopes are tropically convex, every subset of P whose tropical convex hull is P is a generating set. However not every set of generators of P in the sense of Definition 3.1 is a set of tropical generators. The tropical convex hull of a generating set may be strictly smaller than P . For example, consider the square with vertices (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1) in R 3 /R(1, 1, 1). The two vertices (0, 0, 1) and (0, 1, 0) generate the square as an alcoved polytope, but their tropical convex hall is strictly smaller. In fact, the square cannot be generated tropically by two points, as the tropical convex hull of any two points is at most one dimensional in R n /R(1, . . . , 1).
For any bounded S ⊆ V and a ∈ Φ the smallest halfspace of the form {x ∈ V : a, x ≤ c} that contains S is {x ∈ V : a, x ≤ sup s∈S a, s }, so the alcoved polytope generated by S is On the other hand, for an alcoved polytope P , a subset S ⊂ P is a generating set if and only if for every a ∈ Φ we have sup x∈P a, x = sup x∈S a, x .
This can be rephrased as follows. Recall that a support hyperplane H of a polytope P is a hyperplane containing P on one side and having non-empty intersection with P . The intersection H ∩ P is the face of P supported by H.
Lemma 3.2. A set S ⊆ P is a generating set for P if and only if for any a ∈ Φ the support hyperplane of P perpendicular to a contains at least one point from S.
It follows that there is a minimal cardinality generating set consisting of vertices of P only. If a point in a generating set lies in the relative interior of a non-vertex face, then the point can be replaced by a vertex of that face, as doing so only enlarges the set of support hyperplanes containing the point.
Question 3.3. Our problem can be phrased as follows.
(1) For each root system Φ, find a number k such that every alcoved polytope of type Φ has a generating set of size at most k.
(2) More generally, let P be a polytope and H be a set of support hyperplanes of P such that all facets of P are contained in H. The containment may be strict. Find the minimal cardinality of a set S ⊂ P such that every support hyperplane in H contains at least one point from S.
There is a nice solution for type A alcoved polytopes, as we will see in Proposition 3.6 below. The general problem seems computationally difficult, as it amounts to solving a set covering problem, which is known to be NP-complete in general.
Remark 3.4. For every root system Φ there exist alcoved polytopes which cannot be generated with less than |Φ|/ dim(V ) vertices. For a fixed set of facet normals, the condition on the coefficients for the perpendicular hyperplanes to meet at a point is described by vanishing of certain determinants. Generic choices of coefficients would avoid this determinantal locus and give polytopes that are simple, i.e. no more than dim(V ) facets can meet at one point. Therefore in this case we need at least |Φ|/ dim(V ) vertices to support all facets.
Proposition 3.5. Let t be the type of a root system Φ in a vector space of dimension 2. Then every alcoved polytope of type t can be generated by |Φ|/2 many vertices.
Proof. Let us cyclically order the roots using their positions in the plane. For any two support hyperplanes corresponding to two consecutive roots, there is a vertex supporting them, so |Φ|/2 vertices are sufficient to support all root hyperplanes.
In Theorem 12], it is shown that every alcoved polytope of type A n is a tropical simplex, i.e. it has d + 1 tropical generators, where d is the dimension of the alcoved polytope (which might be smaller than n if the polytope lies in a root hyperplane). In particular it can be generated by n + 1 elements in the sense of Definition 3.1. The next proposition provides an easy proof of this fact. This does not imply the Joswig-Kulas result because the set of tropical generators may be strictly larger.
Proposition 3.6. Every alcoved polytope of type A n is generated by n + 1 elements.
Proof. Let P be a nonempty alcoved polytope of type A n defined by x i − x j ≤ c ij for some c ij ∈ R where 0 ≤ i, j ≤ n and i = j. By Lemma 3.2, we need to consider the support hyperplanes of P normal to the roots, so we may assume that all inequalities are tight, i.e. for each pair (i, j), there is a point p ∈ P such that p i − p j = c ij . Then we must have c ik − c jk ≤ c ij for any distinct i, j, k. Otherwise, for any p ∈ P , p i − p k = (p i − p j ) + (p j − p k ) ≤ c ij + c jk < c ik , contradicting our assumption that the inequality using the inequality seen above. Therefore the point p (k) lies in the polytope P . The n + 1 points p (0) , p (1) , . . . , p (n) all lie in P and support all the root hyperplanes, so they form a generating set of size n + 1.
Note that n + 1 is the Coxeter number of the root system of type A n . We will see below in Theorem 4.3 that almost all Weyl group symmetric alcoved polytopes can be generated by a set of vertices whose cardinality is the Coxeter number of the root system.
However, for types of root systems other than A n this is no longer true if we drop the symmetry hypothesis, as seen in the following example.
Example 3.7. Consider the D 4 alcoved polytope in R 4 defined by the inequalities given in the columns of the following matrix in homogeneous coordinates. For example, the first column represents the inequality 100 + x 1 + x 2 ≥ 0. A computer check shows that this alcoved polytope requires eight generators, while the Coxeter number of D 4 is six. The polytope is simple and has f -vector (96,192,120,24,1).

