A rank-three condition for invariant (1, 2)-symplectic almost Hermitian structures

on flag manifolds

Nir Cohen^{∗} Department of Applied Mathematics

Caio J. C. Negreiros Department of Mathematics

Luiz A. B. San Martin^{†} Department of Mathematics Imecc Universidade Estadual de Campinas Cx. Postal 6065 13.083-970, Campinas – SP, Brasil

September 27, 2001

Abstract

This paper considers invariant (1, 2)-symplectic almost Hermitian structures on the maximal flag manifod associated to a complex semisimple Lie group G. The concept of cone-free invariant almost complex structure is introduced. It involves the rank-three subgroups of G, and generalizes the cone-free property for tournaments related to Sl (n, ℂ) case. It is proved that the cone-free property is necessary for an invariant almost-complex structure to take part in an invariant (1, 2)-symplectic almost Hermitian structure. It is also sufficient if the Lie group is not B_{l}, l ≥ 3, G_{2} or F_{4}. For B_{l} and F_{4} a close condition turns out to be sufficient.

∗Supported by CNPq grant no. 300019/96-3 †Supported by CNPq grant n\underline{∘} 301060/94-0

1 Introduction

The subject matter of this paper is the invariant almost Hermitian structures on the generalized flag manifolds associated to semi-simple complex Lie algebras and groups. Let G be a complex semi-simple Lie group and denote by 𝔽 = G/P the maximal flag manifold of G, where P is a Borel (minimal parabolic) subgroup of G. Alternatively, 𝔽 = U/T where U is a compact real form of G and T = P ∩ U a maximal torus.

The U-invariant almost Hermitian structures on 𝔽 have been studied recently in [2] and [10] with different methods. First, in [2] the group G is specialized to be Sl (n, ℂ), so that U = SU (n) and 𝔽 is identified with the manifold of complete flags of subspaces of ℂ^{n}. In this case there exists a natural bijection between the set of U-invariant almost complex structures on 𝔽 and n-player tournaments. Taking advantage of this bijection in [2] the invariant structures were studied with the aid of the combinatorics of tournaments (see also [3]).

On the other hand, [10] adopts the general set up, and studies invariant structures on the flag manifold associated to an arbitrary semi-simple complex group G. The methods of [10] are intrinsic in the sense that the combinatorial questions are resolved within the framework of root systems and Weyl groups.

In both papers the basic issue is the description of the (1, 2)-symplectic Hermitian structures. One of the main results is the derivation of a standard form for the corresponding invariant almost-complex structures. In [2] the standard form is given in terms of stair-shaped incidence matrices of tournaments, while in the general setting of [10] it is proved that the (1, 2)- symplectic Hermitian structures can be put in correspondence to the abelian ideals of a Borel subalgebra. Although the results of [10] extend those of [2] the proofs are completely independent. In particular, the notion of cone-free tournament – which plays a central role in [2] as a necessary and sufficient condition – does not appear in [10], leaving a gap in the development of the theory.

The purpose of this paper is to fill this gap, by extending the cone-free concept to the context of semi-simple Lie algebras, and analyzing its relation to the (1, 2)-symplectic structures. The cone-free property for the A_{l} series can be translated into a condition involving quadruples of roots, and thus makes sense in general (see Definition 3.1). We maintain the name of conefree for the property stated in terms of roots. It is related to the (1, 2)- symplectic structures as follows: An invariant Hermitian structure is a pair (J, Λ) with J a U-invariant almost complex structure and Λ an invariant Riemannian metric. The cone-free property refers to the invariant almost complex structures. Such a structure is said to be (1, 2)-admissible if there exists Λ such that (J, Λ) is (1, 2)-symplectic. We prove in Theorem 3.3 that the cone-free property is necessary for J to be (1, 2)-admissible. It is also sufficient if the semi-simple Lie algebra does not contain components of the types B_{l}, l ≥ 3, G_{2} or F_{4}. The point is that for the Lie algebras with rank ≥ 3, the cone-free property concerns the restriction of J to the rank-three subalgebras, and is equivalent to (1, 2)-admissibility in A_{3} and C_{3} but not in B_{3}. For this reason the correct condition for the Lie algebras B_{l} and F_{4} (which are the only ones which contain B_{3}) is that the restriction of J to any rank-three subalgebra is (1, 2)-admissible.

We regard our approach here as an application of the affine Weyl group characterization of the (1, 2)-symplectic structures, proved in [10]. Indeed we check that a certain J is (1, 2)-admissible by showing that it belongs to the class of affine invariant almost complex structures, which are defined by means of alcoves of the affine Weyl group (see Definition 6.3 below). It was proved in [10] that affine structures are (1, 2)-admissible and conversely. Through the affine structures we have access to the algebra of integer alcove coordinates developed by Shi [11]. This algebra is used to solve the combinatorial problems arising in the study of invariant structures.

The relation of cone-free tournaments with (1, 2)-symplectic structures on the classical flag manifolds is discussed in Mo and Negreiros [7] and Paredes [9]. The necessity of the cone-free property for (1, 2)-admissibility was first stated and proved in [7], with the aid of the moving frame method, while evidence for sufficiency was provided in [9], by checking small-sized tournaments. A general proof of sufficiency for tournaments of arbitrary size was given in [2].

Our attempt to understand the (1, 2)-symplectic structures was motivated by the study of harmonic maps into flag manifolds. However, after studying them in [10] it became clear that among the invariant almost Hermitian structures on the flag manifolds the (1, 2)-symplectic ones form an outstanding class, allowing the classification of the invariant structures given in [10].

2 Preliminaries

Let 𝔤 and 𝔥 be a simple complex Lie algebra and a Cartan subalgebra. Denote by Π the set of roots of the pair (𝔤, 𝔥), and let

𝔤_{α} = {X ∈ 𝔤 : ∀H ∈ 𝔥, [H, X] = α (H) X}

be the one-dimensional root space corresponding to α ∈ Π. Given α ∈ 𝔥^{∗} we let H_{α} be defined by α (·) = ⟨H_{α}, ·⟩, where ⟨·, ·⟩ stands for the Cartan-Killing form of 𝔤 and define 𝔥_{ℝ} to be the subspace spanned over ℝ by H_{α}, α ∈ Π. We fix once and for all a Weyl basis of 𝔤 which amounts to choosing for each α ∈ Π an element X_{α} ∈ 𝔤_{α} such that ⟨X_{α}, X_{−α}⟩ = 1, and [X_{α}, X_{β}] = m_{α,β}X_{α+β} with m_{α,β} ∈ ℝ, m_{−α,−β} = −m_{α,β} and m_{α,β} = 0 if α + β is not a root (see Helgason [4], Chapter IX).

Given a choice of positive roots Π^{+} ⊂ Π, denote by Σ the corresponding simple system of roots and let 𝔭 = 𝔥 ⊕ ∑_{α∈Π^{+}} 𝔤_{α} be the Borel subalgebra generated by Π^{+}. Let 𝔽 = G/P be the associated maximal flag manifold, where G is any connected complex Lie group with Lie algebra 𝔤 and P is the normalizer of 𝔭 in G. Let 𝔲 be the compact real form of 𝔤 spanned by i𝔥_{ℝ} and A_{α}, iS_{α}, α ∈ Π, where A_{α} = X_{α} − X_{−α} and S_{α} = X_{α} + X_{−α}. Denote by U the corresponding compact real form of G. By the transitive action of U on 𝔽 we can write 𝔽 = U/T where T = P ∩ U is a maximal torus of U.

If b_{0} stands for the origin of 𝔽, the tangent space at b_{0} identifies naturally with the subspace 𝔮 ⊂ 𝔲 spanned by A_{α}, iS_{α}, α ∈ Π. Analogously, the complex tangent space of 𝔽 is identified with 𝔮_{ℂ} = 𝔤 ⊖ 𝔥 ⊂ 𝔲, spanned by the root spaces 𝔤_{α}, α ∈ Π. Clearly, the adjoint action of T on 𝔤 leaves 𝔮 invariant.

