(1, 2)-Symplectic metrics on flag manifolds and tournaments

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

November 7, 2000

Abstract

It has been recently shown by Mo and Negreiros [11] that a necessary condition for an invariant almost-complex structure on the complex full flag manifold 𝔽 (n) to admit a (1, 2)-symplectic invariant metric is that its associated tournament is cone-free.

In this paper we find a canonical stair-shaped form for such tournaments and apply it to show that the condition is also sufficient. In doing this we describe all the associated (1, 2)-symplectic metrics, and get, in particular, a different and self-contained proof of a theorem of Gray and Wolf [17] asserting that the Cartan-Killing metric on 𝔽 (n) is not (1, 2)-symplectic for n > 3.

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

AMS 2000 subject classification: 53C55, 05C20, 58E20 Key words: Flag manifold, Tournament, Harmonic maps and Hermitian geometry.

1 Introduction

Denote by 𝔽 (n) the usual manifold of full flags of subspaces of ℂ^{n} and endow it with an almost-complex structure J and a Riemannian metric ds^{2}. Let M^{2} be an arbitrary closed Riemann surface. A theorem due to Lichnerowicz [9], proved independently by Gray [6], states: “If ϕ : M^{2} → (𝔽 (n) , J, ds^{2}) is J-holomorphic and ds^{2} is (1, 2)-symplectic with respect to J, then ϕ is harmonic”. The importance of this theorem rests in the fact that it furnishes solutions of the second order Euler-Lagrange partial differential equations, satisfied by the harmonic maps from the solutions of first order partial differential equations, the Cauchy-Riemann equations. This theorem drives the attention to the problem of understanding the (1, 2)-symplectic metrics on the flag manifolds. Apart from the relation to harmonic maps the (1, 2)- symplectic metrics appear in twistor theory, as shown, for example, in Eells and Salamon [5].

Our approach to this problem is based on a method derived by Burstall and Salamon [4], relating harmonic maps on flag manifolds to tournament theory. It is well known that an invariant almost-complex structure J on the flag manifold 𝔽 (n) is defined by means of a skew-hermitian sign matrix. This matrix can be considered as the incidence matrix of an n-player tournament T_{J}, providing a one-to-one correspondence between the set of invariant almost-complex structures on 𝔽 (n) and the tournaments with n players. Relying on this natural association the method consists in studying properties of J through the combinatorics of T_{J}.

Recently Mo and Negreiros [10], [11] singled out the class of the so-called cone-free tournaments in relation to the (1, 2)-symplectic metric problem. Namely, it was proved in [11] that a necessary condition for J to admit an invariant (1, 2)-symplectic metric ds^{2} is that the tournament T_{J} is cone-free, a property which involves the 4-subtournaments of T_{J} (see Definition 3.1 below for a precise statement). In this paper we prove that this condition is also sufficient, thus arriving at the following characterization.

Theorem 1.1 (𝔽 (n) , J) admits an invariant (1, 2)-symplectic metric Λ if and only if the associated tournament T_{J} is cone-free.

The sufficiency of the condition was studied by Paredes [14], [15], where an affirmative answer was obtained for certain classes of tournaments, including all tournaments with 5 ≤ n ≤ 7 players. For n = 3 and 4 this result was proved in [11]. In Section 4 we offer a proof for arbitrary n (see Theorem 4.4).

In addition, given J consistent with Theorem 1.1), we exhibit an explicit n-dimensional parametrization for all the possible invariant (1, 2)-symplectic metrics for (𝔽 (n) , J).

Our approach is based on a concrete characterization of the structure of cone-free tournaments. Specifically, in Theorem 3.5 it is shown that a tournament T is cone-free if and only if its incidence matrix is permutationsimilar to a stair-shaped incidence matrix of the following type

[figure: ⋮; ⋮; …; …; −; −; −; +; +; +; −; −; −; +; +; +]  (1)

The problem discussed here has a natural extension to the framework of generalized flag manifolds associated to arbitrary complex semi-simple Lie groups. Similar results can be proved in this more general set up, and will appear elsewhere.

2 Preliminaries

2.1 Tournaments

We define an n-player tournament as a complete directed graph T = (N, E) (no loops or multiple edges) where N is an ordered set and |N| = n. For concreteness we shall always assume that N = {1, …, n}. Two tournaments are isomorphic if one is obtained from the other via a rearrangement of N. See Moon [12] for further details.

With each tournament T we assign its incidence matrix ε = ε_{T}, which is a real skew-symmetric matrix with all off-diagonal entries ±1 (see [12]).

Tournament isomorphism amounts to permutation similarity between the associated incidence matrices.

If (a, b) ∈ E we say that a wins against b and set ε_{ab} = 1, ε_{ba} = −1. By our conventions, T would then contain an arrow pointing from vertex a into vertex b.

