TENSOR PRODUCT DECOMPOSITIONS AND OPEN ORBITS IN MULTIPLE FLAG VARIETIES

VLADIMIR L. POPOV^{∗}

Abstract. For a connected semisimple algebraic group G, we consider some special infinite series of tensor products of simple G-modules whose G-fixed point spaces are at most one-dimensional. We prove that their existence is closely related to the existence of open G-orbits in multiple flag varieties and address the problem of classifying such series.

Let G be a connected simply connected semisimple algebraic group. In this paper we establish a close interrelation between some special series of tensor products of simple G-modules whose G-fixed point spaces are at most one-dimensional and multiple flag varieties of G that contain open G-orbit. Motivated by this intimate connection with geometry, we then address the problem of classifying such series. Starting with the basic definition and examples in Sections 1 and 2, we introduce necessary notation in Section 3 and then formulate our main results in Section 4. Other results and proofs are contained in the remaining part of paper.

Below all algebraic varieties are taken over an algebraically closed field k of characteristic zero.

1. Basic definition Fix a choice of Borel subgroup B of G and maximal torus T ⊂ B. Let P_{++} be the

additive monoid of dominant characters of T with respect to B. Put

P_{≫} := P_{++} \ {0}.

For λ ∈ P_{++}, denote by E_{λ} a simple G-module of highest weight λ and by λ^{∗} the highest weight of dual G-module E_{λ}^{∗}. Let P_{λ} be the G-stabilizer of unique B-stable line in E_{λ}. If μ, λ_{1}, …, λ_{d} ∈ P_{++}, denote by c_{λ_{1},...,λ_{d}}^{μ} the multiplicity of E_{μ} inside E_{λ_{1}} ⊗ … ⊗ E_{λ_{d}}, i.e., the Littlewood–Richardson coefficient dim(Hom_{G}(E_{μ}, E_{λ_{1}} ⊗ … ⊗ E_{λ_{d}})). Denote respectively by ϖ_{1}, …, ϖ_{r} and α_{1}, …, α_{r} the systems of fundamental weights of P_{++} and simple roots of G with respect to T and B enumerated as in [Bo_{2}]. Let respectively ℤ_{⩾0} and ℤ_{>0} be the sets of all nonnegative and all positive integers. We write P_{≫}^{d} in place of (P_{≫})^{d}, etc.

Date: August 25, 2006.

2000 Mathematics Subject Classification. 20G05, 14L30.

Key words and phrases. Semisimple group, tensor product of modules, orbit, weight, Weyl group.

∗ Supported by ETH, Zürich, Switzerland, Russian grants ���� �����������, ��������������, and program Contemporary Problems of Theoretical Mathematics of the Mathematics Branch of the Russian Academy of Sciences.

Definition 1. We call a d-tuple (λ_{1}, …, λ_{d}) ∈ P_{≫}^{d} primitive if

c_{n_{1}λ_{1},...,n_{d}λ_{d}}^{0} ⩽ 1 for all (n_{1}, …, n_{d}) ∈ ℤ_{⩾0}^{d}.  (1)

Schur’s lemma and the isomorphism (E_{μ} ⊗ E_{ν})^{G} ≃ Hom_{G}(E_{μ^{∗}}, E_{ν}) imply that

c_{μ,ν}^{0} = \begin{cases}1 & if μ^{∗} = ν, \\ 0 & otherwise; \end{cases}  (2)

whence condition (1) is equivalent to the following:

c_{n_{1}λ_{1},...,n̂_{j}λ_{j},...,n_{d}λ_{d}}^{n_{j}λ_{j}^{∗}}⩽ 1 for all (n_{1}, …, n_{d}) ∈ ℤ_{⩾0}^{d} and some (equivalently, every) λ_{j}. (3)

The set of primitive elements of P_{≫}^{d} is clearly stable with respect to permutation of coordinates and automorphisms of P_{≫}^{d} induced by automorphisms of the Dynkin diagram of G. If (λ_{1}, …, λ_{d}) ∈ P_{≫}^{d} is primitive, then (λ_{i_{1}}, …, λ_{i_{s}}) ∈ P_{≫}^{s} is primitive for every subset {i_{1}, …, i_{s}} of {1, …, d}.

Remark 1. The notion of primitive d-tuple admits a natural generalization: we call a d-tuple (λ_{1}, …, λ_{d}) ∈ P_{≫}^{d} primitive at μ ∈ P_{++} if

c_{n_{1}λ_{1},...,n_{d}λ_{d}}^{μ} ⩽ 1 for all (n_{1}, …, n_{d}) ∈ ℤ_{⩾0}^{d}.  (4)

Then “primitive” means “primitive at 0”.

2. Examples Clearly, for d = 1, every element of P_{≫}^{d} is primitive. By (2) the same is true for d = 2.

For d ⩾ 3, the existence of primitive elements in P_{≫}^{d} is less evident.

Example 1. Let G = SL_{2}. Then P_{++} = ℤ_{⩾0}ϖ_{1} and G/P_{λ} for λ ≠ 0 is isomorphic to the projective line P^{1}. From Definition 1 and the Clebsch–Gordan formula

E_{sϖ_{1}} ⊗ E_{tϖ_{1}} ≃ ⨁_{0⩽i⩽t} E_{(s+t−2i)ϖ_{1}}, s ⩾ t,

it is not difficult to deduce that an element of P_{≫}^{d} is primitive if and only if d ⩽ 3. Theorems 1 and 4 below imply that this is equivalent to the classical fact that for the diagonal action of SL_{2} on (P^{1})^{d}, an open orbit exists if and only if d ⩽ 3.

Example 2. If, for (λ_{1}, …, λ_{d}) ∈ P_{≫}^{d}, condition (4) holds for every μ ∈ P_{++}, then by (3) the (d + 1)-tuple (ν, λ_{1}, …, λ_{d}) is primitive for every ν ∈ P_{≫}. Such d-tuples (λ_{1}, …, λ_{d}) exist. For instance, if G is of type A, B, C, D, or E_{6}, then the explicit decomposition formulas for E_{m_{1}ϖ_{1}} ⊗E_{m_{2}ϖ_{1}} (see [Li, 1.3] or, for the types A, B, C, D, [OV, pp. 300–302]) imply that (ϖ_{1}, ϖ_{1}) shares this property. For G = SL_{n}, the classification of all d-tuples (λ_{1}, …, λ_{d}) sharing this property can be deduced from [St], where the classification of all multiplicity free tensor products of simple SL_{n}-modules is obtained.

Example 3. Let G be the group of type E_{6}. By [Li, 1.3], for every s, t ∈ ℤ_{⩾0}, the following decomposition holds:

E_{sϖ_{1}} ⊗ E_{tϖ_{1}} ≃ \begin{aligned}& ⨁ \\ \{ & \begin{matrix}a_{1}, …, a_{4} ∈ ℤ_{⩾0} \\ a_{1} + a_{3} + a_{4} = s \\ a_{2} + a_{3} + a_{4} = t \end{matrix} \end{aligned} E_{(a_{1}+a_{2})ϖ_{1}+a_{3}ϖ_{3}+a_{4}ϖ_{6}}.  (5)

Since ((a_{1} +a_{2})ϖ_{1} +a_{3}ϖ_{3} +a_{4}ϖ_{6})^{∗} = a_{4}ϖ_{1} +a_{3}ϖ_{5} +(a_{1} +a_{2})ϖ_{6}, it follows from (5) and (2) that dim(⨂_{1⩽i⩽4} E_{n_{i}ϖ_{i}})^{G} is equal to the number of solutions in ℤ_{⩾0} of the following system of eight linear equations in eight variables a_{1}, …, a_{4}, b_{1}, …, b_{4}:

\{ \begin{matrix}a_{4} & = b_{1} + b_{2}, \\ a_{3} & = 0, \\ b_{3} & = 0, \\ a_{1} & + a_{2} = b_{4}, \\ a_{1} & + a_{3} + a_{4} = n_{1}, \\ a_{2} & + a_{3} + a_{4} = n_{2}, \\ b_{1} & + b_{3} + b_{4} = n_{3}, \\ b_{2} & + b_{3} + b_{4} = n_{4}. \end{matrix}

Since this system is nondegenerate, it has at most one such solution. Thus for G of type E_{6}, the 4-tuple (ϖ_{1}, ϖ_{1}, ϖ_{1}, ϖ_{1}) is primitive. By Theorems 1 and 4 below (see also [P_{4}, Theorem 6]) this is equivalent to the existence of an open G-orbit in (G/P_{ϖ_{1}})^{4}. Observe that this example in not in the range of Example 2: for instance, c_{4ϖ_{1},4ϖ_{1},3ϖ_{1}}^{ϖ_{1}+3ϖ_{3}+ϖ_{5}} = 2 (this may be verified, e.g., utilizing LiE).

Example 4. The following definition singles out a natural subset in the set of all primitive d-tuples. Theorem 10 below shows that this subset admits a geometric characterization as well.

