Séminaire Lotharingien de Combinatoire 73 (2015), Article B73a

POLAR ROOT POLYTOPES THAT ARE ZONOTOPES

PAOLA CELLINI AND MARIO MARIETTI

Abstract. Let P_{Φ} be the root polytope of a finite irreducible crystallographic root system Φ, i.e., the convex hull of all roots in Φ. The polar of P_{Φ}, denoted P_{Φ}^{∗}, coincides with the union of the orbit of the fundamental alcove under the action of the Weyl group. In this paper, we determine which polytopes P_{Φ}^{∗} are zonotopes and which are not. The proof is constructive.

1. Introduction

Let Φ be a finite irreducible crystallographic root system in a Euclidean space V with scalar product ( , ), W the Weyl group of Φ, and 𝔤_{Φ} a simple Lie algebra having Φ as root system. Let P_{Φ} be the root polytope associated with Φ, i.e., the convex hull of all roots in Φ.

Motivated by the connections of the root polytope P_{Φ} with 𝔤_{Φ} (more precisely, with the Borel subalgebras of 𝔤_{Φ} and their abelian ideals), in [3] we study P_{Φ} for a general Φ. Among other things, we give a presentation of P_{Φ} as an intersection of half-spaces, and describe its faces as special subposets of the root poset, up to the action of W. In [4], we develop these general results, obtaining several special results for the root types A_{n} and C_{n}. One of the special properties of these two root types is that the cones on the facets of P_{Φ} are the closures of the regions of a hyperplane arrangement. This means that P_{Φ} is combinatorially dual to a zonotope (see [8, §7.3], or [5, §2.3.1]). More precisely, if H_{P_{Φ}} is the arrangement of all the hyperplanes through the origin containing some (n − 2)-dimensional faces of P_{Φ}, then the complete fan associated with H_{P} is equal to the face fan associated with P_{Φ}. This property is satisfied by the polytopes whose polar polytopes are zonotopes (see [8, §7.3]). One of the referees of [4] asked if the polars of the types A_{n} and C_{n} root polytopes are actually zonotopes, and if so which.

In this paper, we answer this question. We denote by P_{Φ}^{∗} the polar polytope of P_{Φ}: we explicitly describe P_{Φ}^{∗} as a zonotope for the types A_{n} and C_{n}, as well as for B_{3} and G_{2}. Moreover, we prove by a direct check that, for all other root types, the set of cones on the facets of P_{Φ} is not equal to the set of closures of the regions of H_{P_{Φ}}. Hence, for all other root types, P_{Φ}^{∗} is not a zonotope.

We point out that P_{Φ}^{∗} is a natural object for the crystallographic root system Φ that can be more familiarly described in terms of alcoves and Weyl groups. Indeed, P_{Φ}^{∗} is the union of the orbit of a fundamental alcove of Φ under the Weyl group, i.e., if we fix any basis of Φ, and denote by A the corresponding fundamental alcove of the affine Weyl group associated with Φ, then P_{Φ}^{∗} = ⋃_{w∈W} wA. Thus, P_{Φ}^{∗} is a fundamental domain for the group of translations by elements in the coroot lattice of Φ.

2010 Mathematics Subject Classification. 17B22 (primary), 05E10, 52B20 (secondary).

Key words and phrases. Root system, Root polytope, Zonotope, Weyl group.

2. Statement of results

Let Φ^{+} be a positive system for Φ, Π the corresponding root basis of Φ, θ the highest root, and Ω^{∨} the dual basis of Π in the dual space V^{∗} of V, i.e., the set of fundamental coweights of Φ.

We set Π = {α_{1}, …, α_{n}}, θ = ∑_{i=1}^{n} m_{i}α_{i}, Ω^{∨} = {ω_{1}^{∨}, …, ω_{n}^{∨}}, so that ⟨ω_{i}^{∨}, α_{j}⟩ = δ_{ij}, where ⟨ , ⟩ : V^{∗} × V ↦ ℝ is the natural pairing of V^{∗} and V, and define

o_{i} = \frac{ω_{i}^{∨}}{m_{i}} , for i = 1, …, n.

Consider the fundamental alcove of the affine Weyl group of Φ (see [2, VI, 2.1–2.2], or [7, 4.2–4.3]):

A = {x ∈ V^{∗} | ⟨x, α⟩ ≥ 0 for all α ∈ Π, ⟨x, θ⟩ ≤ 1}.