We define the valency of a vertex a of T as the number of its wins. We also define, in a strict sense, a winner in T as a (unique) vertex with valency n − 1, and a loser as a (unique) vertex with valency 0. Not every tournament has a winner or a loser; however, when it exists, it is not part of any cycle.

The descending sequence of valencies of T is called the score vector associated with T. Every non-increasing sequence of n non-negative integers with

total sum ( \begin{matrix}n \\ 2 \end{matrix} ) is a score vector for some n-player tournament T. However,

T need not be uniquely determined by its score vector.

Since T is complete, |E| = n(n − 1)/2. Moreover, E splits as a disjoint union

E = E_{1} ∪ E_{2}, E_{1} = {(a, b) ∈ E : a < b}, E_{2} = {(a, b) ∈ E : a > b}. (2)

2.2 Flag manifolds

In this section we discuss the correspondence between invariant almostcomplex structures on the full flag manifold and tournaments or, equivalently, their incidence matrices which are real skew-symmetric matrices with off-diagonal entries in {±1}.

Consider the complex full flag manifold

𝔽 (n) = {(V_{1}, …, V_{n}) : V_{j} ⊂ V_{j+1}, dim V_{j} = j}.

The natural action of the unitary group U (n) on 𝔽 (n) is transitive, turning the flag manifold into the homogeneous space U (n) /M where M is any maximal torus of U (n), i.e., M ≅ U (1) × ⋯ × U (1).

Let 𝔲 (n) be the Lie algebra of skew-hermitian matrices. It decomposes as

𝔲 (n) = 𝔭 ⊕ 𝔲 (1) ⊕ ⋯ ⊕ 𝔲 (1)

where 𝔭 ⊂ 𝔲 (n) is the subspace of zero-diagonal matrices. Let E_{jk} be the canonical basis matrix E_{jk} = (a_{rs}) with a_{rs} = 1 if (r, s) = (j, k) and zero otherwise. Put

𝔭_{jk} = (ℂE_{jk} + ℂ𝔼_{kj}) ∩ 𝔲 (n) .

Of course, 𝔭 = ⨁_{j≠k} 𝔭_{jk}.

In order to give an invariant almost complex structure on 𝔽 (n) it is enough to present J : 𝔭 → 𝔭, J^{2} = −1, which commutes with the adjoint representation of the torus M on 𝔭. This condition implies that J (𝔭_{jk}) = 𝔭_{jk} for all j ≠ k, which in turn guarantees that J (A) = A′ with A′_{jk} = ε_{jk}\sqrt{−1}A_{jk} such that ε_{jk} = ±1 and satisfies ε_{kj} = −ε_{jk}. Thus an invariant almost complex structure is completely determined by a skew-symmetric matrix (ε_{jk}), with off-diagonal entries in {±1}.

For the computations we work in the complexification V of 𝔭. It is easy to check that V is the subspace of complex matrices with zero-diagonal entries. It decomposes as V = ⨁_{j≠k} V_{jk}, where V_{jk} = span_{ℂ}{E_{jk}}.

An almost complex structure J on 𝔭 extends to a ℂ-linear operator on V, also denoted by J. Its eigenvalues are ±\sqrt{−1}. If J is given by the incidence matrix ε_{jk} then the corresponding eigenspaces are

V^{10} = ⨁\{V_{jk} : ε_{jk} = 1\} for \sqrt{−1}

and

V^{01} = ⨁\{V_{jk} : ε_{jk} = −1\} for − \sqrt{−1}.

Let ds^{2} be an U (n)-invariant Riemannian metric on 𝔽 (n). Like the invariant almost complex structures, ds^{2} is completely determined by its value at the origin, that is, by an inner product (·, ·) in 𝔭, which is invariant under the adjoint action of M. To describe these inner products, start with the Cartan-Killing form on 𝔭:

⟨X, Y ⟩ = −tr (XY) ,

which is an M-invariant inner product. Any other inner product on 𝔭 is of the form (X, Y)_{Λ} = ⟨ΛX, Y ⟩ with Λ : 𝔭 → 𝔭 positive-definite with respect to ⟨·, ·⟩. Furthermore, (·, ·)_{Λ} is M-invariant if and only if the elements of the standard basis \sqrt{−1} (E_{jk} + E_{kj}), E_{jk} − E_{kj} are eigenvectors of Λ. Thus Λ (E_{jk}) = λ_{jk}E_{jk} with λ_{jk} > 0 and λ_{kj} = λ_{jk}, and

(X, Y)_{Λ} = −tr (Λ (X) Y) .

We denote by ds_{Λ}^{2} the invariant metric given by Λ. Remark: Consider the n × n symmetric matrix, say A, whose entries are λ_{jk}, j ≠ k, the eigenvalues of Λ, and λ_{jj} = 0. Of course, the inner product depends only on the entries A. Furthermore, Λ (X) = A ∘ X where ∘ stands for the Hadamard product of two n × n matrices. Therefore, (X, Y)_{Λ} = −tr ((A ∘ X) Y). In the sequel we also denote by Λ the matrix A.