Symmetric alcoved polytopes
In this section we define symmetric alcoved polytopes and prove our main result stating that in all cases except F 4 they can be generated by a set of cardinality equal to the Coxeter number. Afterwards we discuss the case of F 4 -alcoved polytopes.
Let Φ be a root system contained in the Euclidean vector space (V, , ) and let W be the associated Weyl group.
Definition 4.1. We call an alcoved polytope in V of type t symmetric if it is invariant under the action of the Weyl group W .
Note that W acts on the faces of a symmetric alcoved polytope. Since every w in W maps the support hyperplane with respect to the root a to the support hyperplane with respect to the root w(a), we can write a symmetric alcoved polytope as with c a = c w(a) for all w in W and for all a ∈ Φ.
Note that an alcoved polytope for a non-reduced root system Φ is also an alcoved polytope of the associated reduced root system Φ red . Moreover, alcoved polytopes for reducible root systems are products of alcoved polytopes for the irreducible components. Therefore we will from now on assume that Φ is irreducible and reduced.
Let us recall some basic facts about root systems from [Bou, Chapter VI]. For every root a we denote by s a the corresponding reflection in V . Then for all roots b we have is a number between −3 and 3 in the reduced case. Proposition 4.2. [Bou, VI, 1.11, Proposition 33] With the notation above, the root system Φ is a disjoint union of the orbits Γθ i for i = 1, . . . n.
Our main result is as follows.
Theorem 4.3. Let P be a symmetric alcoved polytope of irreducible and reduced type t = F 4 . Then P can be generated by h vertices, where h is the Coxeter number of t.
Proof. If dim V = 2, our claim follows from Lemma 3.5. The simply laced case is proved below in Theorem 4.4. The case B n and C n will be shown in Theorem 4.5.
Symmetric alcoved polytopes of type F 4 may require up to 24 generators while the Coxeter number is 12; see Theorem 4.6 for a detailed analysis.
Note that by Remark 3.4 the bound from Theorem 4.3 is the candidate for the best possible result for general (not necessarily symmetric) alcoved polytopes, since h = |Φ|/ dim(V ). This works in the A n case, as explained above. However, in the case of the root system D 4 there are exceptions of non-symmetric alcoved polytopes where more generators are needed; see Example 3.7. It is an interesting question for which types of root system the bound from Theorem 4.3 can be generalized to the non-symmetric case. 4.1. Simply laced case: types A, D, and E. As indicated before, we will prove Theorem 4.3 in several steps. We begin with the simply laced case. Recall that a root system Φ is called simply laced if all roots have the same length, i.e. if the number a, a is constant for all a ∈ Φ. The types of the simply laced root systems are precisely the types A, D and E.
Theorem 4.4. Let P be a symmetric alcoved polytope of type t, where t is irreducible, reduced and simply laced, i.e. either A n for n ≥ 1, D n for n ≥ 4 or E 6 , E 7 or E 8 . Then P can be generated by h vertices.
Proof. We assume first that t is not E 8 . We choose a basis b 1 , . . . , b n of Φ and look at the corresponding Dynkin diagram. We number the basis elements in the following way: Note that the numbering in the D n case is opposite to the numbering often found in the literature. Recall that every vertex in the Dynkin diagram represents a basis element b i and that b i , b j is − b j , b j if the vertices for b i and b j are connected, and 0, if they are not. Since the Dynkin diagram is a tree, for all i and j there exists a geodesic [b i , b j ] connecting the vertices b i and b j . We first show the following claim.
Let us prove the claim by induction on j. If j is equal to i there is nothing to prove. Assume that the claim holds for some j ≥ i with j < n. Then we calculate which finishes the proof of the claim.
Recall that for all i = 1, . . . n we have From the claim we deduce that bn] b ν except possibly in the cases t = D n and i = 1 or t = E 6 or E 7 and i = 1, 2. It can easily be checked directly that the statement ( * ) also holds in these remaining cases.
Since in all cases b n is a neighbour of b n−1 in the Dynkin diagram, ( * ) implies that In a simply laced root system, the Weyl group acts transitively on the set of roots, see [Bou,VI,1.3,Proposition 11]. Hence every symmetric alcoved polytope is of the form We will show now that x is a vertex of P λ . Note that the highest root in Φ is of the following form: A n : b 1 + · · · + b n D n : b 1 + b 2 + 2b 3 + · · · + 2b n−1 + b n E 6 : b 1 + 2b 2 + 3b 3 + 3b 4 + 2b 5 + b 6 E 7 : 2b 1 + 2b 2 + 3b 3 + 4b 4 + 3b 5 + 2b 6 + b 7 Therefore for every positive root a ∈ Φ we find that a, x = 0 or a, x = b n , x = λ. For the negative roots there is nothing to do, since they assume a non-positive value on x. Hence we find that x lies in P λ . Since θ 1 , . . . , θ n are independant (see [Bou], VI, 1.11, Proposition 33), and x lies in all hyperplanes { θ i , − = λ}, we find that x is a vertex.