If x is any element or subset of V or V^{∗}, we denote by W · x the orbit of x by the action of W. It is well-known, and easy to see, that A is the n-simplex with vertices the null vector and o_{1}, …, o_{n} [2, VI, Corollaire in 2.2], and that

(2.1)  ⋃_{w∈W} wA = {x ∈ V^{∗} | ⟨x, β⟩ ≤ 1 for all β ∈ Φ}.

We denote by P_{Φ} the root polytope associated with Φ, i.e., the convex hull of all roots in Φ. For short, we write P for P_{Φ} when the root system is clear from the context. It is easy to see that P is the convex hull of the long roots in Φ. Indeed, this is directly checked if the rank of Φ is 2. In the general case, we observe that, since Φ is irreducible, the set of all long roots and the set of all short roots cannot be mutually orthogonal, hence there exist a short root β and a long root β′ such that (β, β′) ≠ 0. Then β belongs to the convex hull of the long roots in the irreducible dihedral root system generated by β and β′, and since W is transitive on the short roots, any short root belongs to the convex hull of some long roots.

We denote by P_{Φ}^{∗} the polar of the root polytope associated with Φ:

P_{Φ}^{∗} := {x ∈ V^{∗} | ⟨x, v⟩ ≤ 1 for all v ∈ P_{Φ}},

and we call it the polar root polytope. Again, for short, we write P^{∗} for P_{Φ}^{∗}. By definition of P^{∗}, we obtain

(2.2) P^{∗} = {x ∈ V^{∗} | ⟨x, β⟩ ≤ 1 for all β ∈ Φ} = {x ∈ V^{∗} | ⟨x, β⟩ ≤ 1 for all β ∈ Φ_{ℓ}},

where Φ_{ℓ} is the set of long roots in Φ. Hence, the polar root polytope satisfies

(2.3)  P^{∗} = ⋃_{w∈W} wA = Conv (⋃_{i=1}^{n} W · o_{i}) .

Our first result is that, for Φ of type A_{n}, C_{n} (hence also for type B_{2} = C_{2}), B_{3}, and G_{2}, the polar root polytope P^{∗} is a zonotope.

A zonotope is by definition the image of a cube under an affine projection. Let U be a vector space, v_{1}, …, v_{k}, p ∈ U, and S = {v_{1}, …, v_{k}}. We set

Zon_{p}(S) = \{p + ∑_{i=1}^{k} t_{i}v_{i} | −\frac{1}{2} ≤ t_{i} ≤ \frac{1}{2}\}

and call Zon_{p}(S) the zonotope generated by S with center p. Thus a zonotope in U is a polytope of the form Zon_{p}(S), for some finite subset S and some vector p in U.

We prove that, for A_{n} and C_{n}, the polar root polytope P^{∗} is the zonotope generated by the orbit of a single o_{i} (with center the null vector \underline{0}), and that a similar result holds for B_{3} and G_{2}. This is done in Section 4, where we find case free conditions for the general inclusions Zon\underline{0}(W · c ω_{i}^{∨}) ⊆ P^{∗} (c ∈ ℝ, i ∈ \{1, …, n\}), and check directly the reverse inclusions that we need in the special cases A_{n}, C_{n}, B_{3}, G_{2}.

The following is a well-known property of zonotopes (see [8, Corollary 7.18]).

Proposition 2.1. Let U be a vector space, U^{∗} its dual, S = \{v_{1}, …, v_{r}\} ⊆ U \ \{\underline{0}\}, H_{i} = {v ∈ U^{∗} | ⟨v, v_{i}⟩ = 0} for i = 1, …, r, and A_{S} be the arrangement of the hyperplanes H_{1}, …, H_{r}. Then the cones on the faces of Zon_{0}(S)^{∗} coincide with the faces of A_{S}.

We denote by H_{P} the central hyperplane arrangement determined by the (n − 2)-faces of P, i.e., H ∈ H_{P} if and only if H is a hyperplane containing the null vector \underline{0} and some (n − 2)-face of P. Since the null vector \underline{0} lies in interior of P, the polytopes P and P^{∗} are combinatorially dual to each other, and P = (P^{∗})^{∗}. By Proposition 2.1, if P^{∗} is a zonotope, then the cones on the proper faces of P should coincide with the faces of the hyperplane arrangement H_{P}. Therefore, P^{∗} cannot be a zonotope if some hyperplane in H_{P} meets the interior of some facet of P. We will see in Section 5 that this happens for all root types other than A_{n}, C_{n}, B_{3}, G_{2}.

We summarize our results in the following theorem, where we number the simple roots, and hence the fundamental coweights, as in Bourbaki’s tables [2]. For any finite subset S = {v_{1}, …, v_{k}}, we denote by ZT the zonotope

ZT(S) = {∑_{i=1}^{k} t_{i}v_{i} | 0 ≤ t_{i} ≤ 1} .