The inner product (·, ·)_{Λ} admits a natural extension to a symmetric bilinear form on the complexification V of 𝔭. We use the same notation (·, ·)_{Λ} for this bilinear form as well as for the corresponding complexified map Λ. Here the two-dimensional real eigenspace 𝔭_{jk} of Λ, whose basis is \sqrt{−1} (E_{jk} + E_{kj}), E_{jk} −E_{kj}, extends to complex spaces having basis E_{jk} and E_{kj} , respectively.

A special class of invariant inner products is given by those Λ satisfying

λ_{ij} + λ_{jk} = λ_{ik}  (3)

for all j between i and k, or equivalently satisfying the relation

λ_{ij} = ∑_{k=i}^{j−1} λ_{k,k+1}.

In this case Λ is defined by an adjoint operation as follows: Consider the real diagonal matrix

H_{Λ} = diag{μ_{1}, …, μ_{n}}  (4)

whose eigenvalues are defined (up to an additive constant) by the conditions μ_{i} − μ_{i+1} = λ_{i,i+1}, i = 1, …, n − 1. Then the action of Λ on the upper triangular matrices is given by

Λ (E_{jk}) = ad (H_{Λ}) (E_{jk}) j < k

if Λ satisfy the conditions (3). Analogously, Λ = −ad (H_{Λ}) on the lower triangular matrices. In view of these relations a metric satisfying the equalities (3 ) is called of adjoint type.

Now, let J and ds_{Λ}^{2} be an invariant almost complex structure and a metric, respectively. Then it is easily checked that ds_{Λ}^{2} is Hermitian, that is,

ds_{Λ}^{2} (JX, JY) = ds_{Λ}^{2} (X, Y) .

Let Ω = Ω_{J,Λ} stand for the corresponding Kähler form

Ω (X, Y) = ds_{Λ}^{2} (X, JY) = −tr (Λ (X) J (Y)) .

This form is also invariant under U (n). Relying on this invariance its exterior differential is easily calculated from the standard formula: If X, Y, Z ∈ 𝔭 are regarded as vector fields in 𝔽 (n) then dΩ at the origin is given by

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

(see [8]).

For all i = 0, 1, 2, 3 define the operator dΩ^{i,3−i} as follows:

1. dΩ^{i,3−i} (X, Y, Z) = dΩ (X, Y, Z) as long as i of the three matrices X, Y, Z are in V^{10} and the remaining are in V^{01};

2. dΩ^{i,3−i}(X, Y, Z) = 0 as long as j of the three matrices X, Y, Z are in V^{10} and the remaining are in V^{01}, with j ≠ i.

Since V^{10} and V^{01} are orthogonal complements in V, it is readily seen that the four operators dΩ^{i,3−i} are well defined and moreover

dΩ = dΩ^{30} + dΩ^{21} + dΩ^{12} + dΩ^{03}.

Moreover, by replacing X, Y, Z by X^{T} , Y^{T} , Z^{T} it can be seen that dΩ^{30} = −[dΩ^{03}]^{∗} and dΩ^{12} = −[dΩ^{21}]^{∗}.

According to the annihilation of these forms the triple (𝔽 (n) , J, ds^{2}) is distinguished as follows:

1. (𝔽 (n) , J, ds^{2}) is almost Kähler if dΩ = 0. It is Kähler if furthermore J is integrable.

2. (𝔽 (n) , J, ds^{2}) is (r, s)-symplectic if dΩ^{rs} = 0.

The following theorem by Burstall and Salamon [2] ensures that a sufficient condition for J to be integrable is that dΩ = 0.

Theorem 2.1 (𝔽 (n) , J, ds^{2}) is an almost Kähler manifold if and only if it is a Kähler manifold.

3 Cone-free tournaments

Up to isomorphism, there are four distinct 4-player tournaments. The two of them which contain a single (directed) 3-cycle are called coned 3-cycles.

[figure]

Each of them contains a cycle and a winner or a loser.

Definition 3.1 An n-player tournament T is called cone-free if its restriction to any four vertices is never a coned 3-cycle.

In this section we show that the incidence matrix of a cone-free tournament is permutation similar to a stair-shaped matrix of the type shown in (1). We start by providing a rigorous definition of stair-shaped tournaments.

We call a tournament T transitive if it contains no 3-cycle. In this case, T is obviously cone-free, and is isomorphic to the canonical transitive tournament T′ for which the subset E_{1} in (2) is E. The incidence matrix of T′ has a triangular form.

In what follows we shall denote by T/U the restriction of T to U, which is an m-player tournament if |U| = m.

Definition 3.2 A tournament T′ is called stair-shaped if there are integers s, t (with 1 ≤ s ≤ t ≤ n) such that the axioms below are satisfied. For all U ⊂ N, with |U| = m < n.