2.1 Invariant metrics

A U-invariant Riemannian metric on 𝔽 is completely determined by its value at b_{0}, that is, by an inner product (·, ·) in 𝔮, which is invariant under the adjoint action of T. Such an inner product has the form (X, Y)_{Λ} = −⟨Λ (X) , Y ⟩ with Λ : 𝔮 → 𝔮 positive-definite with respect to the Cartan-Killing form. The inner product (·, ·)_{Λ} admits a natural extension to a symmetric bilinear form on the complexification 𝔮_{ℂ} of 𝔮. These complexified objects are denoted the same way as the real ones. The T -invariance of (·, ·)_{Λ} amounts to the elements of the standard basis A_{α}, iS_{α}, α ∈ Π, being eigenvectors of Λ, for the same eigenvalue. Thus, in the complex tangent space we have Λ (X_{α}) = λ_{α}X_{α} with λ_{α} = λ_{−α} > 0. We denote by ds_{Λ}^{2} the invariant metric associated with Λ. In the sequel we allow abuse of notation and write simply Λ instead of ds_{Λ}^{2}.

2.2 Invariant almost complex structures

In the sequel we use the abbreviation iacs for U-invariant almost complex structure on 𝔽. An iacs is completely determined by its value J : 𝔮 → 𝔮 in the tangent space at the origin. The map J satisfies J^{2} = −1 and commutes with the adjoint action of T on 𝔮. We denote by the same letter the real valued structure J and its complexification to 𝔮_{ℂ}. The invariance of J entails that J (𝔤_{α}) = 𝔤_{α} for all α ∈ Π. The eigenvalues of J are ±i and the eigenvectors in 𝔮_{ℂ} are X_{α}, α ∈ Π. Hence J (X_{α}) = iε_{α}X_{α} with ε_{α} = ±1 satisfying ε_{α} = −ε_{−α}. As usual the eigenvectors associated to +i are said to be of type (1, 0) while −i-eigenvectors are of type (0, 1). Thus the (1, 0) vectors are linear combinations of X_{α}, ε_{α} = +1, and the (0, 1) vectors are spanned by X_{α}, ε_{α} = −1.

An iacs on 𝔽 is completely prescribed by a set of signs {ε_{α}}_{α∈Π} with ε_{−α} = −ε_{−α}. In the sequel we allow some abuse of notation and identify the invariant structure on 𝔽 with J = {ε_{α}}.

Since 𝔽 is a homogeneous space of a complex Lie group it has a natural structure of a complex manifold. The associated integrable iacs J_{c} is given by ε_{α} = +1 if α < 0. The conjugate structure −J_{c} is also integrable. These are called the standard iacs.

2.3 Kähler form

It is easy to see that any invariant metric ds_{Λ}^{2} is almost Hermitian with respect to any iacs J, that is, ds_{Λ}^{2} (JX, JY) = ds_{Λ}^{2} (X, Y) (cf. [13], Section 8). Let Ω = Ω_{J,Λ} stand for the corresponding Kähler form

Ω (X, Y) = ds_{Λ}^{2} (X, JY) = −⟨ΛX, JY ⟩.

This form extends naturally to a U-invariant 2-form defined on the complexification 𝔮_{ℂ} of 𝔮, which we also denote by Ω. Its values on the basic vectors are:

Ω (X_{α}, X_{β}) = −iλ_{α}ε_{β}⟨X_{α}, X_{β}⟩.

Since ⟨X_{α}, X_{β}⟩ = 0 unless β = −α, Ω is not zero only on the pairs (X_{α}, X_{−α}), at which Ω takes the value iλ_{α}ε_{α}.

The following formula is well known (see [6]).

Lemma 2.1 Let ω be an invariant k-differential form on the homogeneous space L/H. Then

dω (X_{1}, …, X_{k+1}) = (k + 1) ∑_{i<j} (−1)^{i+j} ω ([X_{i}, X_{j}], X_{1}, …, X̂_{i}, … X̂_{j}, …, X_{k+1}) .

for X_{1}, …, X_{k+1} in the Lie algebra 𝔩 of L.

Specializing this lemma to the form Ω we get

−\frac{1}{3}dΩ (X, Y, Z) = −Ω ([X, Y], Z) + Ω ([X, Z], Y) − Ω ([Y, Z], X) (1)

From (1) an easy computation yields that dΩ (X_{α}, X_{β}, X_{γ}) is zero unless α + β + γ = 0. In this case

dΩ (X_{α}, X_{β}, X_{γ}) = −i3m_{α,β} (ε_{α}λ_{α} + ε_{β}λ_{β} + ε_{γ}λ_{γ})  (2)

with m_{α,β} as in Section 2 (cf. [10], Proposition 2.1).

Taking into account (2) we make the following distinction between two types of roots triples.

Definition 2.2 Let J = {ε_{α}} be an iacs. A triple of roots α, β, γ with α + β + γ = 0 is said to be a {0, 3}-triple if ε_{α} = ε_{β} = ε_{γ}, and a {1, 2}-triple otherwise.

Recall that an almost Hermitian manifold is said to be (1, 2)-symplectic (or quasi-Kähler) if

dΩ (X, Y, Z) = 0

when one of the vectors X, Y, Z is of type (1, 0) and the other two are of type (0, 1).The structure is (2, 1)-symplectic if the roles of (1, 0) and (0, 1) are interchanged. Accordingly, the structure is (i, j)-symplectic if the (i, j) component dΩ^{(i,j)} of dΩ is zero.

In our invariant setting we have the following criterion for an invariant pair (J, Λ) to be (1, 2)-symplectic, which follows immediately from formula (2), and the fact that X_{α} has type (1, 0) if ε_{α} = +1 and (0, 1) if ε_{α} = −1 (see [10], Proposition 2.3 and [13], Theorem 9.15).

Proposition 2.3 The invariant pair (J = {ε_{α}}, Λ = {λ_{α}}) is (1, 2)-symplectic if and only if

ε_{α}λ_{α} + ε_{β}λ_{β} + ε_{γ}λ_{γ} = 0

for every {1, 2}-triple {α, β, γ}.

In the sequel J is said to be (1, 2)-admissible if there exists Λ such that the pair (J, Λ) is invariant and (1, 2)-symplectic.

3 The cone-free property

Given a set of four roots q = {α, β, γ, δ} with α + β + γ + δ = 0 we say that a triple of roots {(u + v) , w_{1}, w_{2}} is extracted from q by u and v if {u, v, w_{1}, w_{2}} = {α, β, γ, δ}. Of course, any such triple satisfies (u + v) + w_{1} + w_{2} = 0. The cone-free condition is stated in terms of such triples.

Definition 3.1 Let J = {ε_{α}} be an iacs. We say that J is cone-free if the following condition is satisfied:

• If q = {α, β, γ, δ} contains no pairs of opposite roots and α+β+γ+δ = 0 then the number of {0, 3}-triples extracted from q is different from 1.

In this definition the hypothesis that the quadruples do not have opposite roots is redundant and is included only for emphasis sake. Indeed, suppose, for instance, that β = −α. Then δ = −γ, and the possible triples extracted from the quadruple are (α + γ, −α, −γ), (α − γ, −α, γ), (−α + γ, α, −γ) and (−α − γ, α, γ). It is easy to see that in this set {0, 3}-triples appear in pairs, independently of J.

Except when the root system is G_{2} the cone-free property is a condition on the rank-three subsystems of the root system. In fact, since there are no opposite roots in {α, β, γ, δ} the subspace V spanned by these roots is either two or three dimensional. However, it is easy to see that in the rank-two root systems A_{1} ⊕ A_{1}, A_{2} and B_{2}, which are different from G_{2}, there are no such sets of roots. Hence, the intersection of Π ∩ V is a rank-three root system if we are not in G_{2} (see Section 5 below for a discussion of G_{2}).

The explanation for the term cone in the above definition comes from the relation between iacs in the flag manifolds of the A_{l} series (the Lie algebras 𝔰𝔩 (n, ℂ), n = l + 1) and tournaments. Recall that an n-player tournament is a complete directed graph T = (N, E) where N is an ordered set, |N| = n, and E stands for the arrows of T. With each tournament T there is assigned its incidence matrix ε = ε_{T}, which is a real skew-symmetric matrix with all off-diagonal entries ±1. If (a, b) ∈ E we say that a wins against b and set ε_{ab} = 1 and ε_{ba} = −1.