If the barycenter of S is the null vector \underline{0}, then ZT(S) coincides with Zon\underline{0} (S) (see Section 4).

Theorem 2.2. (1) For Φ of type A_{n} or C_{n},

P^{∗} = ZT(W · o_{1}).

(2) For Φ of type B_{3}, P^{∗} = ZT (W · \frac{o_{3}}{2} ).

(3) For Φ of type G_{2}, P^{∗} = ZT (W · \frac{o_{1}}{2} ).

(4) For all other root types, P^{∗} is not a zonotope.

3. Preliminaries

In this section, we fix our further notation and collect some basic results on root systems and Weyl groups. Some of these results are well-known (see [1], [2], or [7]), while other results are more unusual and their proofs will be sketched.

Let Φ be a finite irreducible (reduced) crystallographic root system in the real vector space V endowed with the positive definite bilinear form ( , ). From now on, for notational convenience, we identify V^{∗} with V through the form ( , ).

We summarize our notation on the root system and its Weyl group in the following list:

n  the rank of Φ,
Π = {α_{1}, …, α_{n}}  the set of simple roots,
Ω^{∨} = {ω_{1}^{∨}, …, ω_{n}^{∨}} the set of fundamental coweights (the dual basis of Π),
Φ^{+}  the set of positive roots w.r.t. Π,
Φ(Γ)  the root subsystem generated by Γ in Φ, for Γ ⊆ Φ,
θ  the highest root in Φ,
m_{i}  the i-th coordinate of θ w.r.t. Π, i.e., θ = ∑_{i=1}^{n} m_{i}α_{i},
o_{i}  = \frac{ω_{i}^{∨}}{m_{i}} , for i = 1, …, n,
Φ^{∨}  = \{β^{∨} = \frac{2β}{(β,β)} | β ∈ Φ\}, the dual root system of Φ,
W  the Weyl group of Φ,
s_{β}  the reflection with respect to the root β.

For each specific type of the root system, we number the simple roots and hence the fundamental coweights as in Bourbaki’s tables [2].

3.1. Root, coroots and partial orderings. We denote by ≤ the usual partial ordering of V determined by the positive system Φ^{+}: x ≤ y if and only if y−x = ∑_{α∈Π} c_{α}α with c_{α} a nonnegative integer for all α ∈ Π. We denote by ≤^{∨} the analogous ordering determined by the dual root system Φ^{∨}. For S ⊆ Φ, we let S^{∨} = {β^{∨} | β ∈ S}. Then Π^{∨} is the basis of Φ^{∨} corresponding to the positive system (Φ^{+})^{∨}. We have:

x ≤^{∨} y if and only if y − x = ∑_{α∈Π} c_{α}α^{∨} with c_{α} ∈ ℤ, c_{α} ≥ 0 for all α ∈ Π.

3.2. Reflection products. In the following lemma, we provide a result on the product of reflections by general roots (possibly not simple).

Lemma 3.1. Let β_{1}, …, β_{k} ∈ Φ and w = s_{β_{1}} ⋯ s_{β_{k}}. Then, for any x ∈ V,

(3.1) w(x) = x − ∑_{i=1}^{k}(x, β_{i}^{∨})ν_{i} = x − ∑_{i=1}^{k}(x, β_{i})ν_{i}^{∨}, where ν_{i} = s_{β_{1}} ⋯ s_{β_{i−1}}(β_{i});

(3.2)  w(x) = x − ∑_{i=1}^{k}(x, η_{i}^{∨})β_{i} = x − ∑_{i=1}^{k}(x, η_{i})β_{i}^{∨}, where η_{i} = s_{β_{k}} ⋯ s_{β_{i+1}}(β_{i}).

Proof. The first equality in Formula (3.1) is easily proved by induction computing s_{β_{1}}(s_{β_{2}} ⋯ s_{β_{k}}(x)); the second one is clear since β_{i} and ν_{i} have the same length, for all i. Formula (3.2) is an application of (3.1) since, by definition of η_{i} and the fact that s_{s_{β}(β′)} = s_{β}s_{β′}s_{β} for all roots β and β′, we have

s_{β_{h}} ⋯ s_{β_{k}} = s_{η_{k}} ⋯ s_{η_{h}}, for h = 1, …, k,

hence w = s_{η_{k}} ⋯ s_{η_{1}} and β_{i} = s_{η_{k}} ⋯ s_{η_{i+1}}(η_{i}). □

For h = 1, …, k, if we define w_{h} = s_{β_{h}} ⋯ s_{β_{k}} = s_{η_{k}} ⋯ s_{η_{h}}, Formula (3.2) yields

(3.3)  w_{h}(x) = x − ∑_{i=h}^{k}(x, η_{i}^{∨})β_{i} = x − ∑_{i=h}^{k}(x, η_{i})β_{i}^{∨}.