A) T′/{1, …, t} is a maximal canonical-transitive subtournament of T′;

B) T′/{s, …, n} is a maximal canonical-transitive subtournament of T′;

C) If (z, x) ∈ E_{2} then x < s and t < z;

D) if x′ ≤ x and z ≤ z′ then (z, x) ∈ E_{2} implies (z′, x′) ∈ E_{2}.

A special case of a stair-shaped tournament is when (s, t) = (1, n). In such a case T′ is a canonical transitive tournament.

Lemma 3.3 Stair-shaped tournaments are cone-free.

Proof: Let T′ be stair-shaped, and let s, t be the associated numbers. If s = 1 and t = n then T′ is transitive, hence contains no coned 3-cycles, and there is nothing to prove. So, we shall assume that 1 < s ≤ t < n and subsequently, due to the maximality clause in axioms A and B, E_{2} in (2) is not empty. The 3-cycles in T′ are exactly the triples x, y, z ∈ N with

x < a ≤ b < z,  x < y < z,  (z, x) ∈ E_{2}.  (5)

Let U = {x, y, z, w} ⊂ N with |U| = 4. Assume that x, y, z form a 3cycle in T′. Due to axiom C, since x < y < z then the cycle arrows are x → y → z → x, and moreover x < s ≤ t < z.

We want to show that T′/U is not a coned 3-cycle. To this end, we first show that w is not a winner in U. Indeed, if z < w then by axiom B w loses to z; if y < w < z then by axiom D w loses to y; if x < w < y then by axiom D w loses to x; and if w < x then by axiom D w loses to z. By a similar argument, w is not a loser in U either. Thus, T′/U is not coned, implying that T′ is cone-free.

Next we show our central result on the representation of cone-free tournaments. It requires the following.

Definition 3.4 A subtournament T/U of a tournament T = (N, E) is said to be 1-transitive if T/ (U ∪ {p}) is transitive for all p ∈ T.

In particular, T/U itself must be transitive.

Theorem 3.5 A tournament T is cone-free if and only if it is isomorphic to a stair-shaped tournament T′.

Proof: The “if” part is covered by Lemma 3.3. We now prove the “only if” part. Let T be a cone-free n-tournament . If T is transitive, it is isomorphic to the canonical-transitive tournament, which is stair-shaped. Thus, we shall assume that T is not transitive, i.e. T contains 3-cycles.

Let T/U be a maximal 1-transitive subtournament of T. Since every 2player tournament is transitive, m := |U| > 0. Since T is not transitive, m < n. By reordering N we obtain a new tournament T′′, isomorphic to T, such that T′′/(U ∪ {p}) is the canonical-transitive m + 1-player tournament for all p ∈ N.

In particular, T′′/U is canonical-transitive. Let s, t ∈ N , s ≤ t, be its winner and loser. Define the subsets U_{1} = {1, 2, …, t} and U_{2} = {s, s + 1, …, n}. It is easy to see that s is a winner in U_{2} and t is a loser in U_{1}.

We claim that the subtournament T′′/U_{1} is transitive. Indeed, assume that there exists a cycle x, y, z with 1 ≤ x < y < z ≤ t. Necessarily we have z < t since t is a loser in U_{1} and cannot form cycles there. But then T′′/{x, y, z, t} forms a coned 3-cycle, in contradiction to the assumption that T is cone-free.

So, T′′/U_{1} is transitive, and by an analogous argument, T′′/U_{2} is transitive. By a suitable reordering of N, T′′ can be made isomorphic to a new tournament T′ , for which T′/U_{1} and T′/U_{2} are canonical-transitive. Note that this reordering leaves unchanged the set U.

We shall show that T′, t, s as constructed above satisfy axioms A-D, namely T′ is stair-shaped.

Let U′ := U_{1} ∩ U_{2}. The subtournament T′/U′ is transitive. In fact, it is 1-transitive, since for all p ∈ N U′ ∪ {p} is a subset of U_{1} or U_{2}. Moreover, since by construction U ⊂ U_{1} ∩ U_{2}, and T/U is a maximal 1-transitive set, we conclude that

U = U′ = {s, s + 1, …, t}.

In fact, T′/U_{1} is a maximal transitive subtournament of T′. Indeed, assume that U_{1} ⊂ U′_{1} and T′/U′_{1} is transitive. It is easy to see that T′/(U′_{1} ∩ U_{2}) is 1-transitive. Since U ⊂ U′_{1} ∩ U_{2}, and T′/U is a maximal 1-transitive subset, we conclude that U′_{1} = U_{1}, proving the maximality of T′/U_{1}.

By a similar argument, T′/U_{2} is a maximal transitive subtournament of T′. Thus, T′ satisfies axioms A and B.

In order to verify axiom C, we observe that without loss of generality every edge of E_{2} belongs to a 3-cycle. Indeed, all the edges of E_{2} without cycle can be inductively inverted without changing the sets U, U_{1}, U_{2} and reducing E_{2}.