Definition 2. We call a d-tuple (λ_{1}, …, λ_{d}) ∈ P_{≫}^{d} invariant-free if

c_{n_{1}λ_{1},...,n_{d}λ_{d}}^{0} = 0 for all (n_{1}, …, n_{d}) ∈ ℤ_{⩾0}^{d}.

Clearly, every 1-tuple is invariant-free. For d = 2, it follows from (2) that

(λ_{1}, λ_{2}) is invariant-free ⟺ ℚ λ_{1} ≠ ℚ λ_{2}^{∗}.

3. Notation and conventions Below we utilize the following notation, conventions, and definitions.

• k[Y] and k(Y) are respectively the algebra of regular functions and field of rational functions of an irreducible algebraic variety Y.

• Cl(Y) is the Weil divisor class group of an irreducible normal variety Y. For a nonconstant function f ∈ k(Y), the Weil divisor, divisor of zeros, and divisor of poles of f are respectively (f), (f)_{0}, and (f)_{∞}.

• If H is an algebraic group, Lie(H) and X (H) are respectively the Lie algebra and the character group Hom_{alg}(H, G_{m}) of H. We utilize additive notation for X (H) and identify in the natural way X (H) with the lattice in rational vector space

X (H)_{ℚ} := X (H) ⊗ ℚ.

If H is a connected reductive group, B_{H} its Borel subgroup, and S ⊆ B_{H} a maximal torus, we identify the set of isomorphism classes of simple algebraic H-modules with a submonoid X (S)_{++} of X (S) assigning to every simple H-module V the S-weight of unique B_{H}-stable line in V.

• Below all algebraic group actions are algebraic. The action of H on H/F is that by left multiplication. If H acts on Y_{1}, …, Y_{n}, the action of H on Y_{1} × … × Y_{n} is the diagonal one.

If H acts on a variety Y, then H · y and H_{y} are respectively the H-orbit and Hstabilizer of a point y ∈ Y, and k[Y]^{H} and k(Y)^{H} are the subalgebra and subfield of H-invariant elements in k[Y] and k(Y)^{H}.

Definition 3. For an irreducible variety Y, we call the action of H on Y ample if k(Y)^{H} is algebraic over the field of fractions of k[Y]^{H}.

Recall the following definition introduced in [P_{1}].

Definition 4. The action of H on Y is called stable if H-orbits of points lying off a proper closed subset of Y are closed in Y.

• The natural action of H on k[Y] is locally finite. If H is a connected reductive group, by k[Y]_{λ}, where λ ∈ X (S)_{++}, we denote the λ-isotypical component of H-module k[Y] and put

S(H, Y) := {λ ∈ X (S)_{++} | k[Y]_{λ} ≠ 0}.  (6)

The set S(H, Y) is a submonoid of X (S) (indeed, S(H, Y) is the set of all weights of the natural action of S on k[Y]^{B_{H}^{u}}, where B_{H}^{u} is the unipotent radical of B_{H}; whence the claim). If Y is an affine variety, then the monoid S(H, Y) is finitely generated (this readily follows from the fact that in this case k[Y]^{B_{H}^{u}} is a finitely generated k-algebra, see, e.g., [PV_{2}, 3.14]).

• If H is a reductive group and Y is an affine variety, we denote by

π_{H,Y} : Y ⟶ Y//H  (7)

the corresponding categorical quotient, i.e., Y//H is an affine variety and π_{H,Y} a dominant (in fact, surjective) morphism such that π_{H,Y}^{∗} (k[Y//H]) = k[Y]^{H}, see, e.g., [PV_{2}, 4.4]. Utilizing π_{H,Y}^{∗}, we identify k[Y//H] with k[Y]^{H} and, if Y is irreducible, k(Y//H) with the field of fractions of k[Y]^{H}.

It follows from Rosenlicht’s theorem [R_{2}, Theorem], see also, e.g., [PV_{2}, Cor. of Theorem 2.3], that the action of H on Y is ample if and only if dim(Y//H) = dim(Y) − max_{y∈Y} dim(H · y) (i.e., in the terminology of [Lu, Sect. 4], “Y//H a la bonne dimension”).

One can also prove that if Y is normal, then the action of H on Y is ample if and only if k(Y)^{H} is the field of fractions of k[Y]^{H}.

If the action of H on Y is stable, it is ample (indeed, since every fiber of π_{H,Y} contains a unique closed H-orbit, see, e.g., [PV_{2}, Cor. of Theorem 4.7], the action of is stable if and only if every general fiber is a closed H-orbit of maximal dimension).

• N_{G}(T) is the normalizer of T in G and W := N_{G}(T)/T is the Weyl group of G. For every w ∈ W we fix a choice of its representative w^{.} in N_{G}(T). The space X (T)_{ℚ} is endowed with the natural W -module structure.

For λ = ∑_{i=1}^{r} a_{i}ϖ_{i}, a_{i} ∈ ℤ_{⩾0}, the support of λ is

supp(λ) := {i ∈ ℤ_{>0} | a_{i} ≠ 0}.

The subgroup P_{λ} is then generated by T and one-dimensional unipotent root subgroups of G corresponding to all positive roots and those negative roots that are linear combinations of −α_{i}’s with i ∉ supp(λ). We have the equivalence

P_{μ} = P_{ν} ⟺ supp(μ) = supp(ν).  (8)

For every subset A of ⊆ P_{++}, we put

A^{∗} := {λ^{∗} | λ ∈ A}.

• We fix a choice of nonzero point v_{λ} of the unique B-stable line in E_{λ} and denote by O_{λ} the G-orbit of v_{λ} and by \overline{O}_{λ} its closure in E_{λ}. We put

O_{λ_{1},...,λ_{d}} := O_{λ_{1}} × … × O_{λ_{d}} and X_{λ_{1},...,λ_{d}} := \overline{O}_{λ_{1}} × … × \overline{O}_{λ_{d}}  (9)

and identify in the natural way X_{λ_{1},...,λ_{d}} with the closed subset of E_{λ_{1}} ⊕ … ⊕ E_{λ_{d}}.

• If M is a subset of a vector space, then conv(M) and cone(M) are respectively the covex hull of M and the convex cone generated by M. If M is a convex set, int(M) is the set its (relative) interior points.

• We put

ℚ_{>0} := {a ∈ ℚ | a > 0} and ℚ_{⩾0} := {a ∈ ℚ | a ⩾ 0}.

• |N| is the cardinality of a finite set N.

4. Main results In this section we formulate main results of this paper. Theorem 1 explicitly formulates the aforementioned remarkable connection of primitive

tuples with geometry.

Theorem 1. Let (λ_{1}, …, λ_{d}) ∈ P_{≫}^{d}.

(i) If G/P_{λ_{1}} × … × G/P_{λ_{d}} contains an open G-orbit, then (λ_{1}, …, λ_{d}) is primitive. (ii) If (λ_{1}, …, λ_{d}) is primitive and the action of G on X_{λ_{1},...,λ_{d}} is ample, then G/P_{λ_{1}}×

… × G/P_{λ_{d}} contains an open G-orbit.

Theorem 1 and equivalence (8) imply

Corollary. Let (λ_{1}, …, λ_{d}) and (μ_{1}, …, μ_{d}) ∈ P_{≫}^{d}. Assume that

supp(λ_{i}) = supp(μ_{i}) for all i.

If (λ_{1}, …, λ_{d}) is primitive and the action of G on X_{λ_{1},...,λ_{d}} is ample, then (μ_{1}, …, μ_{d}) is primitive as well.

Theorem 1 clarifies relation of classifying primitive d-tuples to the following problems.

Problem 1. Classify multiple flag varieties G/P_{λ_{1}} × … × G/P_{λ_{d}} that contain open G-orbit.

Problem 2. For what d-tuples (λ_{1}, …, λ_{d}) ∈ P_{≫}^{d} is the action of G on X_{λ_{1},...,λ_{d}} ample?

Regarding Problem 1, obvious dimension reason yields the finiteness statement about length of possible d-tuples: d in Problem 1 cannot exceed a constant depending only on G. A more thorough analysis leads to the following upper bounds.

Theorem 2. Let G be a simple group. If G/P_{λ_{1}} ×…×G/P_{λ_{d}} contains an open G-orbit, then d ⩽ b_{G} for the following b_{G}:

Table 1

type of G A_{l}, l ⩾ 1 B_{l}, l ⩾ 3 C_{l}, l ⩾ 2 D_{l}, l ⩾ 4 E_{6} E_{7} E_{8} F_{4} G_{2}
b_{G}  l + 2  l + 1  l + 1  l  4 4 4 3 2

Notice that, by the Bruhat decomposition, every multiple flag variety G/P_{λ_{1}} × G/P_{λ_{2}} contains only finitely many G-orbits (one of which therefore is open).

In [P_{4}] a complete solution to Problem 1 for d-tuples of the form

(λ_{1}, …, λ_{d}) = (m_{1}ϖ_{i}, …, m_{d}ϖ_{i}), 1 ⩽ i ⩽ r, (m_{1}, …, m_{d}) ∈ ℤ_{>0}^{d}.  (10)

is obtained; the answer is the following.