On the other hand, in the standard realization, the roots of A_{l} are α_{jk}, 1 ≤ j ≠ k ≤ l + 1, with α_{kj} = −α_{jk}. Thus an iacs on the corresponding flag manifold is given by the signs ε_{jk} = ε_{α_{jk}} = ±1, j ≠ k. These numbers are assembled to form the incidence matrix ε of some tournament, establishing a one-to-one correspondence between the iacs on the maximal flag manifold of A_{l} and n-players tournaments.

A 3-cycle in a tournament is a 3-players subtournament {i, j, k} which forms the loop i → j → k → i. When T is the tournament associated to the iacs J, a 3-cycle {i, j, k} corresponds to the {0, 3}-triple {α_{ij}, α_{jk}, α_{ki}} (see [2]).

Now, up to isomorphism, there are four distinct 4-player tournaments. The two of them which contain a single 3-cycle are called cones. Each of them contains a cycle and a winner or a loser. The other equivalence classes of 4-player tournaments contain an even number of cycles (zero or two).

Proposition 3.2 In the maximal flag manifold associated to A_{n−1} = 𝔰𝔩 (n, ℂ), an iacs is cone-free in the sense of Definition 3.1 if and only if no 4-player subtournament of the associated tournament is a cone.

Proof: Assume first that an iacs J with corresponding tournament T is cone-free in the sense of Definition 3.1. Let {i, j, k, l} be a 4-player subtournament, and consider the corresponding set of four roots {α_{ij}, α_{jk}, α_{kl}, α_{li}} which satisfies

α_{ij} + α_{jk} + α_{kl} + α_{li} = 0.

From this set we extract the four triples {α_{ik}, α_{kl}, α_{li}}, {α_{jl}, α_{li}, α_{ij}}, {α_{ki}, α_{ij}, α_{jk}} and {α_{lj}, α_{jk}, α_{kl}}. Each one of these triples corresponds to a 3-player subtournament (e.g. {α_{ik}, α_{kl}, α_{li}} is associated to {i, k, l}), in such a way that {0, 3}-triples correspond to 3-cycles. Hence, by our generalized cone-free condition {i, j, k, l} is not a cone.

For the converse, note that a set of four roots {α, β, γ, δ} with α+β +γ + δ = 0 which do not contain opposite roots spans a rank-three root subsystem, and hence the set has the form {α_{ij}, α_{jk}, α_{kl}, α_{li}} for 1 ≤ i, j, k, l ≤ n. Repeating the above argument we get the generalized cone-free condition if the tournament has no cones.

We proceed now to prove that the cone-free condition is necessary for an iacs to be (1, 2)-admissible. Write dΩ^{{0,3}} = dΩ^{(0,3)} + dΩ^{(3,0)} and dΩ^{{1,2}} = dΩ^{(1,2)} + dΩ^{(2,1)}, so that

dΩ = dΩ^{{0,3}} + dΩ^{{1,2}}.

We get a necessary condition for dΩ^{{1,2}} = 0 by exploiting the fact that d^{2} = 0, computing formally d^{2}Ω (X_{α}, X_{β}, X_{γ}, X_{δ}). Analogous to the case of dΩ the only quadruples {α, β, γ, δ} of interest are those satisfying α + β + γ + δ = 0. Using the exterior derivative formula of Lemma 2.1, we get for these quadruples, that d^{2}Ω is the sum of the following six terms:

1. +m_{α,β}m_{γ,δ} (ε_{α+β}λ_{α+β} + ε_{γ}λ_{γ} + ε_{δ}λ_{δ})

2. −m_{α,γ}m_{β,δ} (ε_{α+γ}λ_{α+γ} + ε_{β}λ_{β} + ε_{δ}λ_{δ})

3. +m_{α,δ}m_{β,γ} (ε_{α+δ}λ_{α+δ} + ε_{β}λ_{β} + ε_{γ}λ_{γ})

4. +m_{β,γ}m_{α,δ} (ε_{β+γ}λ_{β+γ} + ε_{α}λ_{α} + ε_{δ}λ_{δ})

5. −m_{β,δ}m_{α,γ} (ε_{β+δ}λ_{β+δ} + ε_{α}λ_{α} + ε_{γ}λ_{γ})

6. +m_{γ,δ}m_{α,β} (ε_{γ+δ}λ_{γ+δ} + ε_{α}λ_{α} + ε_{β}λ_{β})

These terms cancel mutually (e.g. the coefficient of ε_{α}λ_{α} is m_{α,β}m_{γ,δ} + m_{β,γ}m_{α,δ} + m_{γ,α}m_{β,δ} which is known to be zero, see [4], Lemma III 5.3). In order to look at them closer let us take, for instance, the first one. The coefficient m_{α,β} is not zero if and only if α + β is a root. But α + β = − (γ + δ), so that both coefficients m_{α,β} and m_{γ,δ} are simultaneously zero or not. The same remark is true for the other terms. Next, in each term the sum appearing in braces has the form dΩ (X_{ξ}, X_{η}, X_{θ}) with (ξ, η, θ) a triple extracted from {α, β, γ, δ} if the coefficients m_{∗,∗} are not zero.

These comments yield an alternative proof of the following result of [7].

Theorem 3.3 A necessary condition for (J, Λ) to be (1, 2)-symplectic is that J is cone-free in the sense of Definition 3.1.

Proof: Let q = {α, β, γ, δ} be a root quadruple such that α + β + γ + δ = 0. Among the six terms above, those corresponding to {1, 2}-triples extracted from q are zero if dΩ^{{1,2}} = 0. On the other hand a term corresponding to an extracted {0, 3}-triple is not zero. Hence, for d^{2}Ω to be zero it is not possible to extract just one {0, 3}-triple.

4 Rank-three Lie algebras

The cone-free condition involves sets of four roots whose sum is zero in such a way that no two roots are opposite to each other. This has the consequence that the subspace spanned by the roots is three dimensional if the root system is not G_{2}. Hence, excluding G_{2} the cone-free condition refers to the rankthree subsystems of roots. The purpose of this preparatory section is to look at those rank-three root systems (mainly the irreducible ones A_{3}, B_{3} and C_{3}) required to study the cone-free condition in general root systems.

Note first that the rank-three reducible root systems are A_{1} ⊕ A_{1} ⊕ A_{1}, A_{1} ⊕ A_{2} and A_{1} ⊕ B_{2}. It is easy to check that any iacs in these root systems are (1, 2)-admissible, and thus satisfy the cone-free condition.

Concerning A_{3} = 𝔰𝔩 (4, ℂ), an iacs J on the maximal flag manifold corresponds to a 4-tournament T. By Proposition 3.2, J satisfies our cone-free condition if and only if T does not contain a cone. We know that such iacs are (1, 2)-admissible (see [2], [3]). Actually, the set of cone-free iacs has two equivalence classes, which are represented by the incidence matrices

( \begin{matrix}0 & 1 & 1 & 1 \\ −1 & 0 & 1 & 1 \\ −1 & −1 & 0 & 1 \\ −1 & −1 & −1 & 0 \end{matrix} )  ( \begin{matrix}0 & 1 & 1 & −1 \\ −1 & 0 & 1 & 1 \\ −1 & −1 & 0 & 1 \\ 1 & −1 & −1 & 0 \end{matrix} ) .  (3)

The class represented by the first matrix consists of the standard iacs.

Now, we look at the more delicate B_{3}. In its standard realization the positive root system is L ∪ S where L = {e_{i} ± e_{j} : 1 ≤ i < j ≤ 3} and S = {e_{i} : 1 ≤ i ≤ 3} are the sets of long and short roots, respectively.

The set L is isomorphic to the positive root system L_{3} = {α_{ij} : 1 ≤ i < j ≤ 3} of A_{3} via the bijection:

• Simple roots: α_{12} ↔ e_{2} − e_{3}; α_{23} ↔ e_{1} − e_{2}; α_{34} ↔ e_{2} + e_{3}.

• Height 2: α_{13} ↔ e_{1} − e_{3}; α_{34} ↔ e_{1} + e_{3}.

• Height 3: α_{14} ↔ e_{1} + e_{2}.