So, let {x, y, z} be a 3-cycle in T, where x < y < z. Since U_{2} is transitive, we have x < s. Since U_{1} is transitive, we have t < z. Independently of the location of y, one of the two statements x, y ∈ U_{1} or y, z ∈ U_{2} must hold. Since T′/U_{1} and T′/U_{2} are canonical-transitive, we conclude that the arrows in the 3-cycle are directed as x → y → z → x. Note that only the rightmost arrow represents an edge in E_{2}. Thus, T′ satisfies axiom C.

Finally, we verify axiom D. If (z, x) ∈ E_{2} and z′ < z, consider any y ∈ U. Since x, y, z, z′ cannot be a coned 3-cycle, z′ cannot lose to all the others. Since z′ loses to y and z, we must have z′ → x as desired. Similarly, if x′ > x an analogous argument shows that z → x′.

Thus, T′ is a stair-shaped tournament isomorphic to T.

A natural question related to the above theorem is whether there exists uniqueness of the stair-shaped structure for a cone-free tournament. An affirmative answer of this question would lead to canonical forms for such tournaments.

In general, a cone-free tournament may have several maximal 1-transitive subtournaments of different sizes, leading to different, possibly pairwise nonsimilar, stair-shaped incidence matrices. For instance, the 4 -tournament

[figure: 1; 2; 3; 4]

has the following incidence matrices

( \begin{matrix}0 & + & − & − \\ − & 0 & + & + \\ + & − & 0 & + \\ + & − & − & 0 \end{matrix} )  and  ( \begin{matrix}0 & + & + & − \\ − & 0 & + & + \\ − & − & 0 & + \\ + & − & − & 0 \end{matrix} )

with respect to the orders (1, 2, 3, 4) and (2, 3, 4, 1), respectively. The maximal 1-transitive subtournaments are {1}, {2}, {3}, {4} as well as {3 → 4}. Both incidence matrices may show up in the construction made in the proof above. The first one appears if one takes {2} as maximal 1-transitive tournament, whereas the second matrix comes out from {3 → 4}.

We do not know whether two maximal 1-transitive tournaments of the same size necessarily lead to equal stair-shaped incidence matrices for a given tournament. The lack of uniqueness in the stair-shaped representation of a cone-free tournament T may be related to the co-existence of several nonisomorphic 1-transitive subtournaments for T.

4 Proof of Theorem 1.1

Recall that a metric is called (i, 3 − i)-symplectic if dΩ^{i,3−i} = 0. In this case, we also have dΩ^{3−i,i} = 0. Theorem 1.1 in the introduction is formulated in terms of (1, 2)-symplectic metrics. A special case of this theorem concerns the standard complex structure in 𝔽 (n) and is already known (see Borel [2] and [11]). Denote by J_{c} the corresponding almost complex structure. Then in this particular case Theorem 1.1 reads:

Proposition 4.1 Let J_{c} standard complex structure on 𝔽 (n). Then J_{c} corresponds to the canonical transitive tournament and Λ is (1, 2)-symplectic if and only if it is of adjoint type (see (3)).

We now attack Theorem 1.1 in its full generality. The proof that the condition is sufficient is one of the main results of this paper. Before embarking on it we outline the necessity proof of [11]: Let ω = (ω_{i\overline{j}}) be the matrix formed by the Maurer-Cartan form of U (n). The space of (1, 0)-cotangent vectors at the origin identifies to 𝔭_{(1,0)} = span\{ω_{i\overline{ȷ}} : i → j\}. The key point in the proof is to compute dΩ using the moving frame method of Cartan. We find a permutation τ such that

dΩ = ∑_{i<j<k} C_{τ(i)τ(j)τ(k)}ψ_{τ(i)τ(j)τ(k)}

where C_{ijk} = μ_{ij} − μ_{ik} + μ_{jk}, ψ_{ijk} = Im (ω_{i\overline{j}} ∧ ω_{\overline{i}k} ∧ ω_{j\overline{k}}) and finally μ_{ij} = ε_{ij}λ_{ij}. From this expression for dΩ it is not hard to prove that the tournament must be cone-free (see [11], Section 4).

Now, we proceed to the proof of the sufficiency of Theorem 1.1, namely that there are invariant (1, 2)-symplectic metrics if the tournament is conefree. First we calculate the conditions on J and Λ to be (1, 2)-symplectic. Using the formulas

[E_{ij}, E_{rs}] = δ_{jr}E_{is} − δ_{si}E_{rj} tr (E_{ij}E_{rs}) = δ_{is}δ_{jr}

a straighforward computation shows that dΩ (E_{ii′}, E_{jj′}, E_{kk′}) = −3\sqrt{−1}αβ where

α = δ_{ij′}δ_{jk′}δ_{ki′} − δ_{ik′}δ_{kj′}δ_{ji′}

and

β = ε_{ii′}λ_{ii′} + ε_{jj′}λ_{jj′} + ε_{kk′}λ_{kk′}.