Theorem 3 ([P_{4}]). Let G be a simple group. Assume that d ⩾ 3 and (λ_{1}, …, λ_{d}) is given by equality (10). Then the multiple flag variety G/P_{λ_{1}} × … × G/P_{λ_{d}} contains an open G-orbit if and only if the following conditions hold:

Table 2

type of G  A_{l}, l ⩾ 1  B_{l}, l ⩾ 3 C_{l}, l ⩾ 2  D_{l}, l ⩾ 4  E_{6}  E_{7}
conditions d < \frac{(l+1)^{2}}{i(l+1−i)}  \begin{matrix}d = 3, \\ i = 1, l \end{matrix} \begin{matrix}d = 3, \\ i = 1, l \end{matrix} \begin{matrix}d = 3, \\ i = 1, l − 1, l \end{matrix} \begin{matrix}d ⩽ 4, \\ i = 1, 6 \end{matrix} \begin{matrix}d = 3, \\ i = 7 \end{matrix}

The next two theorems concern Problem 2.

Theorem 4. Let (λ_{1}, …, λ_{d}) ∈ P_{≫}^{d}. If every λ_{s} is a multiple of a fundamental weight,

λ_{s} ∈ ℤ_{>0}ϖ_{i_{s}}, s = 1, …, d,  (11)

then the action of G on X_{λ_{1},...,λ_{d}} is ample.

Theorem 5. Let G be a simple group and let (λ_{1}, …, λ_{d}) ∈ P_{≫}^{d}. If

d ⩾ sep(G),

where sep(G) is the separation index of the root system of G with respect to T (see Definition 5 in Section 9), then the action of G on X_{λ_{1},...,λ_{d}} is stable (and hence ample) and the G-stabilizer of a point in general position in X_{λ_{1},...,λ_{d}} is finite.

Other results on stability of the G-action on X_{λ_{1},...,λ_{d}} are obtained in Theorem 13 in Section 15; they are based on some results from [V]. We show that stability imposes some constraints on configuration of the set {λ_{1}, …, λ_{d}} and link the problem with some monoids that generalize Littelwood–Richardson semigroups [Z] whose investigation during the last decade culminated in solving several old problems, in particular, proving Horn’s conjecture, cf. survey [F].

We apply Theorems 1–5 to studying primitive d-tuples. Theorem 4, Definition 1, and Corollary of Theorem 1 immediately imply the following saturation property.

Theorem 6. Let (λ_{1}, …, λ_{d}) ∈ P_{≫}^{d} and (m_{1}, …, m_{d}) ∈ ℤ_{>0}^{d}. If condition (11) holds, then the following properties are equivalent:

(i) (λ_{1}, …, λ_{d}) is primitive; (ii) (m_{1}λ_{1}, …, m_{d}λ_{d}) is primitive.

Utilizing Theorems 1 and 2 we prove the following finiteness theorem about length of primitive d-tuples:

Theorem 7. Let G be a simple group. Then for every primitive d-tuple in P_{≫}^{d},

d ⩽ sep(G) + 1.

In view of inequality (41) below this implies

Corollary. Let G be a simple group. Then d ⩽ |W | + 1 for every primitive d-tuple in P_{≫}^{d}.

¿From Theorem 7 we deduce that for every simple group G,

prim(G) := sup{d ∈ ℤ_{>0} | P_{≫}^{d} contains a primitive element}

is a natural number not exceeding sep(G) + 1. Discussion in Section 2 and the last Corollary imply that 2 ⩽ prim(G) ⩽ |W | + 1.

Example 5. By Example 1 we have prim(SL_{2}) = 3. Theorem 8 below implies that if G is respectively of type A_{l}, B_{l}, C_{l}, D_{l}, E_{6}, and E_{7}, then prim(G) ⩾ l + 2, 3, 3, 3, 4, and 3.

For G = SL_{n}, we can apply to our problem the representation theory of quivers. This leads to a characterization of primitive d-tuples of fundamental weights in terms of canonical decomposition of dimension vectors of representations of some graphs and yields a fast algorithm for verifying whether such a d-tuple is primitive or not (see Theorem 14 and discussion in Section 16).

¿From Theorems 1, 3, and 4 we deduce a complete classification of primitive d-tuples of form (10):

Theorem 8. Let G be a simple group. A d-tuple

(m_{1}ϖ_{i}, …, m_{d}ϖ_{i}), where d ⩾ 3, (m_{1}, …, m_{d}) ∈ ℤ_{>0}^{d},

is primitive if and only if the conditions specified in Table 2 hold.

Combining Theorem 1 with the results of [Li], [MWZ_{1}], and [MWZ_{2}], we prove that the following 3-tuples are primitive.

Theorem 9. Let (λ_{1}, λ_{2}, λ_{3}) ∈ P_{≫}^{d}. Put s_{i} := supp(λ_{i}). Then (λ_{1}, λ_{2}, λ_{3}) is primitive in either of the following cases:

Table 3

no. type of G  condition

1 2 3 4 5 6 7  A_{l}  \begin{matrix}s_{1} = \{1\} \\ |s_{1}| = |s_{2}| = 1 \\ |s_{1}| = |s_{2}| = 1; |s_{3}| = 2 \\ |s_{1}| = 1, |s_{2}| = 2; |s_{3}| = 3 \\ |s_{1}| = 1; |s_{2}| = 2; |s_{3}| = 4 \\ s_{1} = \{2\}; |s_{2}| = 2; |s_{3}| ⩾ 2 \\ |s_{1}| = 1; s_{2} = \{i, i + 1\} or \{1, j\}, i < l, j ≠ 1; |s_{3}| ⩾ 2 \end{matrix}

{8 \atop 9}  B_{l}  \begin{matrix}s_{1} = \{1\}; s_{2} = \{1, …, l\} \\ s_{1} = s_{2} = \{l\}; s_{3} = \{1, …, l\} \end{matrix}

10 11 12 13 14  C_{l}  \begin{matrix}s_{1} = s_{3} = \{l\} \\ s_{1} = \{l\}; s_{2} = \{i\}, i ≠ l; s_{3} = \{j\}, j ≠ l \\ s_{1} = \{l\}; s_{2} = \{i\}, i ≠ l; s_{3} = \{j, m\}, j ≠ m \\ s_{1} = \{l\}; s_{2} = \{1\}; s_{3} ≠ \{l\} \\ s_{1} = \{1\}; s_{2} = \{i\}, i ≠ l; s_{3} ≠ \{l\} \end{matrix}

15 16 17 18  D_{l}  \begin{matrix}s_{1} = \{1\}; s_{2} = \{1, …, l\} \\ s_{1} = \{l − 1\}; s_{2} = \{l\}; s_{3} = \{1, …, l\} \\ s_{1} = \{l\}; s_{2} = \{l\}; s_{3} = \{1, …, l\} \\ s_{1} = \{3\}; s_{2} = \{l\}; s_{3} = \{1, …, l\} \end{matrix}

19  E_{6}  s^{1} = {1}; s^{2} = {i}, i ≠ 4; s^{3} = {1, …, 6}
20  E_{7}  s_{1} = {1} or {2} or {7}; s_{2} = {7}; s_{3} = {1, …, 7}

Finally, Theorem 10 below explains geometric meaning of invariant-freeness of d-tuples and establishes a saturation property for them. Theorem 11 shows that invariant-freeness of (λ_{1}, …, λ_{d}) ∈ P_{≫}^{d} imposes some constraints on configuration of the set {λ_{1}, …, λ_{d}} and gives an upper bound of length of invariant-free d-tuples.

Theorem 10. Let (λ_{1}, …, λ_{d}) ∈ P_{≫}^{d} and (m_{1}, …, m_{d}) ∈ ℤ_{>0}^{d}. The following properties are equivalent:

(i) (λ_{1}, …, λ_{d}) is invariant-free;

(ii) (m_{1}λ_{1}, …, m_{d}λ_{d}) is invariant-free; (iii) the closure of every G-orbit in X_{λ_{1},...,λ_{d}} contains (0, …, 0) ∈ E_{λ_{1}} ⊕ … ⊕ E_{λ_{d}}; (iv) k[X_{λ_{1},...,λ_{d}}]^{G} = k.

Theorem 11. Let (λ_{1}, …, λ_{d}) ∈ P_{≫}^{d} be an invariant-free d-tuple. Then (i) ℚ_{>0}λ_{i}^{∗} ∉ cone({λ_{1}, …, λ̂_{i}, …, λ_{d}}) for every i; (ii) d ⩽ rk(G) if G is a simple group.

5. Primitiveness and open orbits Let (λ_{1}, …, λ_{d}) ∈ P_{≫}^{d}. In this section we establish a connection between the primi-

tiveness of (λ_{1}, …, λ_{d}) and some properties of the G-actions on X_{λ_{1},...,λ_{d}} and G/P_{λ_{1}} × … × G/P_{λ_{d}}.