Now, let J = {ε_{α}} be a cone-free iacs in B_{3}. Its restriction J^{l} to L is also cone-free so that we can assume that it is represented by one of the two matrices in (3). It remains to see what happens at the short roots e_{1}, e_{2} and e_{3}. Regarding e_{3}, we can assume without loss of generality that ε_{e_{3}} = +1. In fact, the reflection r_{3} with respect to e_{3} leaves L_{3} invariant fixes the highest root e_{1} + e_{2}. Hence, we can replace J by r_{3} · J without affecting its values in L_{3} if J^{l} is represented by one of the matrices in (3). As to e_{1} and e_{2} we have

Lemma 4.1 ε_{e_{1}} = ε_{e_{2}}.

Proof: Consider the quadruple (−e_{1}) + (e_{1} − e_{2}) + (e_{2} − e_{3}) + e_{3} = 0. The triples extracted from it are {−e_{2}, e_{2} − e_{3}, e_{3}}, {−e_{1} + e_{3}, e_{1} − e_{2}, e_{2} − e_{3}}, {e_{2}, −e_{1}, e_{1} − e_{2}} and {e_{1} − e_{3}, −e_{1}, e_{3}}.

Note that {−e_{1} + e_{3}, e_{1} − e_{2}, e_{2} − e_{3}} is a {1, 2}-triple. Suppose that ε_{e_{2}} = −1. Then {−e_{2}, e_{2} − e_{3}, e_{3}} is a {0, 3}-triple, and {e_{2}, −e_{1}, e_{1} − e_{2}} is a {1, 2}-triple, forcing the last triple to be {0, 3}, which implies ε_{e_{1}} = −1. The root e_{1} + e_{2} does not appear in the extracted triples, ensuring that our arguments are independent of the choice of J^{l}.

On the other hand from the quadruple (e_{1} − e_{2}) + (e_{2} + e_{3}) + (−e_{1}) + (−e_{3}) = 0, the only extracted triple which is not automatically of type {0, 3} is {e_{2}, e_{1} − e_{2}, −e_{1}}. Hence, this set must be a {1, 2}-triple, so that ε_{e_{2}} = +1 implies ε_{e_{1}} = +1. Again the extracted triples do not involve e_{1} + e_{2}, hence it is immaterial which of the J^{l}’s we consider.

We arrive at the following description of the cone-free iacs on B_{3}.

Proposition 4.2 Denote by M (J) the set of positive roots α of B_{3} such that ε_{α} = −1. Fixing the choices of J^{l} given by (3) and ε_{e_{3}} = +1, the possible iacs satisfying the cone-free condition are:

1. M (J_{1}) = ∅.

2. M (J_{2}) = {e_{1} + e_{2}}.

3. M (J_{3}) = {e_{1}, e_{2}}.

4. M (J_{4}) = {e_{1}, e_{2}, e_{1} + e_{2}}.

Among them the only (1, 2)-admissible iacs are J_{1} and J_{2}.

Proof: The (1, 2)-admissibility of J_{1} and J_{2} is a consequence of the abelian ideal shape of [10]. On the other hand, J_{3} and J_{4} are not (1, 2)-admissible. To see this consider the triples {e_{1}, e_{1} + e_{3}, −e_{3}} and {e_{1}, e_{3}, −e_{1} − e_{3}}. They are {1, 2}-triples for both J_{3} and J_{4}. Now, assume that Λ = {λ_{α}} is (1, 2)- symplectic with respect to J_{3} or J_{4}. Then λ_{e_{1}+e_{3}} = λ_{e_{1}} + λ_{e_{3}} and λ_{e_{3}} = λ_{e_{1}} + λ_{e_{1}+e_{3}}, forcing λ_{e_{1}} = 0, a contradiction.

Finally, it is straighforward but cumbersome to verify that J_{3} and J_{4} indeed satisfy the cone-free condition. One must write down the quadruples of roots of B_{3} summing up zero, and their extracted triples, and check that the {0, 3}-triples do not appear isolated.

The discussion of C_{3} follows the same pattern as that of B_{3}. In the standard realization of C_{3}, its short roots coincide with the long roots of B_{3}, whereas the long roots are given by ±2e_{i}, i = 1, 2, 3. Again we can assume that the restriction J^{s} of a cone-free iacs J to the short roots has one of the incidence matrices (3). Also, after applying the reflection with respect to e_{3} we can assume that ε_{2e_{3}} = +1. With the aid of these choices we can check the quadruples of C_{3} and prove the

Proposition 4.3 Denote, as before, by M (J) the set of positive roots α of C_{3} such that ε_{α} = −1. Fixing the above choices of J^{s} and ε_{e_{3}} = +1, the possible iacs satisfying the cone-free condition are:

1. M (J_{1}) = ∅.

2. M (J_{2}) = {e_{1} + e_{2}, 2e_{1}}.

3. M (J_{3}) = {2e_{2}, e_{1} + e_{2}, 2e_{1}}.

Each M (J_{i}), i = 1, 2, 3, is an abelian ideal of the set of positive roots, so that the cone-free iacs are (1, 2)-admissible.

Proof: The proposition is a consequence of the following implications:

ε_{e_{1}+e_{2}} = +1 ⇒ ε_{2e_{2}} = ε_{2e_{1}} = +1, ε_{e_{1}+e_{2}} = −1 ⇒ ε_{2e_{1}} = −1.

which are easy consequences of the cone-free property applied to the quadruples {e_{1} − e_{2}, 2e_{2}, −e_{2} + e_{3}, −e_{1} − e_{3}}, {e_{1} − e_{2}, e_{1} − e_{3}, e_{2} + e_{3}, −2e_{1}} and {e_{1} − e_{2}, e_{2} − e_{3}, e_{1} + e_{3}, −2e_{1}}, respectively.

5 G_{2}

As mentioned above, G_{2} is the only rank-two root system where the cone-free condition is not vacuous. For the sake of completeness we analyze here the iacs on G_{2} which satisfy this condition. We write the positive roots as

\begin{matrix}α_{1} \\ α_{2} \end{matrix} α_{1} + α_{2} α_{1} + 2α_{2} α_{1} + 3α_{2} 2α_{1} + 3α_{2}.

The set of short roots {±α_{2}, ± (α_{1} + α_{2}) , ± (α_{1} + 2α_{2})} is an A_{2}-root system. Let J be an iacs on G_{2} and denote by J^{s} its restriction to the set of short roots. In A_{2} there are two equivalence classes of iacs, so that we can assume without loss of generality that J^{s} is one of the following two iacs:

1. J_{1}^{s} = {ε_{α_{2}} = +1, ε_{α_{1}+α_{2}} = +1, ε_{α_{1}+2α_{2}} = +1}.

2. J_{2}^{s} = {ε_{α_{2}} = +1, ε_{α_{1}+α_{2}} = +1, ε_{α_{1}+2α_{2}} = −1}.

Denote by r the reflection with respect to α_{1}. It satisfies rα_{2} = α_{1} + α_{2} and r (α_{1} + 2α_{2}) = α_{1} + 2α_{2}. This implies that r leaves J^{s} invariant. Hence, we may assume that ε_{α_{1}} = +1.

Now, assuming that J satisfies the cone-free condition, it remains to determine the values of ε_{α_{1}+3α_{2}} and ε_{2α_{1}+3α_{2}}. Up to change of signs there are the following three zero-sum root quadruples:

1. q_{1}: (α_{1}) + (α_{2}) + (α_{1} + 2α_{2}) + (−2α_{1} − 3α_{2}) = 0.

2. q_{2}: (α_{2}) + (α_{1} + α_{2}) + (α_{1} + α_{2}) + (−2α_{1} − 3α_{2}) = 0.

3. q_{3}: (α_{1}) + (α_{1} + 3α_{2}) + (−α_{1} − α_{2}) + (−α_{1} − 2α_{2}) = 0.

First suppose that J^{s} = J_{1}^{s}. Writing down the triples extracted from q_{3}, it is straighforward to check that ε_{α_{1}+3ε_{2}} = −1 implies that ε_{2α_{1}+3ε_{2}} = −1.

Hence, the possible cone-free iacs are {+ \atop +} ++++, {+ \atop +} +++− and {+ \atop +} ++−−.

By the abelian ideal property stated in [10], these iacs are (1, 2)-admissible, and hence they are indeed cone-free.