The expression for α shows that it is nonzero only when i′ = k, j′ = i, k′ = j or i′ = j, j′ = k, k′ = i. Hence dΩ is completely determined by its values in the triples

(E_{ij}, E_{jk}, E_{ki}) and (E_{ik}, E_{kj}, E_{ji}) ,  (6)

given by the triples {i, j, k}. Note that these sets are transposed to each other so that if one of them contains a vector in V^{10} then the other contains a vector in V^{01}. Let us say that the triple {i, j, k} is of type {3, 0} if one of the corresponding sets (6) is contained in V^{10}. The triple is of type {1, 2} otherwise. The proof of the following lemma is immediate from the definitions.

Lemma 4.2 The triple {i, j, k} is of type {3, 0} if and only if the restriction to {i, j, k} of the tournament corresponding to J is a cycle.

Equivalently, {i, j, k} is of type {1, 2} if and only if the tournament is transitive in {i, j, k}.

Therefore, in order to check that an invariant metric Λ is (1, 2)-symplectic, it is enough to compute dΩ in one set of vectors (6), only for the transitive subtournaments {i, j, k}.

Proposition 4.3 The invariant metric Λ is (1, 2) -symplectic if and only if for all transitive subtournaments T_{3} = {i, j, k} the following holds

λ_{ik} = λ_{ij} + λ_{jk}  (7)

where i is the winner of T_{3} while k is the loser.

Proof: For fixing ideas let us write T_{3} = {a, b, c} with a < b < c. From the computations performed we have dΩ (E_{ab}, E_{bc}, E_{ca}) = \sqrt{−1}αβ with

β = ε_{ab}λ_{ab} + ε_{bc}λ_{bc} + ε_{ca}λ_{ca}.  (8)

In order to have the values of ε_{rs}, we must run through the six transitive tournaments having vertices (a, b, c). Below we list the signs of (ε_{ab}, ε_{bc}, ε_{ca}) with the winners (w) and losers (l) for each transitive tournament:

1. (+, +, −), w = a, l = c.

2. (+, −, −), w = a, l = b.

3. (+, −, +), w = c, l = b.

4. (−, +, −), w = b, l = c.

5. (−, +, +), w = b, l = a.

6. (−, −, +), w = c, l = a.

By a direct inspection at these tournaments we see that in the right hand side of (8), λ_{wl} appears with a sign different from the other terms. Therefore, β = 0 if and only if λ_{ik} = λ_{ij} + λ_{jk} with w = i and l = k. Now the statement follows by Lemma 4.2.

Our central result gives an n-parameter description for all the (1, 2)- symplectic metrics on 𝔽 (n). We only cover the non-transitive case, given that the transitive case is covered by Proposition 4.1.

Theorem 4.4 Let ε be the incidence matrix of a given complex structure J on 𝔽 (n). Assume that ε is stair-shaped and not canonical-transitive. Then Λ corresponds to a (1, 2)-symplectic metric on (𝔽 (n) , J) if and only if there are n positive values λ_{1} through λ_{n} such that for all i < j:

(i) λ_{ij} = ∑_{k=i}^{j−1} λ_{k} if ε_{ij} = 1, and

(ii) λ_{ij} = λ_{n} + ∑_{k=1}^{i} λ_{k} + ∑_{k=j}^{n−1} λ_{k}, otherwise. Equivalently,

λ_{ij} = q − ∑_{k=i+1}^{j−1} λ_{k}

where q = ∑_{k=1}^{n} λ_{k}.

Proof: Our starting point is Proposition 4.3. Dividing N into three subsets: {1, …, s − 1}, {s, …, t}, and {t + 1, …, n}, we see that each triple X = E_{ij}, Y = E_{jk}, Z = E_{ki} of basic matrices may be located in one of nine rectangular regions. The requirement X, Y ∈ V^{10} and Z ∈ V^{01} reduces the number of cases to the following (see ( 1)):

1. i < j < k < s;

2. i < j < s ≤ k ≤ t; 3. i < s ≤ j < k ≤ t;

4. i < j ≤ s ≤ t < k;

5. i < s ≤ j ≤ t < k;

6. i < s ≤ t < j < k;

7. s ≤ i < j < k ≤ t;

8. s ≤ i < j ≤ t < k;

9. s ≤ i ≤ t < j < k;

10. t < i < j < k;

11. k < s ≤ t < i < j;

12. j < k < s ≤ t < i.

It can be seen that (i) is equivalent to the totality of conditions on Λ which emanate from cases 1-10, while (ii) emanates equally from cases 11 and 12.