By [PV_{1}, Theorem 1] the variety X_{λ_{i}} is a cone in E_{λ_{i}}, i.e., is stable with respect to the action of G_{m} on E_{λ_{i}} by scalar multiplications, and

X_{λ_{i}} = O_{λ_{i}} ∪ {0}.  (12)

This G_{m}-action commutes with the G-action and determines a G-stable k-algebra ℤ_{⩾0}- grading of k[X_{λ_{i}}]:

k[X_{λ_{i}}] = ⨁_{n_{i}∈ℤ_{⩾0}} k[X_{λ_{i}}]_{n_{i}},  (13)

where k[X_{λ_{i}}]_{n_{i}} is the space of G_{m}-semi-invariants of the weight t ↦ t^{n_{i}}. By [PV_{1}, Theorem 2] there is an isomorphism of G-modules

k[X]_{n_{i}} ≃ E_{n_{i}λ_{i}^{∗}}.  (14)

The group G^{d} × G_{m}^{d} acts on X_{λ_{1},...,λ_{d}} in the natural way, and O_{λ_{1},...,λ_{d}} is a G_{m}^{d}-stable open G^{d}-orbit in X_{λ_{1},...,λ_{d}}. By restriction this action entails the action of G × G_{m}^{d}, where G is diagonally embedded in G^{d}. The action of G_{m}^{d} on X_{λ_{1},...,λ_{d}} determines a G-stable k-algebra ℤ_{⩾0}^{d}-grading of k[X_{λ_{1},...,λ_{d}}],

k[X_{λ_{1},...,λ_{d}}] = ⨁_{n_{1},...,n_{d}∈ℤ_{⩾0}} k[X_{λ_{1},...,λ_{d}}]_{(n_{1},...,n_{d})},  (15)

where k[X_{λ_{1},...,λ_{d}}]_{(n_{1},...,n_{d})} is the space of G_{m}^{d}-semi-invariants of the weight (t_{1}, …, t_{d})↦ t_{1}^{n_{1}} ⋯ t_{d}^{n_{d}}. Since, by (9), k[X_{λ_{1},...,λ_{d}}] and ⨂_{i=1}^{d} k[X_{λ_{i}}] are G-isomorphic k-algebras, (13), (14), and (15) yield that, for every (n_{1}, …, n_{d}) ∈ ℤ_{⩾0}^{d}, there is an isomorphism of Gmodules

k[X_{λ_{1},...,λ_{d}}]_{(n_{1},...,n_{d})} ≃ ⨂_{1⩽i⩽d} k[X_{λ_{i}}]_{n_{i}} ≃ ⨂_{1⩽i⩽d} E_{n_{i}λ_{i}^{∗}}.  (16)

Consider now the categorical quotient (7) for H = G and Y = X_{λ_{1},...,λ_{d}} and denote π_{H,Y} by π_{λ_{1},...,λ_{d}}. The field of fractions of k[X_{λ_{1},...,λ_{d}}]^{G} is then π_{λ_{1},...,λ_{d}}^{∗}(k(X_{λ_{1},...,λ_{d}}//G)). Since the action G_{m}^{d} on X_{λ_{1},...,λ_{d}} commutes with that of G, it descends to X_{λ_{1},...,λ_{d}}//G. The corresponding action of G_{m}^{d} on the k-algebra k[X_{λ_{1},...,λ_{d}}//G] determines its ℤ_{⩾0}^{d}- grading

\begin{matrix}k[X_{λ_{1},...,λ_{d}}//G] = ⨁_{n_{1},...,n_{d}∈ℤ_{⩾0}} k[X_{λ_{1},...,λ_{d}}//G]_{(n_{1},...,n_{d})}, where \\ π_{λ_{1},...,λ_{d}}^{∗}(k[X_{λ_{1},...,λ_{d}}//G]_{(n_{1},...,n_{d})}) = k[X_{λ_{1},...,λ_{d}}]^{G} ∩ k[X_{λ_{1},...,λ_{d}}]_{(n_{1},...,n_{d})}. \end{matrix}  (17)

From this, (15), and (16) we deduce that

k[X_{λ_{1},...,λ_{d}}//G]_{(n_{1},...,n_{d})} ≃ (⨂_{1⩽i⩽d} E_{n_{i}λ_{i}^{∗}})^{G}.  (18)

Lemma 1. The following properties are equivalent:

(i) k(X_{λ_{1},...,λ_{d}}//G)^{G_{m}^{d}} = k; (ii) there is an open G_{m}^{d}-orbit in X_{λ_{1},...,λ_{d}}//G; (iii) (λ_{1}, …, λ_{d}) is primitive.

Proof. (i)⇔(ii) follows from Rosenlicht’s theorem [R_{2}], cf., e.g., [PV_{2}, Cor. of Theorem 2.3].

Assume that (λ_{1}, …, λ_{d}) is not primitive. Then dim(⨂_{1⩽i⩽d} E_{n_{i}λ_{i}})^{G} ⩾ 2 for some (n_{1}, …, n_{d}) ∈ ℤ_{⩾0}^{d}. Since for all (μ_{1}, …, μ_{d}) ∈ P_{++}^{d} and (m_{1}, …, m_{d}) ∈ ℤ_{⩾0}^{d}, we have

dim(⨂_{1⩽i⩽d} E_{m_{i}μ_{i}})^{G} = dim(⨂_{1⩽i⩽d} E_{m_{i}μ_{i}^{∗}})^{G},  (19)

it then follows from (18) that dim(k[X_{λ_{1},...,λ_{d}}//G]_{(n_{1},...,n_{d})}) ⩾ 2. This means that the algebra k[X_{λ_{1},...,λ_{d}}//G] contains two nonproportional G_{m}^{d}-semi-invariant functions f_{1} and f_{2} of the same weight. Hence f_{1}/f_{2} ∈ k(X_{λ_{1},...,λ_{d}}//G)^{G_{m}^{d}}, f_{1}/f_{2} ∉ k. This proves (i)⇒(iii).

Conversely, let f ∈ k(X_{λ_{1},...,λ_{d}}//G)^{G_{m}^{d}}, f ∉ k. Since X_{λ_{1},...,λ_{d}}//G is an affine variety, k(X_{λ_{1},...,λ_{d}}//G) is the field of fractions of k[X_{λ_{1},...,λ_{d}}//G]. As G_{m}^{d} is a connected solvable group, by [PV_{2}, Theorem 3.3] this implies that in k[X_{λ_{1},...,λ_{d}}//G] there are two G_{m}^{d}-semi-invariant elements of the same G_{m}^{d}-weight, say, f_{1}, f_{2} ∈ k[X_{λ_{1},...,λ_{d}}//G]_{(n_{1},...,n_{d})}, such that f = f_{1}/f_{2}. Since f_{1} and f_{2} are nonproportional, dim(k[X_{λ_{1},...,λ_{d}}//G]_{(n_{1},...,n_{d})}) ⩾ 2. By (18) and (19) this yields dim(⨂_{i=1}^{d} E_{n_{i}λ_{i}})^{G} ⩾ 2. Hence (λ_{1}, …, λ_{d}) is not primitive. This proves (iii)⇒(i). □

Remark 2. By [PV_{2}, Theorem 3] every X_{λ_{i}} is a normal variety. From (9) we then conclude that X_{λ_{1},...,λ_{d}} is normal as well. Hence property (ii) in Lemma 1 means that X_{λ_{1},...,λ_{d}}//G is a toric G_{m}^{d}-variety.

Lemma 2. The following properties are equivalent:

(i) k(X_{λ_{1},...,λ_{d}})^{G×G_{m}^{d}} = k; (ii) there is an open G × G_{m}^{d}-orbit in X_{λ_{1},...,λ_{d}}; (iii) there is an open G-orbit in G/P_{λ_{1}} × … × G/P_{λ_{d}}.

Proof. The aforementioned Rosenlicht’s theorem yields the equivalencies (i)⇔(ii) and

k(G/P_{λ_{1}} × … ×G/P_{λ_{d}})^{G}=k ⟺ G/P_{λ_{1}} × … ×G/P_{λ_{d}} contains an open G-orbit. (20)

The natural projection ρ_{λ_{i}}: O_{λ_{i}} → G/P_{λ_{i}} is a rational quotient for the G_{m}-action on O_{λ_{i}}, cf. [PV_{2}, 2.4]. Hence the G-equivariant morphism ρ_{λ_{1}} × … × ρ_{λ_{d}}: O_{λ_{1},...,λ_{d}} → G/P_{λ_{1}} × … × G/P_{λ_{d}} is a rational quotient for the G_{m}^{d}-action on O_{λ_{1},...,λ_{d}}. Therefore it induces an isomorphism of invariant fields

k(G/P_{λ_{1}} × … × G/P_{λ_{d}})^{G} ⟶^{≃} k(O_{λ_{1},...,λ_{d}})^{G×G_{m}^{d}}.  (21)

But k(O) = k(X_{λ_{1},...,λ_{d}}) since O is open in X_{λ_{1},...,λ_{d}}. This, (20), and (21) now imply (i)⇔(iii). □

Lemma 3. Let (λ_{1}, …, λ_{d}) ∈ P_{≫}^{d}, s ∈ ℤ_{>0}, and

M_{d}(s) := {(n_{1}, …, n_{d}) ∈ ℤ_{⩾0}^{d} | c_{n_{1}λ_{1},...,n_{d}λ_{d}}^{0} ⩾ s}.