Suppose now that J^{s} = J_{2}^{s}. Looking at the triples extracted from q_{1} it is easy to see that ε_{α_{1}+3α_{2}} = +1 implies ε_{2α_{1}+3α_{2}} = +1. Since there are no

other restrictions, the cone-free iacs are {+ \atop +} + − + +, {+ \atop +} + − − + and

{+ \atop +} + − − −. The last one is (1, 2)-admissible, whereas, similar to the B_{3}

case, one can check that the first two are not (1, 2)-admissible. (We remark that in checking the cone-free property the quadruple q_{2} is irrelevant, since in it each extracted triple appears twice.)

6 The affine Weyl group

In this section we recall the definition of the affine iacs introduced in [10]. These structures are constructed by counting hyperplanes separating a given alcove and the basic one. We refer to Humphreys [5] as a basic source for the affine Weyl group. Consider the subspace 𝔥_{ℝ} introduced in Section 2. To conform with the usual notation we often identify 𝔥_{ℝ} with its dual 𝔥_{ℝ}^{∗} and write ⟨x, α⟩ instead of α (x), x ∈ 𝔥_{ℝ}, α ∈ 𝔥_{ℝ}^{∗}. Given α ∈ Π and k ∈ ℤ define the affine hyperplane

H (α, k) = {x ∈ 𝔥_{ℝ} : ⟨x, α⟩ = k}.

The complement A of the set of hyperplanes H (α, k), α ∈ Π, k ∈ ℤ, is the disjoint union of connected open simpleces called alcoves. Given an alcove A and a root α, by definition there exists an integer k_{α} = k_{α} (A) such that

k_{α} < ⟨x, α⟩ < k_{α} + 1 x ∈ A.

Of course, k_{α} = [α (x)] for any x ∈ A where [a] denotes the integer part of the real number a. According to Shi [11], the integers k_{α} (A) are called the coordinates of the alcove A. An alcove is completely determined by its coordinates. A necessary and sufficient condition for k_{α}, α ∈ Π, to be the coordinates of an alcove are given by the inequalities below. In writing down these inequalities we must look Π as the set of co-roots of another root system Π̃:

Π = \{α^{∨} = \frac{2α}{⟨α, α⟩} : α ∈ Π̃\}.

The root system is normalized so that |α| = 1 if α is a short root.

Proposition 6.1 A set of integers k_{α}, α ∈ Π̃^{+}, form the coordinates of an alcove if and only if for every pair of roots α, β ∈ Π̃ such that α + β ∈ Π̃, the following inequalities hold:

\begin{aligned}|α|^{2}k_{α} + |β|^{2}k_{β} + 1 & ≤ & |α + β|^{2} (k_{α+β} + 1) \\ & ≤ & |α|^{2}k_{α} + |β|^{2}k_{β} + |α|^{2} + |β|^{2} + |α + β|^{2} − 1. \end{aligned} (4)

Proof: See [11], Lemma 1.2 and Proposition 5.1.

Remark: It is easy to see that the inequalities in this proposition are equivalent to

|α|^{2} k_{α} + |β|^{2} k_{β} + |γ|^{2} k_{γ} ≤ 1.

For later reference we note also the following easy necessary condition.

Lemma 6.2 A necessary condition for the integers k_{α} ∈ ℤ, α ∈ Π, to be the coordinates of an alcove is that k_{α+β} is either k_{α} +k_{β} or k_{α} +k_{β} +1 whenever α, β and α + β are roots.

Proof: We have, for all x ∈ A, k_{α} < ⟨x, α⟩ < k_{α}+1 and k_{β} < ⟨x, β⟩ < k_{β} +1, so that

k_{α} + k_{β} < ⟨x, α + β⟩ < k_{α} + k_{β} + 2.

Hence, the integer part of ⟨x, α + β⟩ is either k_{α+β} = k_{α} + k_{β} or k_{α+β} = k_{α} + k_{β} + 1.

Definition 6.3 Given an alcove A with coordinates {k_{α} : α ∈ Π}, the iacs J (A) = {ε_{α} (A)} is defined by ε_{α} (A) = (−1)^{k_{α}}. We say that J is an affine iacs if it has the form J = J (A) for some alcove A.

Note that J (A) is indeed an iacs, since k_{−α} = −k_{α} −1, so that ε_{−α} (A) = −ε_{α} (A). The following theorem is one of the main results in [10]. It provides the criterion which will be used in the sequel for ensuring that iacs are (1, 2)- admissible.

Theorem 6.4 An iacs J is (1, 2)-admissible if and only if it is affine.

7 Simply-laced root systems

In this section we prove that the cone-free condition is sufficient for an iacs to be (1, 2)-admissible, in case the algebra 𝔤 has a simply-laced Dynkin diagram, i.e. Π = A_{l}, D_{l}, E_{6}, E_{7} or E_{8}. The doubly-laced case will be treated in Section 8. We use the equivalence between the affine and (1, 2)-admissible iacs, as stated in Theorem 6.4, and construct an alcove A such that J = J (A) if J satisfies the cone-free condition. Thus the purpose of this section is to prove the following statement.

Theorem 7.1 Let Π be a simply-laced root system, and suppose that J = {ε_{α}} is a cone-free iacs on 𝔽. Then J is affine.

The proof will consist of several steps. By definition of affine iacs we must find a set of integers {k_{α} : α ∈ Π} satisfying the inequalities of Shi (4) such that ε_{α} = (−1)^{k_{α}}, α ∈ Π. In a simply-laced root system the roots have the same length, simplifying these inequalities. In fact, we have the following equivalent condition for a set k_{α} to be the coordinates of an alcove.

Lemma 7.2 Let Π be simply-laced. Then the integers k_{α} ∈ ℤ, α ∈ Π, form the coordinates of an alcove if and only if either k_{α+β} = k_{α} + k_{β} or k_{α+β} = k_{α} + k_{β} + 1 when α, β and α + β are roots.

Proof: The condition is necessary by Lemma 6.2. Conversely, if Π is simplylaced, the |·|^{2} appearing in inequalities (4) are equal to 1, hence they reduce to

k_{α} + k_{β} ≤ k_{α+β} ≤ k_{α} + k_{β} + 1.

Therefore, these inequalities are satisfied by k_{α}, α ∈ Π, if they are under the conditions of the statement.

Before proceeding we prove some lemmas.

Lemma 7.3 Let Π_{∗} ⊂ Π be a root subsystem of Π. Then Π_{∗}^{+} = Π_{∗} ∩ Π^{+} is a choice of positive roots in Π_{∗}.

Proof: There exists γ ∈ 𝔥_{ℝ}^{∗} such that

Π^{+} = {α ∈ Π : ⟨α, γ⟩ > 0}.

Of course, ⟨α, γ⟩ ≠ 0 for all α ∈ Π. Let γ_{1} be the orthogonal projection of γ onto the subspace of 𝔥_{ℝ}^{∗} spanned by Π_{∗}. For β ∈ Π_{∗} we have ⟨β, γ⟩ = ⟨β, γ_{1}⟩, so that ⟨β, γ_{1}⟩ ≠ 0 for all β ∈ Π_{∗}. Hence, γ_{1} is regular for Π_{∗}, implying that

Π_{∗}^{+} = {β ∈ Π_{∗} : ⟨β, γ_{1}⟩ > 0}

is a choice of positive roots in Π_{∗}. Using again ⟨β, γ⟩ = ⟨β, γ_{1}⟩, β ∈ Π_{∗}, it follows that Π_{∗}^{+} = Π_{∗} ∩ Π^{+}, proving the lemma.

Lemma 7.4 Fix a simple system of roots Σ, and let J = {ε_{α}} be an affine iacs. Suppose that a set of integers m_{α} ∈ ℤ, α ∈ Σ, satisfies ε_{α} = (−1)^{m_{α}}. Then there exists an alcove A such that J = J (A) and k_{α} (A) = m_{α}, α ∈ Σ.