The sufficiency of the condition stated in Theorem 1.1 now follows as an immediate corollary of Theorem 4.4. Remark: Any metric Λ arising in the above theorem decomposes as Λ = Λ_{1} + Λ_{2}, where Λ_{1} corresponds to the entries {(i, j) : i < j, ε_{ij} = +1} and Λ_{2} to region where ε_{ij} = −1, i < j. The entries λ_{ij} of both Λ_{1} and Λ_{2} are eigenvalues of ad (H) for some diagonal H. For Λ_{1} take H = H_{Λ} as in (4) while for Λ_{2} take H = H_{Λ_{2}} with

H_{Λ_{2}} = diag{ν_{1}, …, ν_{n}}

with ν_{n} − ν_{1} = λ_{n} and ν_{i+1} − ν_{i} = −λ_{i,i+1} if i = 1, …, s or i = t, …, n − 1. Therefore the (1, 2)-symplectic metrics are combinations of metrics of adjoint type, restricted to the proper regions.

Is is clear that the possible (1, 2)-symplectic metrics Λ built in Theorem 4.4 cannot have all entries λ_{ij} = 1 if n ≥ 4. As a consequence our theorem provides an independent proof of the following result by Wolf and Gray.

Corollary 4.5 The Cartan-Killing metric on 𝔽 (n) is not (1, 2)-symplectic for any J if n ≥ 4.

Proof: In fact, the Cartan-Killing metric is given by Λ = (λ_{ij}) with λ_{ij} = 1 if i ≠ j.

Remark: Notice that for invariant forms the annihilation of both dΩ^{30} and dΩ are equivalent to the nonexistence of cycles in the corresponding tournament. Therefore Ω is (3, 0) (or (0, 3)) symplectic if and only if it is Kähler.

5 Partial flag manifolds

Let 𝔽 = 𝔽^{N} (n_{1}, …, n_{k}) denote the manifold of generalized flags of subspaces of ℂ^{n} viewed as a homogeneous space as U (N) /U (n_{1}) × ⋯ × U (n_{k}), where n_{1} + ⋯ + n_{k} = N. Its tangent space at the origin identifies with a subspace 𝔭 of the Lie algebra 𝔲 (N), which complements the Lie algebra 𝔲 (n_{1})⊕⋯⊕𝔲 (n_{k}) of the isotropy group. We can easily work with this apparently more general situation like in the full flag manifold case. We write the subspace of zero-diagonal complex matrices in blocks according to the partition determined by n_{1}, …, n_{k}. Let V_{ij} be the subspace having nonzero block only in the position ij and put V = ⨁_{ij} V_{ij}, which is the complexification of 𝔭.

An invariant almost complex structure J : 𝔭 → 𝔭 is given by multiplication by ±\sqrt{−1} on each block. To such a structure we can associate, in the same way, a k-player tournament T_{J} as follows. Put J (A_{ij}) = (A′_{ij}) and for i < j choose the orientation i → j if A′_{ij} = \sqrt{−1}A_{ij} and i ← j if A′_{ij} = −\sqrt{−1}A_{ij}.

Analogous to the full flag manifold, an invariant metric is determined by its value at the origin, which is of the form (X, Y)_{Λ} = ⟨ΛX, Y ⟩ with Λ : 𝔭 → 𝔭 positive-definite with respect to the Cartan-Killing metric ⟨·, ·⟩. Again Λ is multiplication by λ_{ij} on each V_{ij} and hence is given by a k × k matrix (λ_{ιj}) which is also denoted by Λ.

By working with the block decomposition of matrices the study of the generalized flag manifold 𝔽^{N} (n_{1}, …, n_{k}) reduces to the understanding of 𝔽 (k). In particular, 𝔽 (2) codifies any Grassmannian of n_{1} subspaces 𝔽^{N} (n_{1}, N − n_{1}). We note in passing that the Grassmannians are symmetric spaces as happens to 𝔽 (2) = ℂP^{1} (with the Fubini-Study metric), the only of the full flag manifolds that is a symmetric space.

6 Parabolic invariant almost complex structures

The canonical parabolic tournament T_{P} is defined by the incidence matrix γ = (γ_{ij}) given by

γ_{ij} = \{ \begin{matrix}−1 if |i − j| is odd \\ +1 if |i − j| is even. \end{matrix}  (9)

A tournament is called parabolic if it is equivalent to T_{P}. Similarly an invariant almost complex structure is parabolic if this happens to its tournament.

It is easy to check that the canonical parabolic tournament is cone-free. In fact, a 4-player subtournament of T_{P} has nodes a < b < c < d which are joined by the following rules: Take i < j. Then i → j if i − j is even and i ← j if i − j is even.

According to the parity value of a, b, c, d one gets up to 16 distinct 4 -player subtournaments, some of which may be pairwise similar. We leave to the reader the amusement of checking that none of these tournaments is a cone. Therefore T_{P} is cone-free. (Alternatively, in [11] (1, 2)-symplectic metrics for parabolic J was exhibited. Hence the cone-free property of T_{P} follows from Theorem 1.1.)