3.3. Reduced expressions. For any w ∈ W, we set

N(w) = \{γ ∈ Φ^{+} | w^{−1}(γ) < \underline{0}\}.

If w = s_{β_{1}} ⋯ s_{β_{k}} is a reduced expression of w (so β_{i} ∈ Π, for all i = 1, …, k), and if we define ν_{i} = s_{β_{1}} ⋯ s_{β_{i−1}}(β_{i}) and η_{i} = s_{β_{k}} ⋯ s_{β_{i+1}}(β_{i}) for i = 1, …, k as in Lemma 3.1, then we have the following result (see [2, VI, 1.6, Corollaire 2]):

(3.4)  N(w) = {ν_{1}, …, ν_{k}} and N(w^{−1}) = {η_{1}, …, η_{k}}.

3.4. Stabilizers. We denote by C the fundamental chamber of W:

C = {x ∈ V | (x, α) ≥ 0 for all α ∈ Π}.

Formula (3.1) implies directly the well-known fact that

(3.5)  Stab_{W} (x) = ⟨s_{α} | α ∈ Π, (x, α) = 0⟩, for all x ∈ C,

and, as an easy consequence, the equally well-known fact that

(3.6)  Stab_{W} (x) = ⟨s_{β} | β ∈ Φ, (x, β) = 0⟩, for all x ∈ V.

For all j ∈ {1, …, n}, we set

W^{j} = ⟨s_{α_{i}} | i ∈ [n] \ {j}⟩,

so that

W^{j} = Stab_{W} (ω_{j}^{∨}).

3.5. Images of fundamental coweights. Let w = s_{β_{1}} ⋯ s_{β_{k}} be a reduced expression of w and η_{i} = s_{β_{k}} ⋯ s_{β_{i+1}}(β_{i}), for i = 1, …, k. Recall that, by definition, the left descents of w are the simple roots in N(w), and the right descents the simple roots in N(w^{−1}). For any j ∈ {1, …, n}, if w is a minimal length representative in the left coset wW^{j}, then α_{j} is the unique right descent of w. Hence, every reduced expression of w ends with s_{α_{j}} and every reduced expression of w^{−1} starts with s_{α_{j}}, i.e., in our notation,

(3.7)  β_{k} = η_{k} = α_{j}.

For γ ∈ Φ, let

Supp(γ) = {α_{i} ∈ Π | (γ, ω_{i}^{∨}) ≠ 0},

and, for all α ∈ Π, let

M_{α} = {γ ∈ Φ^{+} | α ∈ Supp(γ)}.

It is clear that if Supp(γ) ∩ N(w^{−1}) = ∅, then γ ∉ N(w^{−1}), by the linearity of w. Hence, if w is the minimal length representative in wW^{j}, we have

(3.8)  {η_{1}, …, η_{k}} ⊆ M_{α_{j}}.

Equivalently, (ω_{j}^{∨}, η_{i}) ≥ 1 for i = 1, …, k. Hence, by (3.2) and (3.3), we have

(3.9)  w_{h}(ω_{j}^{∨}) ≤^{∨} ω_{j}^{∨} − β_{k}^{∨} ⋯ − β_{h}^{∨}.

In particular, by (3.7), for each w ∉ Stab_{W} (ω_{j}^{∨})

(3.10)  w(ω_{j}^{∨}) ≤^{∨} ω_{j}^{∨} − α_{j}^{∨}.

4. Polar root polytopes that are zonotopes

In this section, we prove items (1), (2), and (3) of Theorem 2.2.

By Proposition 2.1, if P^{∗} is a zonotope, then the cones on the proper faces of P coincide with the faces of the hyperplane arrangement H_{P}, and P^{∗} = Zon\underline{0}(S), where S is a complete set of orthogonal vectors to the hyperplanes of H_{P}. Recall from [4, Proposition 3.2] that the hyperplanes in the arrangement H_{P} are of a very special form: there exists a subset H_{Φ} ⊆ {1, …, n} (depending on Φ) such that H_{P} = {w(ω_{k}^{∨})^{⊥} | w ∈ W, k ∈ H_{Φ}}. The sets H_{Φ} are given in [4, Table 2]. We will see that, when P^{∗} is a zonotope, the set S generating it is the W -orbit of a multiple of a single coweight.