Since the Weyl group maps faces to faces, the elements ω k x for k = 0, . . . , h − 1 are also vertices of P λ , where ω = s b 1 • . . . • s bn is the corresponding Coxeter element. Since x satisfies θ i , x = λ for all i = 1, . . . , n, the vertex y = ω k (x) satisfies ω k θ i , y = λ for all i = 1, . . . , n. Since every root is of the form ω k θ i for suitable i and k, every hyperplane appearing at the boundary of P λ contains one of the vertices ω k x for k = 0, . . . , h − 1. This implies our claim for all types except possibly E 8 .

4.2.
Types B and C. Since the symmetric alcoved polytopes of type C n coincide with the symmetric alcoved polytopes of type B n , it suffices to consider the B n case. Here we may assume V = R n and Φ = {±a i : i = 1, . . . n} ∪ {±a i ± a j : 1 ≤ i < j ≤ n}, where a i is the canonical basis of V . The Coxeter number for types B n and C n is 2n.
Theorem 4.5. Let P be a symmetric alcoved polytope of type B n or C n for n ≥ 2. Then P can be generated by 2n vertices.
Proof. We choose as a basis b 1 = a 1 − a 2 , b 2 = a 2 − a 3 , . . . b n−1 = a n−1 − a n and b n = a n of Φ. The Dynkin diagram of type B n is the following: We then calculate the roots θ 1 , . . . , θ n as θ n = a n and θ i = a n + a i for i = 1, . . . , n − 1.
Recall that the union of the orbits of the θ i under the group Γ generated by the Coxeter element ω = s 1 • . . . • s n is the whole root system. Since Weyl group elements map long roots to long roots and short roots to short roots, we deduce that the set of short roots is equal to Γb n , and that the set of long roots is equal to Γb 1 ∪ . . . ∪ Γb n−1 . A type B n or C n symmetric alcoved polytope P is of the form for positive numbers λ and µ satisfying µ ≤ λ ≤ 2µ. In the extreme cases µ = λ and λ = 2µ, only long or short root hyperplanes appear as facets, and the others support lower dimensional faces. However, this would not affect our reasoning. Let x be the point in V defined by a 1 , x = . . . = a n−1 , x = λ − µ and a n , x = µ.
Then x is a point in P λ,µ lying in the n independent hyperplanes { θ n , − = µ} and { θ i , − = λ} for i = 1, . . . , n − 1. Therefore x is a vertex. As in the proof of Theorem 4.4, the h vertices ω k x for k = 0, . . . , h − 1 touch every boundary hyperplane. Hence they form a generating set of cardinality h. (±a 1 ± a 2 ± a 3 ± a 4 )} is a root system of type F 4 in V . It consists of 24 long roots (of the form ±a i ± a j ) and 24 short roots. A symmetric alcoved polytope of type F 4 is of the form

Type
for positive numbers λ and µ satisfying µ ≤ λ ≤ 2µ. This condition ensures that all root hyperplanes are support hyperplanes. When µ = λ or λ = 2µ, either only long roots or only short roots appear as facet normals and the others support faces of lower dimension. When µ < λ < 2µ, all root hyperplanes support facets. The convex hull of short roots of F 4 is a regular polytope called the 24-cell. The long roots are the vertices of its dual. The 24-cell is combinatorially equivalent to P 1,1 . The root system of type F 4 is self-dual. By doubling the lengths of the short roots, and then rescaling everything by 1 √ 2 , we get back F 4 , rotated, with the roles of long and short roots switched. More concretely, the matrix A = 1 2 1 1 0 0 1 −1 0 0 0 0 1 1 0 0 1 −1 maps the long roots to short roots and the doubled short roots to long roots. Then A −T is a linear isomorphism from P λ,µ to P 2µ,λ , so these two polytopes have exactly the same combinatorics, in particular, the same number of generators.
Theorem 4.6. Let P λ,µ be an alcoved polytope of type F 4 , using the notation above.
A computation using Gfan [Jen] shows that all values of λ and µ satisfying 4 3 µ < λ < 3 2 µ give the same combinatorial type for P λ,µ . A Polymake [Ga-Jo] computation shows that the polytope is simple in this case, i.e. each vertex lies in exactly 4 facets. Hence at least 12 vertices are necessary to meet all 48 facets. The polytope has f -vector (288,576,336,48,1).
Computation by Gfan [Jen] shows that all values of λ and µ satisfying 3 2 µ < λ < 2µ give the same the combinatorial type for P λ,µ , which is simple and has f -vector (192,384,240,48,1). The 192 vertices of P λ,µ are obtained from x 1 (or equivalently x 2 ) by permuting coordinates in any of the 24 ways and changing signs in any of the 8 ways, for a total of 192 distinct combinations. It can then be seen directly that each vertex lies in exactly one long-root facet and there are 24 long-root facets, so P λ,µ needs at least 24 generators.