We propose here to find a stair-shaped parabolic tournaments. Such a tournament has an incidence matrix of the form

ε = ( \begin{matrix}ct_{a} & + & ∗_{u} \\ − & ct_{b} & + \\ ∗_{l} & − & ct_{c} \end{matrix} )  (10)

where a + b + c = n and ct_{k} stands for the canonical transitive k -player tournament. By the construction in Theorem 3.5, b is the size of a maximal 1-transitive tournament. Our method is based in the following remarks:

1. Let δ and wδw^{−1} be the incidence matrix of two equivalent tournaments, where w is a permutation matrix. Let v^{δ} = (v_{1}^{δ}, …, v_{n}^{δ}) be the row sum vector of δ:

v_{j}^{δ} = ∑_{k=1}^{n} δ_{jk}.

Clearly, v^{δ} = δe, where e = (1, …, 1). Then the entries of v^{wδw^{−1}} are obtained form v^{δ} by the permutation w. In fact, it is clear that v^{δ} = δe, where e = (1, …, 1). Hence,

v^{wδw^{−1}} = wδw^{−1}e = w (δe) .

2. The sum of the entries of the column a + j, 1 ≤ j ≤ b, of the matrix ε

in (10) is

a − b − c + 2 (j − 1) + 1.  (11)

Now, the entries of the columns of the canonical parabolic tournament add up to 0 if n is odd, and to ±1 if n is even. Hence, the expression in (11) ensures that, for a stair-shaped parabolic ε, we must have b = 1 if n is odd and b = 1 or 2 if n is even.

Lemma 6.1 If n is even then the stair-shaped ε has b = 2.

Proof: As mentioned above, b is the number of elements in a 1-maximal subtournament of T_{P}. Hence, it is enough to exhibit such a tournament having two elements. This is given by {1, n}. In fact in T_{P} the following 3-tournaments hold, for any even e < n and odd o > 1:

\begin{aligned}1 & ← & e \\ & ↖ & ↓ \\ & & n \end{aligned}  \begin{aligned}1 & → & o \\ & ↖ & ↑ \\ & & n \end{aligned}

These tournaments are transitive, showing that {1, n} is 1-transitive.

Now, we can describe stair-shaped parabolic tournaments.

Proposition 6.2 Given the parabolic tournament T_{P} put n = 2k + 1 if n is odd and n = 2 (k + 1) if n is even. Order {1, …, n} as follows:

1. {2, 4, …, n − 1, 1, 3, …, n − 2, n} if n is odd.

2. {2, 4, …, n − 2, n, 1, 3, …, n − 1} if n is even.

In these orderings T_{P} has the following stair-shaped incidence matrices

ε = ( \begin{matrix}ct_{k} & + & U \\ − & δ & + \\ L & − & ct_{k} \end{matrix} ) .

Here δ = (0)_{1×1} if n = 2k + 1 and δ = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) if n = 2 (k + 1). Also,

U = (U_{ij})_{k×k} is U_{ij} = +1 if i > j and −1 otherwise, and L = −U^{T}.

Proof: The choice of the orderings was made after the construction of T′ in the proof of Theorem 3.5. Here we took as maximal 1-transitive tournaments {1} for n odd and {n → 1} for n even. Then the even numbers are the winners of 1 in the first case and of n in the second. In both cases the odd numbers are the losers of 1. It is straighforward to check that the incidence matrices are as stated.

Once we have the permutation realizing the stair-shaped equivalence the n -th parameter family of invariant (1, 2)-symplectic is readily obtained from Theorem 4.4. Therefore we can recover Theorem 3.2 of [11] and get the invariant (1, 2)-symplectic metrics for the parabolic almost complex structures. In order to write down these metrics we consider the permutations leading to the orderings of Proposition 6.2:

σ_{o} (i) = \{ \begin{matrix}2i if 1 ≤ i ≤ k \\ 2 (i − k) − 1 if k + 1 ≤ i ≤ n \end{matrix}

for n = 2k + 1 and

σ_{e} (i) = \{ \begin{matrix}2i if 1 ≤ i ≤ k \\ 2 (i − k) − 1 if k + 1 ≤ i ≤ n \end{matrix}

for n = 2k.

Theorem 6.3 Let J be the canonical parabolic invariant almost complex structure in 𝔽 (n). Then the invariant (1, 2)-symplectic metrics Λ = (λ_{ij}) are given by n parameters a_{1}, …, a_{n} as follows: Given i < j,

λ_{σ(i)σ(j)} = \{ \begin{matrix}a_{i} + ⋯ + a_{j−1} if |i − j| ≤ k + 1 \\ a_{j} + ⋯ + a_{n} + ⋯ + a_{i} if |i − j| > k + 1 \end{matrix}

where σ = σ_{o} if n = 2k + 1 and σ = σ_{e} if n = 2k.

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