Then

M_{d}(1) + M_{d}(s) ⊆ M_{d}(s).

Proof. By (18) and (19)

(n_{1}, …, n_{d}) ∈ M_{d}(s) ⟺ dim(k[X_{λ_{1},...,λ_{d}}//G]_{(n_{1},...,n_{d})}) ⩾ s.  (22)

Let α ∈ M_{d}(1) and β ∈ M_{d}(s). Pick a nonzero function f ∈ k[X_{λ_{1},...,λ_{d}}//G]_{α} and linear independent functions h_{1}, …, h_{s} ∈ k[X_{λ_{1},...,λ_{d}}//G]_{β}: by (22) this is possible. Then the functions fh_{1}, …, fh_{d} are linearly independent since k[X_{λ_{1},...,λ_{d}}//G] is an integral domain. They lie k[X_{λ_{1},...,λ_{d}}//G]_{α+β} since (17) is a grading. So dim(k[X_{λ_{1},...,λ_{d}}//G]_{α+β}) ⩾ s, whence α + β ∈ M_{d}(s) by (22). □

6. Proof of Theorem 1 If the assumption of (i) holds, Lemma 2 implies that

k(X_{λ_{1},...,λ_{d}})^{G×G_{m}^{d}} = (k(X_{λ_{1},...,λ_{d}})^{G})^{G_{m}^{d}} = k.

Since

π_{λ_{1},...,λ_{d}}^{∗}: k(X_{λ_{1},...,λ_{d}}//G) ↪ k(X_{λ_{1},...,λ_{d}})^{G},  (23)

this yields k(X_{λ_{1},...,λ_{d}}//G)^{G_{m}^{d}} = k; whence (λ_{1}, …, λ_{d}) is primitive by Lemma 1. This proves (i).

If the assumption of (ii) holds, let

ϱ: X_{λ_{1},...,λ_{d}} −−→ X_{λ_{1},...,λ_{d}}_{▪} G^{▪}^{▪}

be a rational quotient for the action of G on X_{λ_{1},...,λ_{d}}. i.e., X_{λ_{1},...,λ_{d}}_{▪} G^{▪}^{▪} is an irreducible variety and ϱ a dominant rational map such that ϱ^{∗}(k(X_{λ_{1},...,λ_{d}}_{▪} G^{▪}^{▪} )) = k(X_{λ_{1},...,λ_{d}})^{G}, cf., e.g., [PV_{2}, 2.4]. By [PV_{2}, Prop. 2.6] the action of G_{m}^{d} on X_{λ_{1},...,λ_{d}} induces a rational G_{m}^{d}-action on X_{λ_{1},...,λ_{d}}_{▪} G^{▪}^{▪} such that ϱ becomes G_{m}^{d}-equivariant. By [PV_{2}, Cor. of Theorem 1.1] replacing X_{λ_{1},...,λ_{d}}_{▪} G^{▪}^{▪} with a birationally isomorphic variety, we may (and shall) assume that the rational action of G_{m}^{d} on X_{λ_{1},...,λ_{d}}_{▪} G^{▪}^{▪} is regular (morphic).

Embedding (23) induces a dominant rational G_{m}^{d}-equivariant map τ: X_{λ_{1},...,λ_{d}}_{▪} G^{▪}^{▪} −−→ X_{λ_{1},...,λ_{d}}//G such that we obtain the following commutative diagram:

\xymatrix{ & X_{λ_{1},...,λ_{d}} \ar@{-->}[dl]_{ϱ} \ar[dr]^{π_{λ_{1},...,λ_{d}}} \\ X_{λ_{1},...,λ_{d}}_{▪} G^{▪}^{▪} \ar@{-->}[rr]^{τ} &  & X_{λ_{1},...,λ_{d}}//G} .  (24)

Since the action of G on X_{λ_{1},...,λ_{d}} is ample, we have

dim(X_{λ_{1},...,λ_{d}}_{▪} G^{▪}^{▪} ) = dim(X_{λ_{1},...,λ_{d}}//G).  (25)

Since (λ_{1}, …, λ_{d}) is primitive, Lemma 1 yields that X_{λ_{1},...,λ_{d}}//G contains an open G_{m}^{d}- orbit. ¿From this, (24), and (25) it then follows that X_{λ_{1},...,λ_{d}}_{▪} G^{▪}^{▪} contains an open G_{m}^{d}- orbit. Hence (k(X_{λ_{1},...,λ_{d}}_{▪} G^{▪}^{▪} ))^{G_{m}^{d}} = k, i.e., (k(X_{λ_{1},...,λ_{d}})^{G})^{G_{m}^{d}} = k(X_{λ_{1},...,λ_{d}})^{G×G_{m}^{d}} = k. Lemma 2 then implies that G/P_{λ_{1}} × … × G/P_{λ_{d}} contains an open G-orbit. This proves (ii). □

7. Proof of Theorem 2 Since dimension of a multiple flag variety G/P_{λ_{1}} × … × G/P_{λ_{d}} containing an open

G-orbit does not exceed dim(G), we have

dim(G) ⩾ ∑_{i=1}^{d}(dim(G) − dim(P_{λ_{i}})) ⩾ d(dim(G) − dim(P)),

where P is a parabolic subgroup of G of maximal dimension. Since dim(G) = dim(L) + 2 dim(P_{u}) and dim(P) = dim(L) + dim(P_{u}), where L and P_{u} are respectively a Levi subgroup and the unipotent radical of P, this yields

d ⩽ \frac{2 dim G}{dim G − dim L}.  (26)

Let L_{i} be a Levi subgroup of P_{i} := P_{ϖ_{i}}. The equality dim(L_{i}) = 2 dim(P_{i}) − dim(G) implies that dim(L_{i}) and dim(P_{i}), as functions in i, attain their absolute maximums at the same values of i; let M be the set of these values. Then the group P is conjugate to P_{i_{0}} for some i_{0} ∈ M.

Since L_{i} is a reductive group of rank rk(G) and the Dynkin diagram of its commutator group is obtained from that of G by removing the ith node, finding all the dim(L_{i})’s and then the set M is a matter of some clear calculations. We skip them (see some details in [P_{4}, Sect. 5–13]). The results are collected in Table 4 below.

Table 4

type of G A_{l}, l ⩾ 1 B_{l}, l ⩾ 3 C_{l}, l ⩾ 2 D_{l}, l ⩾ 4 E_{6} E_{7} E_{8} F_{4} G_{2}
M  1, l  1  1  1  1, 6 7  8 1, 4 1, 2

\frac{2 dim G}{dim G−dim L} l + 2 \frac{2l^{2}+l}{2l−1} \frac{2l^{2}+1}{2l−1} \frac{2l^{2}−l}{2l−2} \frac{39}{8} \frac{133}{27} \frac{248}{57} \frac{52}{15} \frac{14}{5}

The claim now immediately follows from (26) and Table 4. □

8. Proof of Theorem 4 We put, for brevity,

X := X_{λ_{1},...,λ_{d}}, O := O_{λ_{1},...,λ_{d}}, v := v_{λ_{1}} × … × v_{λ_{d}} ∈ O.

Clearly, dim(\overline{O}_{λ}) ⩾ 2 for every λ ∈ P_{≫}, hence by (9) and (12)

codim_{X}(X \ O) ⩾ 2.  (27)

Recall from Remark 2 that X is normal. From (27) we conclude that

Cl(X) ≃ Cl(O).  (28)

Since G_{v}^{d} = G_{v_{λ_{1}}} × … × G_{v_{λ_{d}}} and G^{d} is a connected simply connected semisimple group, we deduce from [P_{2}, Prop. 1; Cor. of Theorem 4] that

Cl(O) ≃ Cl(G^{d}/G_{v}^{d}) ≃ X (G_{v}^{d}) ≃ X (G_{v_{λ_{1}}} ) ⊕ … ⊕ X (G_{v_{λ_{d}}}).  (29)

On the other hand, (11) and [PV_{1}, §1, no. 5] imply that

X (G_{v_{λ_{i}}}) ≃ ℤ/m_{i}.  (30)

¿From (28), (29), and (30) we obtain that

Cl(X) ≃ ⨁_{i=1}^{d} ℤ/m_{i}.  (31)

Further, since G is semisimple, we have

X (G) = {0};  (32)

whence every invertible element of k[G] is constant, see [R_{1}]. Therefore the same holds for k[O] as well. As O is open in X, this yields that every invertible element of k[X] is constant.

Take now a nonconstant function f ∈ k(X)^{G}. Then (f) ≠ 0. For, otherwise, the normality of X would imply (see, e.g., [M, Theorem 38]) that f is invertible element of k[X], hence a constant, a contradiction.