From the definitions (see Section 2), it is clear that ZT(S) = Zon_{p}(S), with p = \frac{1}{2} ∑_{s∈S} s. For any λ ∈ V, we get

ZT(W · λ) = Zon\underline{0}(W · λ),

since \frac{1}{2} ∑_{w∈W} w(λ) is fixed by all elements in W and so must be the null vector \underline{0}. To prove items (1), (2), and (3) of Theorem 2.2, we need the following lemmas.

Lemma 4.1. Let S be a W -stable finite subset of V.

(1) ZT(S) ⊆ P^{∗} if and only if, for each X ⊆ S, (∑_{x∈X} x, θ) ≤ 1.

(2) If for each i ∈ {1, …, n} there exists X ⊆ S such that ∑_{x∈X} x = o_{i}, then P^{∗} ⊆ ZT(S).

Proof. (1) It is easy to see that the set of vertices of ZT(S) is a subset of {∑_{x∈X} x | X ⊂ S} (see for example [5, §2.3]). Hence, the claim follows from (2.2) and the stability of S under W, since all long roots are in the same W -orbit.

(2) By (2.3) and the stability of S under W, the assumption that for each i ∈ {1, …, n}, o_{i} = ∑_{x∈X} x, with X ⊆ S, implies that the set of vertices of P^{∗} is contained in the set of vertices of ZT(S), hence the claim. □

For each j ∈ {1, …, n}, we set

r_{j} = \frac{∥θ∥^{2}}{∥α_{j}∥^{2}} .

Lemma 4.2. Let j ∈ {1, …, n}. For each long root β, we have

(β, ω_{j}^{∨}) ≡ m_{j} mod r_{j}.

Proof. The claim is obvious if α_{j} is long, in which case r_{j} = 1. We recall that, if α_{j} is short, then (γ, α_{j}^{∨}) ∈ {−r_{j}, 0, r_{j}} for each long root γ. Hence the claim follows by induction on the length of w ∈ W such that w(θ) = β. □

For each subset S of Π, we denote by Φ(S) the standard parabolic subsystem of Φ generated by S, and by W (S) the Weyl group of Φ(S). We set

Φ_{0} = Φ(Π ∩ θ^{⊥}), W_{0} = W (Φ_{0}).

For each α ∈ Π, α ⊥ θ if and only if α is not connected to α_{0}, the extra root added to Π in the extended Dynkin diagram of Φ (see [7, §4.7], or [2, Chapter VI, n^{o} 4.3]). Since θ is in the fundamental chamber of W, we have

W_{0} = Stab_{W} (θ).

For each j ∈ {1, …, n}, we set

Φ_{0}^{j} = Φ((Π ∩ θ^{⊥}) \ {α_{j}}),  W_{0}^{j} = W (Φ_{0}^{j}),  q_{j} = [W_{0} : W_{0}^{j}].

It is clear that

W (Φ_{0}^{j}) = Stab_{W} (ω_{j}^{∨}) ∩ Stab_{W} (θ) = Stab_{W_{0}}(ω_{j}^{∨}).

Lemma 4.3. Let j ∈ {1, …, n} and x ∈ W · ω_{j}^{∨}. Then, (x, θ) = m_{j} if and only if x ∈ W_{0} · ω_{j}^{∨}. In particular,

|{x ∈ W · ω_{j}^{∨} | (x, θ) = m_{j}}| = q_{j}.

Proof. It is obvious that, if w ∈ W_{0}, then (w(ω_{j}^{∨}), θ) = m_{j}, since (w(ω_{j}^{∨}), θ) = (ω_{j}^{∨}, w^{−1}(θ)) = (ω_{j}^{∨}, θ). Conversely, assume x ∈ W · ω_{j}^{∨} and (x, θ) = m_{j}. Let w be the minimal length element in W such that x = w(ω_{j}^{∨}), and w = s_{β_{1}} ⋯ s_{β_{k}} be a reduced

expression of w. Then, by (3.9), we have x ≤^{∨} ω_{j}^{∨} − ∑_{i=1}^{k} β_{i}^{∨}, and hence, since θ is in

the fundamental chamber and (ω_{i}^{∨}, θ) = m_{i}, we obtain (θ, β_{i}^{∨}) = 0 for i = 1, …, k, i.e., w ∈ W_{0}. □

Proposition 4.4. Let j ∈ {1, …, n}. If r_{j} = m_{j}, then for all c ∈ ℝ,

ZT (W · c ω_{j}^{∨}) ⊆ P^{∗} if and only if c ≤ \frac{1}{q_{j}m_{j}} .