Proof: Put Σ = {α_{1}, …, α_{l}} and define {ω_{1}, …, ω_{l}} by ⟨α_{i}, ω_{j}⟩ = δ_{ij}. Also, let A^{1} be an alcove such that J = J (A^{1}), that is, ε_{α} = (−1)^{k_{α}(A^{1})}. Since ε_{α} = (−1)^{m_{α}}, the integers m_{α_{i}} − k_{α_{i}} (A^{1}) are even. Now, a translation t_{λ} with λ spanned over ℤ by ω_{i}, i = 1, …, l, maps alcoves into alcoves, and the coordinates are changed according to

k_{α} (t_{λ}A) = ⟨α, λ⟩ + k_{α} (A) .  (5)

Take λ = d_{α_{1}}ω_{1}+⋯+d_{α_{l}}ω_{l}, with d_{α_{i}} = m_{α_{i}} −k_{α_{i}} (A^{1}). Then the coordinates of A = t_{λ}A^{1} are k_{α} (A) = ⟨α, λ⟩ + k_{α} (A^{1}), and since ⟨α, λ⟩ is even for all α, we conclude that J = J (A). Furthermore, for a simple roots α_{i} we have k_{α_{i}} (A) = d_{α_{1}} + k_{α} (A^{1}) = m_{α_{i}}, proving the lemma.

Now, for proving Theorem 7.1 we construct k_{α}, α ∈ Π, by induction on the height of α. Thus let us fix once and for all a simple system of roots Σ with Π^{+} the corresponding set of positive roots. Then given J = {ε_{α}} define:

1. Let α ∈ Σ. Then k_{α} = (1 − ε_{α}) /2.

2. Let α, β, γ ∈ Π^{+} be such that α = β + γ. Then

k_{α} = k_{β} + k_{γ} + \frac{1 − (−1)^{k_{β}+k_{γ}} ε_{α}}{2} .  (6)

3. Let α ∈ −Π^{+}. Then k_{α} = −k_{−α} − 1.

A case by case analysis shows easily that the coordinates {k_{α}} so defined satisfy ε_{α} = (−1)^{k_{α}}. Also, the condition of Lemma 7.2 is readily satisfied. The point is to show that k_{α} is independent of the decomposition α = β + γ used in (6). We prove this by induction on the height h (α) of α ∈ Π^{+}. If h (α) = 1, the root is simple, and no decomposition α = β + γ, β, γ ∈ Π^{+} exists, hence k_{α} is well defined.

Now take α ∈ Π^{+} such that α = β_{1} + γ_{1} = β_{2} + γ_{2}, β_{i}, i = 1, 2, positive roots, and hence having height smaller than h (α). By the inductive hypothesis k_{β_{i}}, k_{γ_{i}}, i = 1, 2 are well defined. We must show that k_{β_{1}} +k_{γ_{1}} = k_{β_{2}} +k_{γ_{2}}.

Denote by V ⊂ 𝔥_{ℝ}^{∗} the subspace spanned by β_{1}, γ_{1}, β_{2} and γ_{2}. We have dim V = 2 or 3.

In case dim V = 2, the subset V ∩ Π is a rank-two system of roots, containing two roots (β_{1} and γ_{1}) whose sum is a root. Hence V ∩ Π is irreducible, and since our original root system is simply-laced, it follows that V ∩ Π is an A_{2} system. Now, in A_{2} a root is written uniquely as a sum of two roots, hence there is nothing to prove.

Suppose then that dim V = 3, and let Π_{∗} = V ∩ Π be the corresponding rank-three system. Since the roots in Π have the same length, either Π_{∗} = A_{1} ⊕ A_{2} or Π_{∗} = A_{3}. Again, there is nothing to prove in the A_{1} ⊕ A_{2} case.

Assuming that Π_{∗} = A_{3}, let J_{∗} be the restriction of J to Π_{∗}. Then J_{∗} is (1, 2)-admissible and hence affine.

Now, by Lemma 7.3, Π_{∗}^{+} = Π_{∗}∩Π^{+} is a positive root system. Let Σ_{∗} ⊂ Π^{+} the corresponding set of simple roots.

Lemma 7.5 α is the highest root in Π_{∗}^{+}.

Proof: Write the positive roots of A_{3} as α_{ij}, 1 ≤ i < j ≤ 4, so that α is one of these roots. It is not a simple root, since α = β_{1} + γ_{1} with β_{1}, γ_{1} ∈ Π_{∗}^{+}. Also, a root of height 2 in A_{3} is written uniquely as a sum of two positive roots. Hence the height of α in Π_{∗}^{+} is not 2, since dim V = 3 and α = β_{1} + γ_{1} = β_{2} + γ_{2}. Therefore, the height of α in Π_{∗}^{+} is three, that is, α is the highest root.

By this lemma and the equality Π_{∗}^{+} = Π_{∗} ∩ Π^{+} we conclude that the height of α in Π^{+} is bigger than the height (in Π^{+}) of any γ ∈ Σ_{∗}. Hence, the inductive hypothesis ensures that k_{γ} is well defined for γ ∈ Σ_{∗}.

Now, by the cone-free assumption, there exists an alcove A^{∗} in the affine system of Π_{∗} such that J_{∗} = J (A^{∗}). By Lemma 7.4 we can choose A^{∗} so that k_{γ} (A^{∗}) = k_{γ} for all γ ∈ Σ_{∗}. The integers k_{δ} (A^{∗}), δ ∈ Π_{∗}, satisfy the conditions of Lemma 7.2. Also, J_{∗} = J (A^{∗}) is the restriction of J to Π_{∗}. Hence starting with k_{γ} (A^{∗}) = k_{γ}, γ ∈ Σ_{∗}, the values of k_{δ} (A^{∗}), δ ∈ Π_{∗}^{+}, are determined according to the rules used to define k_{α}. This means that within Π_{∗}^{+}, k_{α} is well defined. However, the decompositions α = β_{1} + γ_{1} = β_{2} + γ_{2} are inside Π_{∗}^{+}, so that the value of k_{α} does not depend upon one of these decompositions, concluding the proof of Theorem 7.1.

Corollary 7.6 In a simply-laced situation let A^{1} and A^{2} be alcoves such that J (A^{1}) = J (A^{2}). Then there exists λ with ⟨λ, α⟩ ∈ 2ℤ for every root α such that A^{2} = t_{λ}A^{1}.

Proof: As in the proof of Lemma 7.4 let {α_{1}, …, α_{l}} be a simple system of roots and {ω_{1}, …, ω_{l}} its dual basis, and put

λ = d_{α_{1}}ω_{1} + ⋯ + d_{α_{l}}ω_{l}

with d_{α_{i}} = k_{α_{i}} (A^{2}) − k_{α_{i}} (A^{1}). The assumption J (A^{1}) = J (A^{2}) implies that d_{α_{i}}, i = 1, …, l, are even integers, so that ⟨λ, α⟩ ∈ 2ℤ for all α ∈ Π. According to the change of coordinates formula (5), to see that A^{2} = t_{λ}A^{1} we must check that k_{α} (A^{2}) = ⟨λ, α⟩ + k_{α} (A^{1}) for every positive root α. This is done by induction on the height of α: If α is simple, the equality holds by definition of λ. On the other hand if α = β + γ with β, γ ∈ Π^{+}, we assume by induction that the equality is true for β and γ. In particular, k_{β} (A^{1}) + k_{γ} (A^{1}) ≡ k_{β} (A^{2}) + k_{γ} (A^{2}) mod2. Now, from the construction performed in the proof of Theorem 7.1, it follows that formula (6) holds for both sets of integers k_{δ} (A^{1}) and k_{δ} (A^{2}), δ ∈ Π, with the same ε_{α}. Therefore, k_{α} (A^{i}) − k_{β} (A^{i}) − k_{γ} (A^{i}) is independent of i = 1, 2. Thus applying the inductive hypothesis we get

k_{α} (A^{2}) = k_{α} (A^{1}) + ⟨λ, β⟩ + ⟨λ, γ⟩ = k_{α} (A^{1}) + ⟨λ, α⟩,

concluding the proof.

Remark: It is worth mentioning that with Theorem 7.1 we get an indirect proof of a result in tournament theory, namely Theorem 3.5 of [2] which asserts that the vertices of a tournament T can be rearranged so that its incidence matrix becomes stair-shaped in case T has no cones. In fact, this result follows by piecing together Proposition 3.2, Theorem 7.1 and the results of [10] on invariant almost complex structures (see [10], Theorem 4.12).

8 Doubly-laced root systems

In this section we look at the cone-free property for the doubly-laced diagrams (B_{l}, C_{l} and F_{4}). The final result for C_{l} differs from B_{l} and F_{4}.