Since, by (31), Cl(X) is a finite group, there is n ∈ ℤ_{>0} such that both divisors n(f)_{0} and n(f)_{∞} are principal, i.e.,

n(f)_{0} = (h_{1}) and n(f)_{∞} = (h_{2}) for some h_{1}, h_{2} ∈ k(X).  (33)

As n(f)_{0} ⩾ 0 and n(f)_{∞} ⩾ 0, the normality of X and (33) imply that h_{1}, h_{2} ∈ k[X] (see, e.g., [M, Theorem 38]). Further, since f is G-invariant, the supports of (f)_{0} and (f)_{∞} are G-stable subsets of X. By [PV_{2}, Theorem 3.1] this and (33) imply that h_{1} and h_{2} are G-semi-invariants. Hence by (32)

h_{1}, h_{2} ∈ k[X]^{G}.  (34)

On the other hand, (f^{n}h_{2}/h_{1}) = 0 by (33), hence f^{n}h_{2}/h_{1} is a constant. By (34) this means that f is algebraic over the field of fractions of k[X]^{G}. Hence, by Definition 3, the action of G on X is ample. This completes the proof. □

9. Separation index of irreducible root system Let R be a root system in a rational vector space L (we assume that L is the linear

span of R) and let W (R) be the Weyl group of R. For any linear function l ∈ L^{∗}, put

l^{+} := {x ∈ L | l(x) ⩾ 0}, l^{0} := {x ∈ L | l(x) = 0}, l^{−} := {x ∈ L | l(x) ⩽ 0}. (35)

Given a subset S of L, denote by \overline{S} the closure of S in L.

Lemma 4. Let R be an irreducible root system. Then for every nonzero linear function l ∈ L^{∗}, there is a Weyl chamber C ⊂ L of R such that

\overline{C} ⊂ l^{+} and \overline{C} ∩ l^{0} = \{0\}.

Proof. First, we prove that R ∩ l^{+} contains a basis of R. If R ∩ l^{0} = ∅, this is proved, e.g., in [Se, §8, Prop. 4]. In general case, fix a choice of Euclidean structure on L^{∗} and let S be a ball in L^{∗} with the center at l. We identify in the natural way every α ∈ R with a linear function on L^{∗}. Taking S small enough, we may (and shall) assume that every α ∈ R \ l^{0} has no zeros on S. On the other hand, since R is finite, S does not lie in the union of hyperplanes defined by vanishing of the roots from R ∩ l^{0}. Hence there is an element s ∈ S such that

R ∩ s^{0} = ∅ and R ∩ l^{+} ⊇ R ∩ s^{+}.  (36)

According to the aforesaid, the equality in (36) implies that R ∩ s^{+} contains a basis of R. Then the inclusion in (36) yields the claim.

Let now β_{1}, …, β_{r} be a basis of R contained in R ∩ l^{+}. Then

∑_{i} ℚ_{>0}β_{i} ⊂ l^{+} \ l^{0}.  (37)

Let π_{1}, …, π_{r} ∈ L be the basis of L dual to β_{1}^{∨}, …, β_{r}^{∨} (i.e., π_{1}, …, π_{r} ∈ L are the fundamental weights corresponding to β_{1}, …, β_{r}). Then

π_{i} = ∑_{j} c_{ij}β_{j},  (38)

where c_{ij} are the elements of inverse Cartan matrix of R. Since R is irreducible,

c_{ij} ∈ ℚ_{>0} for all i and j,  (39)

see, e.g., [OV]. Consider now the Weyl chamber C := ∑_{i} ℚ_{>0}π_{i}. Since \overline{C} := ∑_{i} ℚ_{⩾0}π_{i}, it follows from (38) and (39), that \overline{C} \\{0\} ⊂ ∑_{i} ℚ_{>0}β_{i}. Now the claim follows from (37). □

Corollary. Let R be an irreducible root system. Then there is a sequence C_{1}, …, C_{n} ⊂ L of the Weyl chambers of R satisfying the following property:

for every nonzero linear function l ∈ L^{∗}, there is a natural i ∈ [1, n] such that

\overline{C_{i}} ⊂ l^{+} and \overline{C_{i}} ∩ l^{0} = \{0\}. (40)

Proof. Let C_{1}, …, C_{n} be a sequence of all Weyl chambers of R. By Lemma 4 it satisfies property (40). □

Definition 5. Let R be an irreducible root system in a rational vector space L. The separation index sep(R) of R is the minimal length of sequences C_{1}, …, C_{n} of Weyl chambers of R satisfying property (40).

Lemma 5. The following inequalities hold:

rk(R) + 1 ⩽ sep(R) ⩽ |W (R)|.  (41)

Proof. Let C_{1}, …, C_{sep(R)} be a sequence of Weyl chambers of R satisfying property (40). For every i, fix a choice of point x_{i} ∈ C_{i}. Arguing on the the contrary, assume that sep(R) ⩽ rk(R). Then there is a nonzero linear function l ∈ L^{∗} such that x_{i} ∈ l^{−} for all i. This contradicts property (40). Thus the left inequality in (41) is proved. The right one follows from the fact that |W (R)| is equal to the cardinality of set of all Weyl chambers of R. □

The example below shows that all equalities and inequalities in (41) are attained for suitable R’s.

Example 6. Clearly, sep(A_{1}) = 2, and it is not difficult to verify that

sep(A_{2}) = 6, sep(B_{2}) = 4, and sep(G_{2}) = 3

(since sep(R) depends only on the type of R, the meaning of notation is clear).

Remark 3. The notion of separation index can be defined in a more general setting.

Namely, let M be a finite set of nonempty subsets of a finite dimensional real vector space L.

Definition 6. We call a subset S of M separating for M if for every nonzero linear function l ∈ L^{∗} there exists a set M ∈ S such that l is strictly positive at every nonzero point of M. If there exists a separating set for M, we say that the separation property holds for M and call the minimum sep(M) of cardinalities of separating sets for M the separation index of M.

Arguing as in the proof of Lemma 5, we obtain dim(L) + 1 ⩽ sep(M) ⩽ |M|.

Example 7. Assume that (a) L = ⋃_{M∈M} M;

(b) there exists a Euclidean inner product ⟨ , ⟩ on L such that for every set M ∈ M,

the angle between every two nonzero vectors of M is acute. Then the separation property holds for M. Indeed, let l ∈ L^{∗}, l ≠ 0. Identify L^{∗} with L by means of ⟨ , ⟩. Then (a) implies that l lies in some M_{0} ∈ M, and (b) implies that l is strictly positive at every nonzero point of M_{0}.

Example 8. Let K ⊂ GL(L) be an irreducible finite reflection group and let M be the set of closures of its chambers in L. Then (a) holds. Let ⟨ , ⟩ be a K-invariant Euclidean inner product on L. Then the irreducibility of K implies that (b) holds as well. Hence, in this case, the separation property holds for M.

Definition 7. In this case we call sep(M) the separation index of K and denote it by sep(K).

Definitions 5 and 7 imply that if K is crystallographic, i.e., K = W (R) for an irreducible root system R, then sep(R) = sep(K). The next example illustrates the noncrystallographic case.

Example 9. It is not difficult to verify that sep(I_{2}(p)) = 3 for p ⩾ 7 and sep(I_{2}(5)) = 4.

It would be interesting to calculate sep(K) for every irreducible finite reflection group K and, in particular, to find sep(Δ) for every irreducible root system Δ.^{1}

10. Proof of Theorem 5 The proof is based on the following lemma.

Lemma 6. Let V = V_{1} ⊕ … ⊕ V_{n}, where V_{1}, …, V_{n} are finite dimensional G-modules, let v_{i} ∈ V_{i} be a T -weight vector of a weight μ_{i} ∈ X (T), and let v := v_{1} + … + v_{n} ∈ V. Then the following properties are equivalent:

(1) G · v is closed; (2) T · v is closed;

1Added in proof. Recently in V. Zhgoon, D. Mironov, Separating systems of Weyl chambers, Math.

Notes, to appear, the following upper bounds have been obtained: sep(A_{l}) ⩽ 2l! + 2, sep(B_{l}) = sep(C_{l}) ⩽ 2^{l+1} − 2, sep(D_{l}) ⩽ 2^{l−1}l!/(l − 1) + 2, sep(F_{4}) ⩽ 30, sep(E_{6}) ⩽ 242, sep(E_{7}) ⩽ 4610, sep(E_{8}) ⩽ 9222, sep(H_{3}) ⩽ 14, sep(H_{4}) ⩽ 30.

(3) 0 ∈ int(conv({μ_{1}, …, μ_{n}})).

Proof. This is proved in [P_{3}, Theorem 1]. □

Passing to the proof of Theorem 5, we first establish the existence of elements w_{1},…,w_{d} of W such that

(i) dim(conv({w_{1} · λ_{1}, …, w_{d} · λ_{d}})) = r(= rk(G)); (ii) 0 ∈ int(conv({w_{1} · λ_{1}, …, w_{d} · λ_{d}})). Let R ⊂ L := X (T)_{ℚ} be the root system of G with respect to T and let C_{1}, …, C_{sep(G)}

be a sequence of Weyl chambers of R satisfying property (40). For every i ⩽ sep(G), let w_{i} be the (unique) element of W such that w_{i} · λ_{i} ∈ \overline{C_{i}}. For every i ⩾ sep(G) + 1, put w_{i} = e. If (i) or (ii) fails, then conv({w_{1} · λ_{1}, …, w_{d} · λ_{d}}) ⊂ l^{−} for some linear function l ∈ L^{∗}. But the choice of C_{1}, …, C_{sep(G)} implies that there is i ⩽ sep(G) such that \overline{C_{i}} ⊂ l^{+} and \overline{C_{i}} ∩ l^{0} = \{0\}. Since λ_{i} ≠ 0, we have w_{i} · λ_{i} ∈ \overline{C_{i}} \ \{0\}, hence w_{i} · λ_{i} ∈ l^{+} \ l^{0}. Therefore w_{i} ·λ_{i} ∉ l^{−}, a contradiction. Thus (i) and (ii) hold, and the existence of desired w_{i}’s is proved.