Equivalently, if r_{j} = m_{j}, then the hyperplane {(x, θ) = q_{j}m_{j}} is a supporting hyperplane for ZT (W · ω_{j}^{∨}).

Proof. By Lemma 4.3, for all w ∈ W \W_{0}, we have (w(ω_{j}^{∨}), θ) < m_{j}. Hence, by Lemma 4.2, we infer (w(ω_{j}^{∨}), θ) ≤ 0. It follows that

( ∑_{x∈W_{0}·ω_{j}^{∨}} x, θ) = q_{j}m_{j},

and that, for any other z ∈ ZT (W · ω_{j}^{∨}), we have (z, θ) ≤ q_{j}m_{j}. This proves the claim. □

For both types A_{n} and C_{n}, we have H_{P} = {w(ω_{1}^{∨})^{⊥} | w ∈ W }, and the values of Φ_{0}, Φ_{0}^{1}, q_{1}, m_{1}, and r_{1} are the following:

A_{n} : Φ_{0} = Φ_{0}^{1} = Φ(Π \ {α_{1}, α_{n}}), q_{1} = m_{1} = r_{1} = 1,
C_{n} : Φ_{0} = Φ_{0}^{1} = Φ(Π \ {α_{1}}), q_{1} = 1, m_{1} = r_{1} = 2.

By Proposition 4.4, from the previous computations, in types A_{n} and C_{n}, we have ZT (W · o_{1}) ⊆ P^{∗}. In both cases, also the other inclusion holds.

Theorem 4.5. Let Φ be of type A_{n} or C_{n}. Then

ZT (W · o_{1}) = P^{∗}.

Proof. To prove that ZT (W · o_{1}) ⊇ P^{∗}, we show that, for each k ∈ {1, …, n}, o_{k} is a sum of distinct elements in W · o_{1} (Lemma 4.1, (2)).

Let w_{i} := s_{i} ⋯ s_{1}, for i ∈ {1, …, n}, and w_{0} := e, the identity element. We show by

induction on k, 0 ≤ k < n, that ∑_{i=0}^{k} w_{i}(o_{1}) = o_{k+1}. This is trivially true for k = 0. If k > 0

and ∑_{i=0}^{h} w_{i}(o_{1}) = o_{h+1}, h < k < n, then w_{h}(o_{1}) = −o_{h} + o_{h+1} (where o_{0} := \underline{0}). We have

∑_{i=0}^{k} w_{i}(o_{1}) = o_{k} + w_{k}(o_{1}) = o_{k} + s_{k}w_{k−1}(o_{1}) = o_{k} + s_{k}(−o_{k−1} + o_{k}) = o_{k} − o_{k−1} + s_{k}o_{k}.

Since (o_{i}, α_{j}) = \frac{1}{m_{i}} δ_{i,j}, we have (s_{k}(o_{k}), α_{j}) = (o_{k}, s_{k}(α_{j})) = \frac{−(α_{k}^{∨},α_{j})}{m_{k}}, and hence

(∑_{i=0}^{k} w_{i}(o_{1}), α_{j}) = 0

for j = k and for all j with |j − k| > 1. Only the two cases j = k − 1 and j = k + 1 are left out. We have

(∑_{i=0}^{k} w_{i}(o_{1}), α_{k−1}) = −\frac{1}{m_{k−1}} − \frac{(α_{k}^{∨}, α_{k−1})}{m_{k}} = 0,

(note that, for type C_{n}, we need k < n), and

(∑_{i=0}^{k} w_{i}(o_{1}), α_{k+1}) = −\frac{(α_{k}^{∨}, α_{k+1})}{m_{k}} = 1.

So we get the assertion. □

While the property ZT (W · o_{1}) = P^{∗} is a property of the root system Φ, the property that P^{∗} is a zonotope is a property of the root polytope P. Hence, in view of P_{B_{3}}^{∗} ≅ P_{A_{3}}^{∗} and P_{G_{2}}^{∗} ≅ P_{A_{2}}^{∗}, we deduce that the polar root polytopes of types B_{3} and G_{2} are also zonotopes. It turns out that, also in these two cases, the polar root polytope is the zonotope generated by the orbit of a single vector, proportional to a coweight. More precisely, the following result holds.

Proposition 4.6. In types B_{3} and G_{2}, we have

P_{B_{3}}^{∗} = ZT (W · \frac{o_{3}}{2} ) ,
P_{G_{2}}^{∗} = ZT (W · \frac{o_{1}}{2} ) .