Theorem 8.1 Let Π be a root system and J an iacs on the corresponding maximal flag manifold.

1. Suppose that Π is C_{l}. Then J is affine (and hence (1, 2)-admissible) if and only if J satisfies the cone-free property.

2. Suppose that Π is B_{l} or F_{4}, and that the restriction of J to any rankthree subsystem is affine. Then J is affine, and hence (1, 2)-admissible.

Remark: The rank-three condition for B_{l} and F_{4} is equivalent to J being cone-free together with the additional assumption that the restriction of J to any B_{3}-subsystem is affine. This assumption is not required for C_{l} because it does not contain B_{3}-subsystems.

The proof of Theorem 8.1 uses the corresponding result for simply-laced diagrams (Theorem 7.1), applied to the set of short roots of Π. Let Π^{s} and Π^{l} denote the sets of short roots and long roots, respectively. We have the disjoint union Π = Π^{s}∪Π^{l}. Both sets Π^{s} and Π^{l} are simply-laced root systems (for example, in Π = B_{l}, Π^{l} is a D_{l} while Π^{s} is reducible with l orthogonal components).

Let J^{s} stand for the restriction of J to Π^{s}. Clearly, under the conditions of Theorem 8.1, J^{s} satisfies the cone-free assumption of Theorem 7.1, so that J^{s} is affine in Π^{s}. Thus there are integers k_{α}, α ∈ Π^{s}, with ε_{α} = (−1)^{k_{α}} such that k_{α} form the coordinates of an alcove in Π^{s}. We shall prove Theorem 8.1 by extending these coordinates to Π^{l}.

For the doubly-laced root systems, we have the following characterization of the coordinates of alcoves, which is obtained from the inequalities of Shi after a case-by-case analysis.

Proposition 8.2 In a doubly-laced root system a set of integers k_{α}, α ∈ ℤ, are the coordinates of an alcove if and only if the following inequalities are satisfied. Each inequality is satisfied by a triple of roots as indicated, where s means short root and l long root.

1. (α, β, α + β) = (l, l, l): k_{α} + k_{β} ≤ k_{α+β} ≤ k_{α} + k_{β} + 1 2. (α, β, α + β) = (s, s, s): 2k_{α} + 2k_{β} + 1 ≤ 2k_{α+β} + 2 ≤ 2k_{α} + 2k_{β} + 5

3. (α, β, α + 2β) = (l, s, l): k_{α} + 2k_{β} ≤ k_{α+2β} ≤ k_{α} + 2k_{β} + 2

4. (α, β, (α + β) /2) = (l, l, s): k_{α} + k_{β} ≤ 2k_{(α+β)/2} + 1 ≤ k_{α} + k_{β} + 2

Proof: See [10], Proposition 5.4.

Lemma 8.3 Suppose the doubly-laced root system Π is irreducible, and let α be a long root. Then there exists a short root β such that ⟨α, β⟩ ≠ 0.

Proof: There are a long root α_{1} and a short root β_{1} such that ⟨α_{1}, β_{1}⟩ ≠ 0 (look e.g. at the Dynkin diagram). The Weyl group W leaves invariant both Π^{l} and Π^{s}, and since Π is irreducible, these subsets are orbits of W. Hence, for a long root α there exists w ∈ W with α = wα_{1}. Thus, ⟨α, wβ_{1}⟩ ≠ 0.

Lemma 8.4 Let α be a long root. Then there are short roots β and γ such that

α = β + γ.

Proof: By the previous lemma there exists a short root β with ⟨α, β⟩ ≠ 0. Let Π_{2} be the intersection of Π with the subspace spanned by α and β. It has rank-two and contains two roots α and β of different length with ⟨α, β⟩ ≠ 0. Hence Π_{2} is a B_{2}. The lemma follows then by looking at the roots of B_{2}.

Now, we write down the conditions for a set of integers to be the coordinates of an alcove in terms of the short and long roots.

Lemma 8.5 In a doubly-laced root system Π the following conditions are necessary and sufficient for a set of integers k_{α}, α ∈ Π, to be the coordinates of an alcove:

1. The integers k_{α}, α ∈ Π^{l}, are the coordinates of an alcove in the root system of the long roots.

2. The integers k_{α}, α ∈ Π^{s}, are the coordinates of an alcove in the root system of the short roots.

3. Take a long root α = β + γ with β and γ short roots. Then either k_{α} = k_{β} + k_{γ} or k_{α} = k_{β} + k_{γ} + 1.

Proof: Suppose first that k_{α}, α ∈ Π, are the coordinates of an alcove. Then the first and second sets of inequalities in Proposition 8.2 together with the corresponding inequalities in the simply-laced case show that the restriction of k_{α} to the long roots as well as to the short roots are coordinates of alcoves. Furthermore, the last condition is necessary by Lemma 6.2.

We prove sufficiency by showing that the three conditions of the lemma imply the inequalities of Proposition 8.2. The first two sets of those inequalities are equivalent to our conditions on the sets of long and short roots, respectively. For the other two we make a case by case analysis. As before l means long root and s short root.

• (α, β, α + 2β) = (l, s, l): k_{α} + 2k_{β} ≤ k_{α+2β} ≤ k_{α} + 2k_{β} + 2. Put γ = α + β and δ = α + 2β. Note that γ is a short root (look at the roots of B_{2}) δ is a long root, and δ = β + γ. Hence by the third condition either k_{δ} = k_{β} + k_{γ} or k_{δ} = k_{β} + k_{γ} + 1. On the other hand, α = −β + γ is a sum of short roots giving rise to a long root. So that either k_{α} = −k_{β} + k_{γ} − 1 or k_{α} = −k_{β} + k_{γ}. Now, we plug these possibilities into 2k_{β} ≤ k_{δ} −k_{α} ≤ 2k_{β} +2. We list below the inequalities that arise:

\begin{matrix}k_{δ} − k_{α} = (k_{β} + k_{γ}) − (−k_{β} + k_{γ} − 1) & 2k_{β} ≤ 2k_{β} + 1 ≤ 2k_{β} + 2 \\ k_{δ} − k_{α} = (k_{β} + k_{γ}) − (−k_{β} + k_{γ}) & 2k_{β} ≤ 2k_{β} ≤ 2k_{β} + 2 \\ k_{δ} − k_{α} = (k_{β} + k_{γ} + 1) − (−k_{β} + k_{γ} − 1) & 2k_{β} ≤ 2k_{β} + 2 ≤ 2k_{β} + 2 \\ k_{δ} − k_{α} = (k_{β} + k_{γ} + 1) − (−k_{β} + k_{γ}) & 2k_{β} ≤ 2k_{β} + 1 ≤ 2k_{β} + 2 \end{matrix}

Hence the third set of inequalities of Proposition 8.2 holds under the conditions of the lemma.

• (α, β, (α + β) /2) = (l, l, s): k_{α} + k_{β} ≤ 2k_{(α+β)/2} + 1 ≤ k_{α} + k_{β} + 2. Put γ = (α + β) /2 and δ = (β − α) /2. Both γ and δ are short roots (again look at B_{2}). We have β = γ + δ and α = γ − δ, so that k_{β} = k_{γ} + k_{δ} or k_{β} = k_{γ} + k_{δ} + 1 and k_{α} = k_{γ} − k_{δ} − 1 or k_{α} = k_{γ} − k_{δ}. Plugging these choices into k_{α} + k_{β} ≤ 2k_{γ} + 1 ≤ k_{α} + k_{β} + 2 we get:

\begin{matrix}k_{α} + k_{β} = (k_{γ} − k_{δ} − 1) + (k_{γ} + k_{δ}) & 2k_{γ} − 1 ≤ 2k_{γ} + 1 ≤ 2k_{γ} + 1 \\ k_{α} + k_{β} = (k_{γ} − k_{δ} − 1) + (k_{γ} + k_{δ} + 1) & 2k_{γ} ≤ 2k_{γ} + 1 ≤ 2k_{γ} + 2 \\ k_{α} + k_{β} = (k_{γ} − k_{δ}) + (k_{γ} + k_{δ}) & 2k_{γ} ≤ 2k_{γ} + 1 ≤ 2k_{γ} + 2 \\ k_{α} + k_{β} = (k_{γ} − k_{δ}) + (k_{γ} + k_{δ} + 1) & 2k_{γ} + 1 ≤ 2k_{γ} + 1 ≤ 2k_{γ} + 3 \end{matrix}

Concluding the proof of the lemma.