Consider now the point

v := w^{.}_{1} · v_{λ_{1}} + … + w^{.}_{d} · v_{λ_{d}} ∈ X_{λ_{1},...,λ_{d}} ⊆ E_{λ_{1}} ⊕ … ⊕ E_{λ_{d}}.  (42)

Since w^{.}_{i} · v_{λ_{i}} ∈ E_{λ_{i}} is a weight vector of weight w_{i} · λ_{i}, it follows from (ii) and Lemma 6 that the orbit G·v is closed in X_{λ_{1},...,λ_{d}}. In turn, this implies, by Matsushima’s criterion, see, e.g., [PV_{2}, Theorem 4.17], that G_{v} is a reductive group. We claim that G_{v} is finite, i.e., that Lie(G_{v}) = 0.

To prove this, observe that since G_{w^{.}_{i}·v_{λ_{i}}} = w^{.}_{i}G_{v_{λ_{i}}} w^{.}_{i}^{−1}, decomposition (42) implies that

G_{v} = ⋂_{i=1}^{d} w^{.}_{i}G_{v_{λ_{i}}} w^{.}_{i}^{−1}.  (43)

Taking into account that w^{.}_{i} ∈ N_{G}(T) and G_{v_{λ_{i}}} is normalized by T, we deduce from (43) that G_{v} is normalized by T as well. Hence, cf., e.g., [TY, 20.7],

Lie(G_{v}) = 𝔥 ⊕ ( ⊕_{α∈S}𝔤_{α}),  (44)

where 𝔤_{α} is the Lie algebra of one-dimensional unipotent root subgroup of G corresponding to the root α ∈ R, S is a subset of R, and 𝔥 is a maximal torus of Lie(G_{v}) contained in Lie(T). Since Lie(G_{v}) is reductive, the conditions Lie(G_{v}) = 0 and 𝔥 = 0 are equivalent. To prove that 𝔥 = 0, observe that

Lie(G_{v_{λ_{i}}}) = Lie (ker λ_{i}) ⊕ ( ⊕_{α∈S_{i}}𝔤_{α})  (45)

for some S_{i} ⊂ R, see [PV_{1}]. From (43) and (45) we then deduce that

Lie(G_{v}) = ⋂_{i=1}^{d}(Lie (ker(w_{i} · λ_{i})) ⊕ ( ⊕_{α∈w_{i}·S_{i}}𝔤_{α})).  (46)

In turn, it follows from (44) and (46) that

𝔥 ⊆ ⋂_{i=1}^{d} Lie (ker(w_{i} · λ_{i})).  (47)

¿From property (i) we deduce that the right-hand side of (47) is equal to 0. Hence 𝔥 = 0, as claimed. Thus we proved that G_{v} is finite.

It follows from dim(G_{v}) = 0 that dim(G·v) = dim(G). Hence maximum of dimensions of G-orbits in X_{λ_{1},...,λ_{d}} is equal to dim(G). But the set of point whose G-orbit has maximal dimension is open in X_{λ_{1},...,λ_{d}}, cf., e.g., [PV_{2}, 1.4]. Hence G-stabilizer of a point in general position in X_{λ_{1},...,λ_{d}} is finite. Finally, since G · v is a closed orbit of maximal dimension, [P_{1}, Theorem 4] implies that the action of G on X_{λ_{1},...,λ_{d}} is stable. □

11. Proof of Theorem 7 Let (λ_{1}, …, λ_{d}) ∈ P_{≫}^{d} be a primitive d-tuple. Assume the contrary, i.e.,

sep(G) + 2 ⩽ d.  (48)

¿From (48) and Theorems 1, 5 we deduce that the multiple flag variety G/P_{λ_{1}} × … × G/P_{λ_{d}} contains an open G-orbit. Theorem 2 then implies that

d ⩽ rk(G) + 2.  (49)

¿From (48), (49) we obtain the inequality sep(G) ⩽ rk(G) that contradicts (41). □

12. Proof of Theorem 9 By Theorem 1 the claim follows from the fact that in either of the cases listed in Table

2 the multiple flag variety G/P_{λ_{1}} × G/P_{λ_{2}} × G/P_{λ_{3}} contains an open G-orbit. The latter is proved as follows.

If G is of type B_{l}, D_{l}, E_{6}, or E_{7}, then s_{3} = {1, …, rk(G)}, hence P_{λ_{3}} = B. Therefore G/P_{λ_{1}} × G/P_{λ_{2}} × G/P_{λ_{3}} contains an open G-orbit if and only if G/P_{λ_{1}} × G/P_{λ_{2}} contains an open B-orbit, cf., e.g., [P_{4}, Lem. 4]. All the pairs of fundamental weights (λ_{1}, λ_{2}) for which the latter holds are classified in [Li, 1.2]. According to this classification, for these types of G, the supports of λ_{1} and λ_{2} are precisely (up to automorphism of the Dynkin diagram) s_{1} and s_{2} specified in Table 2.

For G of types A_{l} and C_{l}, in [MWZ_{1}] and [MWZ_{2}] it is given a classification of all the products G/P_{λ_{1}} × G/P_{λ_{2}} × G/P_{λ_{3}} that contain only finitely many G-orbits. One of these orbits is then open in G/P_{λ_{1}} × G/P_{λ_{2}} × G/P_{λ_{3}}. The triples (λ_{1}, λ_{2}, λ_{3}) arising in these classifications are precisely (up to automorphism of the Dynkin diagram) the ones whose supports satisfy the conditions of cases listed in Table 2 for these types of G. (Actually, in [MWZ_{1}] and [MWZ_{2}], flag varieties are described in terms of “compositions”, i.e., essentially, dimension vectors of corresponding flags. The information in Table 2 is obtained by reformulating results of [MWZ_{1}] and [MWZ_{2}] in terms of supports of the corresponding dominant weights; obtaining this reformulation is not difficult: for instance, for G of type A_{l}, one deduces it from the fact that cardinality of the set of nonzero parts of a composition is equal to cardinality of the support of corresponding dominant weight plus 1.) □

13. Proof of Theorem 10 Since (0, …, 0) is a fixed point for the action of G on X_{λ_{1},...,λ_{d}}, the equivalence

(iii)⇔(iv) follows from the property that for every reductive group action on affine variety, disjoint invariant closed subsets are separated by the algebra of invariants, see, e.g., [PV_{2}, Theorem 4.7].

The equivalence (i)⇔(iv) follows from (17), (18), (19), and Definition 2.

The implication (i)⇒(ii) follows from Definition 2.

Arguing on the contrary, assume that (ii) holds, but (i) does not. The latter means that c_{s_{1}λ_{1},...,s_{d}λ_{d}}^{0} ⩾ 1 for some (s_{1}, …, s_{d}) ∈ ℤ_{⩾0}^{d}. Lemma 3 then implies that c_{ms_{1},...,ms_{d}}^{0} ⩾ 1 for every m ∈ ℤ_{>0}. Taking m = m_{1} … m_{d}, we obtain

c_{n_{1}m_{1}λ_{1},...,n_{d}m_{d}λ_{d}}^{0} ⩾ 1 where n_{i} = m_{1} … m̂_{i} … m_{d}s_{i}.  (50)

Definition 2 now shows that property (ii) contradicts (50). □.

14. Proof of Theorem 11 We utilize the following lemma.

Lemma 7. Let R be an irreducible reduced root system in an n-dimensional rational vector space L. Then there are the Weyl chambers C_{1}, …, C_{n+1} ⊂ L of R such that

0 ∈ conv(\{x_{1}, …, x_{n+1}\}) for every choice of points x_{1} ∈ \overline{C_{1}}, …, x_{n+1} ∈ \overline{C_{n+1}}. (51)

Proof (R. Suter). Let R^{∨} ⊂ L^{∗} be the dual root system. Take a basis l_{1}, …, l_{n} of R^{∨} and let −l_{n+1} be the corresponding maximal root of R^{∨}. Utilizing notation (35), put

Z_{i} := ⋂_{j∈[1,n+1], j≠i} l_{j}^{+}.  (52)

We claim that 0 ∈ conv({x_{1}, …, x_{n+1}}) for every choice of points x_{i} ∈ Z_{i}, i = 1, …, n+1. Indeed, if 0 ∉ conv({x_{1}, …, x_{n+1}}) for some x_{i} ∈ Z_{i}, i = 1, …, n + 1, then there is a nonzero linear function l ∈ L^{∗} such that

l(x_{i}) < 0 for every i = 1, …, n + 1.  (53)

Since L^{∗} = ⋃_{j=1}^{n+1} cone({l_{1}, …, l̂_{j}, …, l_{n+1}}), there is i_{0} such that

l ∈ cone({l_{1}, …, l̂_{i_{0}}, …, l_{n+1}}).  (54)

¿From (52) and (54) we deduce that l(x_{i_{0}}) ⩾ 0, contrary to (53). A contradiction.