Proof. One inclusion follows by Proposition 4.4 since we have

B_{3} : Φ_{0} = Φ(Π \ {α_{2}}) ≅ A_{1} × A_{1},
Φ_{0}^{3} = Φ(Π \ {α_{2}, α_{3}}) ≅ A_{1}, q_{3} = 2, m_{3} = r_{3} = 2.
G_{2} : Φ_{0} = Φ(Π \ {α_{2}}) ≅ A_{1}, Φ_{0}^{1} = Φ(Π \ {α_{1}, α_{2}}) = ∅, q_{1} = 2, m_{1} = r_{1} = 3.

The other inclusion can be directly proved using Lemma 4.1, (2). □

5. Polar root polytopes that are not zonotopes

In this section, we prove item (4) of Theorem 2.2, i.e., that, for all root systems Φ other than those of types A_{n}, C_{n}, B_{3} and G_{2}, the polar root polytope P^{∗} is not a zonotope.

In fact, we show that, for all such root systems, the set of cones on the facets of the root polytope P is not equal to the set of closures of the regions of the hyperplane arrangement H_{P}. This is enough to show that P^{∗} cannot be a zonotope by Proposition 2.1, noting that:

(1) being the convex hull of the long roots in Φ, the polytope P is centrally symmetric with respect to the null vector \underline{0};

(2) the polar of a polytope which is centrally symmetric with respect to \underline{0} is centrally symmetric with respect to the null vector in the dual space;

(3) every zonotope which is centrally symmetric with respect to the null vector \underline{0} is of the form Zon\underline{0}(S), for an appropriate set S.

Recall that H_{P} is the central hyperplane arrangement determined by the (n − 2)-faces of P, i.e., H ∈ H_{P} if and only if H is a hyperplane containing \underline{0} and some (n − 2)-face of P. We show that some hyperplane in H_{P} meets the interior of some facet of P, for all root types other than A_{n}, C_{n}, B_{3}, G_{2}. More precisely, for each of the root systems Φ we are considering, we identify a hyperplane of H_{P} containing the barycenter of a facet of P and hence cutting that facet.

For the reader’s convenience, we recall from [3] that P has certain distinguished faces, which we call standard parabolic, which give a complete set of representatives of the W - orbits of faces of P (see [3, Corollary 4.3 and Theorem 5.11]). For each I ⊆ {1, …, n}, we let

F_{I} := Conv{α ∈ Φ^{+} | (ω_{i}^{∨}, α) = m_{i}, for all i ∈ I}

be the standard parabolic face associated with I. Here we need the standard parabolic facets, which are all of the form F_{i} := F_{{i}}, for i ∈ {1, …, n} (not all standard parabolic faces indexed by singletons are facets but all standard parabolic faces which are facets are indexed by singletons). The numbers i such that F_{i} are facets are those such that the extended Dynkin diagram is still connected after removing α_{i} (see [3, Section 5]). Moreover, we will use the fact that the barycenter of a standard parabolic facet F_{i} is a multiple of the corresponding fundamental weight ω_{i} (see [3, Lemma 4.2]).

As we already recalled in Section 4, there exists a subset H_{Φ} ⊆ {1, …, n} such that H_{P} = {w(ω_{k}^{∨})^{⊥} | w ∈ W, k ∈ H_{Φ}}. We call the hyperplanes (ω_{k}^{∨})^{⊥}, k ∈ H_{Φ}, standard hyperplanes.

For each irreducible root type other than A_{n}, C_{n}, B_{3}, G_{2}, we will exhibit a standard hyperplane containing the barycenter of a facet. Since each facet is of the form wF_{i}, and the barycenter of wF_{i} is a scalar multiple of w(ω_{i}), it suffices to find i, k ∈ {1, …, n} and w ∈ W such that: F_{i} is a facet, (ω_{k}^{∨})^{⊥} is a standard hyperplane, and w(ω_{i}) ⊥ ω_{k}^{∨}. In the following table, beside each root type other than A_{n}, C_{n}, B_{3} and G_{2}, in the first row we list all the standard parabolic facets and all the standard hyperplanes; in the further rows we write down explicitly a particular triple i, k, w such that i, k ∈ {1, …, n}, w ∈ W and w(ω_{i}) ⊥ ω_{k}^{∨}.