We return now to Theorem 8.1. Let J = {ε_{α}} be an iacs in the doublylaced root system Π, which satisfies the cone-free property. Then the restriction J^{s} of J to the short roots Π^{s} is cone-free. Hence, by Theorem 7.1, J^{s} is affine, so that there are integers k_{β}, β ∈ Π^{s}, forming the coordinates of an alcove in Π^{s}, such that ε_{β} = (−1)^{k_{β}} for all β ∈ Π^{s}.

Maintaining this choice of alcove in Π^{s} we intend to extend the integers k_{α} to the long roots. Taking into account the third condition of Lemma 8.5, we must define k_{α}, α ∈ Π^{l}, by the expression (6) already used in the simply-laced case, but now with α a long root and β and γ short roots, such that α = β + γ. Again, the very expression for k_{α} ensures that ε_{α} = (−1)^{k_{α}}. Hence, in order to proceed we must prove that the integers k_{α}, α ∈ Π^{l}, are well defined, and form the coordinates of an alcove.

Lemma 8.6 Let k_{β}, β ∈ Π^{s}, be the coordinates of an alcove in Π^{s}, representing J^{s}. Let α > 0 be a long root with α = β + γ, β and γ short roots, and put

k_{α} = k_{β} + k_{γ} + \frac{1 − (−1)^{k_{β}+k_{γ}} ε_{α}}{2} .  (7)

Then k_{α} is independent of the short roots β and γ.

Proof: Let α = β_{1} + γ_{1} be another sum with β_{1} and γ_{1} short roots. Denote by V the subspace spanned by the roots β, γ, β_{1} and γ_{1}, and let Π_{∗} = V ∩ Π be the corresponding subsystem. The possible dimensions of V are 2 or 3. If dim V = 2, Π_{∗} is a B_{2} system, so that the components in the two sums are equal. Similarly, in B_{3} there is only one way of writing a long root as a sum of two short roots. Hence we can assume that Π_{∗} is C_{3}.

By Proposition 4.3 any cone-free iacs in C_{3} is affine. Of course, the restriction J_{∗} of J to Π_{∗} is cone-free. Hence, there are integers, say m_{δ}, δ ∈ Π_{∗}, which are the coordinates of an alcove in C_{3}, such that ε_{δ} = (−1)^{δ} for all δ ∈ Π_{∗}. In particular k_{δ} ≡ m_{δ}mod2 for every short root δ ∈ Π_{∗}. The set of short roots in C_{3} forms an A_{3}-root system, so that we can apply Corollary 7.6, to get λ such that for every short root δ ∈ Π_{∗} we have k_{δ} = m_{δ} + ⟨λ, δ⟩ and ⟨λ, δ⟩ ∈ 2ℤ. Since the long roots in C_{3} are linear combinations of short roots with integer coefficients, it follows that ⟨λ, δ⟩ ∈ 2ℤ for the long roots in Π_{∗} as well. Therefore, m_{δ} +⟨λ, δ⟩, δ ∈ Π_{∗}, are the coordinates of an alcove A^{∗} such that J_{∗} = J (A^{∗}).

Now, let α = β + γ be as in the statement. By the third condition in Lemma 8.5 we have

m_{α} + ⟨λ, α⟩ = k_{β} + k_{γ} + \frac{1 − (−1)^{k_{β}+k_{γ}} ε_{α}}{2} .

Since the left hand side is independent of the way α is written as a sum of short roots, the lemma follows.

It remains to prove that the extension of k_{α} to the long roots given by (7) form the coordinates of an alcove. For this we use Lemma 8.5, and verify that the three conditions of that lemma are satisfied. Firstly, the integers k_{α} were chosen so that they form the coordinates of an alcove on the short roots. Also, the compatibility condition (3) follows immediately from the definition of k_{α} in (7). Hence, the point is to show that the integers k_{α} are coordinates of an alcove on the long roots. At this point we consider C_{l} separately. In fact, the set Π^{l} of long roots of C_{l} is a decomposable root system with l orthogonal positive roots. In such a root system there are no restrictions on the integers to be the coordinates of an alcove. In particular, for C_{l} the condition of Lemma 8.5 regarding the long roots is vacuous. Therefore, Theorem 8.1 is true in the C_{l} case.

To consider B_{l} and F_{4} we prove first the following easy lemma.

Lemma 8.7 Let J = {ε_{α}} be an affine iacs in the root system

B_{l} = {± (e_{i} ± e_{j}) : 1 ≤ i < j ≤ l} ∪ {±e_{i} : 1 ≤ i ≤ l}.

Suppose we are given integers k_{i}, i = 1, …, l such that ε_{e_{i}} = (−1)^{k_{i}}. Then there exists an alcove A with coordinates k_{α} (A) satisfying k_{e_{i}} = k_{i}, i = 1, …, l and such that J = J (A).

Proof: Is similar to the proof of Lemma 7.4, after taking into account that the short roots e_{i}, i = 1, …, l, span B_{l} over ℤ.

Finally, we can conclude the proof of Theorem 8.1 for B_{l} and F_{4}, by showing that the extension defined in Lemma 8.6 are indeed the coordinates of an alcove on the set of long roots.

Lemma 8.8 Given k_{β}, β ∈ Π^{s}, define k_{α}, α ∈ Π^{l} as in Lemma 8.6. Then {k_{α}} are the coordinates of an alcove in Π^{l}.

Proof: By Lemma 7.2 we must show that if α, β and α + β are long roots then either k_{α+β} = k_{α} + k_{β} or k_{α+β} = k_{α} + k_{β} + 1.

Write β = γ_{1} + γ_{2} as a sum of short roots and denote by V the subspace spanned by {α, γ_{1}, γ_{2}}. Let Π_{∗} be the root system V ∩ Π. We claim that Π_{∗} is a B_{3}-subsystem. In fact, dim V > 1 because β ≠ ±α and we cannot have dim V = 2, since this would imply that Π_{∗} is a B_{2}-system, because it contains short and long roots and a pair of roots (γ_{1} and γ_{2}) whose sum is a root. But in B_{2} the sum of two long roots is not a root. Hence, dim V = 3. Analogous arguments show that Π_{∗} is irreducible. Now, Π_{∗} has roots of different length, so that either Π_{∗} = B_{3} or C_{3}. However, in C_{3} no sum of two long roots is a root. Therefore, Π_{∗} = B_{3}, as claimed.

By looking at the roots of B_{3} we can ensure that, since α + β is a root, one of the roots in the decomposition of β, say γ_{1}, is such that there exists a short root δ with α = −γ_{1} + δ. Hence, α + β = γ_{2} + δ, and we have

• k_{α+β} = k_{γ_{2}} + k_{δ} + (1 − (−1)^{k_{γ_{2}}+k_{δ}} ε_{α+β}),

• k_{α} = −k_{γ_{1}} − 1 + k_{δ} + (1 + (−1)^{k_{γ_{1}}+k_{δ}} ε_{α}), and

• k_{β} = k_{γ_{1}} + k_{γ_{2}} + (1 − (−1)^{k_{γ_{1}}+k_{γ_{2}}} ε_{β}).

These formulae imply that the dependence of k_{α+β} −(k_{α} + k_{β}) on the integers k_{γ}, γ ∈ Π, is only mod2.

Now, we use the cone-free condition to get an alcove A^{∗} in V such that J_{∗} = J (A^{∗}), where J_{∗} is the restriction of J to Π_{∗}. For a root γ ∈ Π_{∗}, ε_{γ} = (−1)^{k_{γ}(A^{∗})} = (−1)^{k_{γ}}, so that k_{γ} (A^{∗}) ≡ k_{γ} (mod2). The formulae above are true with k_{γ} (A^{∗}), γ ∈ Π, in place of k_{γ}. But we know that for the coordinates of an alcove either k_{α+β} (A^{∗}) − (k_{α} (A^{∗}) + k_{β} (A^{∗})) = 0 or 1. Hence, either k_{α+β} − (k_{α} + k_{β}) = 0 or 1, concluding the proof of the lemma.

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