\begin{matrix}B_{n} \\ n ≥ 4 \end{matrix}  \begin{matrix}F_{1}, F_{n},  (ω_{1}^{∨})^{⊥}, (ω_{n}^{∨})^{⊥} \\ ω_{1} = α_{1} + ⋯ + α_{n},  s_{1}(ω_{1}) = (ω_{1} − α_{1}) ⊥ ω_{1}^{∨} \end{matrix}

\begin{matrix}D_{n} \\ n ≥ 4 \end{matrix}  \begin{matrix}F_{1}, F_{n−1}, F_{n},  (ω_{1}^{∨})^{⊥}, (ω_{n−1}^{∨})^{⊥}, (ω_{n}^{∨})^{⊥} \\ ω_{1} = α_{1} + ⋯ + α_{n}, s_{1}(ω_{1}) = (ω_{1} − α_{1}) ⊥ ω_{1}^{∨} \end{matrix}

E_{6}  \begin{matrix}F_{1}, F_{6},  (ω_{1}^{∨})^{⊥}, (ω_{6}^{∨})^{⊥} \\ ω_{1} = \frac{4}{3}α_{1} + α_{2} + \frac{5}{3}α_{3} + 2α_{4} + \frac{4}{3}α_{5} + \frac{2}{3}α_{6} \\ s_{2}s_{4}s_{3}s_{1}(ω_{1}) = ω_{1} − α_{1} − α_{3} − α_{4} − α_{2} ⊥ ω_{2}^{∨} \end{matrix}

E_{7}  \begin{matrix}F_{2}, F_{7},  (ω_{1}^{∨})^{⊥}, (ω_{2}^{∨})^{⊥} \\ ω_{7} = α_{1} + \frac{3}{2}α_{2} + 2α_{3} + 3α_{4} + \frac{5}{2}α_{5} + 2α_{6} + \frac{3}{2}α_{7} \\ s_{1}s_{3}s_{4}s_{5}s_{6}s_{7}(ω_{7}) = ω_{7} − α_{7} − α_{6} − α_{5} − α_{4} − α_{3} − α_{1} ⊥ ω_{1}^{∨} \end{matrix}

E_{8} F_{1}, F_{2}, (ω_{2}^{∨})^{⊥}, (ω_{8}^{∨})^{⊥} ω_{1} = 4α_{1} + 5α_{2} + 7α_{3} + 10α_{4} + 8α_{5} + 6α_{6} + 4α_{7} + 2α_{8} s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{1}(s_{2}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3})s_{1}(ω_{1}) = s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{1}(s_{2}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{3})(ω_{1} − α_{1}) = s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{1}(ω_{1} − θ_{6}) = (ω_{1} − α_{1} + α_{3} + α_{4} + α_{5} + α_{6} + α_{7} + α_{8}) − (θ_{6} − α_{7} − α_{8}) ⊥ ω_{8}^{∨} [here θ_{6} is the highest root of the type E_{6} root system generated by α_{1} …, α_{6}]

F_{4}  \begin{matrix}F_{4},  (ω_{4}^{∨})^{⊥} \\ ω_{4} = α_{1} + 2α_{2} + 3α_{3} + 2α_{4} \\ s_{4}s_{3}(s_{2}s_{3}s_{4})(ω_{4}) = s_{4}s_{3}(ω_{4} − α_{4} − α_{3} − α_{2}) = ω_{4} − 2α_{4} − 2α_{3} − α_{2} ⊥ ω_{4}^{∨} \end{matrix}

References

[1] A. Björner, F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, 231, Springer-Verlag, New York, 2005.

[2] N. Bourbaki, Groupes et Algèbres de Lie, Chapitres 4–6, Hermann, Paris, 1968.

[3] P. Cellini, M. Marietti, Root polytopes and Borel subalgebras, Internat. Math. Res. Notices, published online May 13, 2014; doi:10.1093/imrn/rnu070.

[4] P. Cellini, M. Marietti, Root polytopes and abelian ideals, J. Algebraic Combin., 39 (2014), no 3, 607-645

[5] C. De Concini, C. Procesi, Topics in Hyperplane Arrangements, Polytopes, and Box-Splines, Universitext, Springer, New York, 2011.

[6] B. Grünbaum, Convex Polytopes, Second edition edited by V. Kaibel, V. Klee and G. M. Ziegler, Graduate Texts in Mathematics, 221, Springer-Verlag, New York, 2003.

[7] J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge Univ. Press, Cambridge, 1990.

[8] G. M. Ziegler, Lectures on Polytopes, Updated seventh printing of the first edition, Graduate Texts in Mathematics, 152, Springer-Verlag, New York, 2007.

Paola Cellini, Dipartimento di Ingegneria e Geologia, Università di Chieti e Pescara, Viale Pindaro 42, 65127 Pescara PE, Italy

 E-mail address: pcellini@unich.it

Mario Marietti, Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona AN, Italy

 E-mail address: m.marietti@univpm.it