Now, since every Z_{i} is a union of the closures of Weyl chambers, we can choose a Weyl chamber C_{i} lying in Z_{i}. Then required property (51) holds for C_{1}, …, C_{n+1}. □

Passing to the proof of Theorem 11 and arguing on the contrary, assume that (i) fails, i.e., for some i,

m_{i}λ_{i}^{∗} = m_{1}λ_{1} + … + { \atop m_{i}λ_{i}} + … + m_{d}λ_{d},  (55)

where (m_{1}, …, m_{d}) ∈ ℚ_{⩾0}^{d} and m_{i} > 0. Multiplying both sides of (55) by an appropriate natural number, we may (and shall) assume that (m_{1}, …, m_{d}) ∈ ℤ_{⩾0}^{d}. The Cartan com-

ponent of E_{m_{1}λ_{1}} ⊗ … ⊗ { \atop E_{m_{i}λ_{i}}} ⊗ … ⊗ E_{m_{d}λ_{d}} is E_{m_{1}λ_{1}+...+{ \atop m_{i}λ_{i}}+...+m_{d}λ_{d}}. Hence

E_{m_{1}λ_{1}} ⊗ … ⊗ { \atop E_{m_{i}λ_{i}}} ⊗ … ⊗ E_{m_{d}λ_{d}} ≃ E_{m_{1}λ_{1}+...+{ \atop m_{i}λ_{i}}+...+m_{d}λ_{d}} ⊕ …,  (56)

where the right-hand side of (56) is a direct sum of simple G-modules. It follows from (55), (56), and (2) that

c_{m_{1}λ_{1},...,m_{d}λ_{d}}^{0} ⩾ 1.  (57)

Since (57) contradicts the assumption that (λ_{1}, …, λ_{d}) is invariant-free, this proves (i). Again arguing on the contrary, assume that (ii) fails, i.e.,

d ⩾ r + 1, where r = rk(G).  (58)

By Lemma 7 there are the Weyl chambers C_{1}, …, C_{r+1} ⊂ X (T)_{ℚ} of the root system of G with respect to T such that property (51) (with n = r) holds. Inequality (58) implies that there are (unique) elements w_{1}, …, w_{r+1} ∈ W such that w_{i} · λ_{i} ∈ \overline{C_{i}} for every i. By (51) we have

0 ∈ conv({w_{1} · λ_{1}, …, w_{r+1} · λ_{r+1}}).

Hence 0 is an interior point of some face of the potytope conv({w_{1} · λ_{1}, …, w_{r+1} · λ_{r+1}}); whence

0 ∈ int(conv({w_{i_{1}} · λ_{i_{1}}, …, w_{i_{m}} · λ_{i_{m}}}))  (59)

for some i_{1}, …, i_{m}. Since w^{.}_{i} · v_{λ_{i}} ∈ E_{λ_{i}} is a weight vector of weight w_{i} · λ_{i}, it follows from (59) and Lemma 6 that the G-orbit of point w^{.}_{i_{1}} · v_{λ_{i_{1}}} + … + w^{.}_{i_{m}} · v_{λ_{im}} is closed in X_{λ_{i_{1}},...,λ_{im}}. But X_{λ_{i_{1}},...,λ_{im}} clearly admits a closed G-invariant embedding in X_{λ_{1},...,λ_{d}}, so this gives a closed G-orbit in X_{λ_{1},...,λ_{d}} as well. Since this orbit is different from (0, …, 0), Theorem 10 yields a contradiction with the assumption that (λ_{1}, …, λ_{d}) is invariant-free. This proves (ii). □

15. Stability of G-action on X_{λ_{1},...,λ_{d}} In this section we prove that several other conditions are sufficient for stability of the

action of G on X_{λ_{1},...,λ_{d}}.

Consider P_{++}^{d} as a submonoid of the group X (T)^{d} that, in turn, is considered as a lattice in the rational vector space X (T)_{ℚ}^{d} := X (T)^{d} ⊗ ℚ. Notice that if A and B are submonoids of P_{++}^{d}, then the condition

int(cone(A)) ∩ int(cone(B)) ≠ ∅

is equivalent to the property that A − B := {a − b | a ∈ A, b ∈ B} is a group.

For (λ_{1}, …, λ_{d}) ∈ P_{++}^{d}, consider the submonoid ⟨λ_{1}, …, λ_{d}⟩ of P_{++}^{d} generated by (λ_{1}, 0, …, 0), …, (0, …, 0, λ_{d}),

⟨λ_{1}, …, λ_{d}⟩ := {(n_{1}λ_{1}, …, n_{d}λ_{d}) | (n_{1}, …, n_{d}) ∈ ℤ_{⩾0}^{d}}.

Put

Γ(G, d) := {(μ_{1}, …, μ_{d}) ∈ P_{++}^{d} | (E_{μ_{1}} ⊗ … ⊗ E_{μ_{d}})^{G} ≠ 0}.  (60)

Example 10. Γ(G, 1) = {0}, and, by (2), we have Γ(G, 2) = {(μ, μ^{∗}) | μ ∈ P_{++}}.

Example 11. Put LR(G, 3) := {(λ_{1}, λ_{2}, λ_{3}) | (λ_{1}, λ_{2}, λ_{3}^{∗}) ∈ Γ(G, 3)}. Then LR(SL_{n}, 3) is the Littelwood–Richardson semigroup of order n, [Z]. It has been intensively studied during the last decade and is now rather well understood. For instance, a minimal system of linear inequalities cutting out cone(LR(SL_{n}, 3)) in X (T)_{ℚ}^{3} is found, the walls of cone(LR(SL_{n}, 3)) are described, and it is proved that LR(SL_{n}, 3) is the intersection of cone(LR(SL_{n}, 3)) with the corresponding lattice in X (T)_{ℚ}^{3} (saturation conjecture), see survey [F] and [Be_{1}], [Be_{2}]. This immediately implies analogous results about Γ(SL_{n}, 3). In [KM] some general structural results for Γ(G, 3) are obtained and Γ(Sp_{4}, 3) and Γ(G_{2}, 3) are computed. Γ(Spin_{8}, 3) is studied in [KKM].

These examples show that Γ(G, d) for d ⩽ 3 is a finitely generated submonoid of P_{++}^{d}. Actually this is true for every d, see Corollary of Theorem 12 below. It would be interesting to understand the structure of this monoid in general case. What are the inequalities cutting out cone(Γ (G, d)) in X (T)_{ℚ}^{d}? What are the generators of Γ(G, d)?

Theorem 12. Consider G as the diagonal subgroup of G^{d}. Then (see (6))

Γ(G, d) = S(G^{d}, G^{d}/G).  (61)

Proof. We can (and shall) identify in the natural way P_{++}^{d} with the monoid of dominant weights of the semisimple group G^{d} with respect to maximal torus T^{d} and Borel subgroup B^{d}. Simple G^{d}-modules are tensor products E_{μ_{1}} ⊗ … ⊗ E_{μ_{d}}, where E_{μ_{i}} is considered as the G^{d}-module via the ith projection G^{d} → G, cf., e.g., [OV, Ch. 4, §3]. This, Frobenius duality (cf., e.g., [PV_{2}, Theorem 3.12]), and formulas (19), (60), (6) now imply the claim. □

Corollary. Γ(G, d) is a finitely generated submonoid of P_{++}^{d}.

Proof. Equality (61) implies that Γ(G, d) is a submonoid of P_{++}^{d}. Since G is a reductive group, Matsushima’s criterion implies that G^{d}/G is an affine variety; whence Γ(G, d) is finitely generated (see the arguments right after formula (6)). □

Theorem 13. Let (λ_{1}, …, λ_{d}) ∈ P_{≫}^{d}. If either of the following conditions holds, then the action of G on X_{λ_{1},...,λ_{d}} is stable:

(i) int(cone(Γ(G, d))) ∩ int(cone({(λ_{1}, 0, …, 0), …, (0, …, 0, λ_{d})})) ≠ ∅; (ii) there is i such that

dim(cone({λ_{1}, …, λ̂_{i}, …, λ_{d}})) = rk(G),
λ_{i}^{∗} ∈ int(cone({λ_{1}, …, λ̂_{i}, …, λ_{d}}));

(62)

(iii) {1, …, d} is a disjoint union of subsets {i_{1}, …, i_{s}} and {j_{1}, …, j_{t}} such that

dim(cone({λ_{i_{1}}, …, λ_{i_{s}}})) = dim(cone({λ_{j_{1}}, …, λ_{j_{t}}})) = rk(G),

int(cone({λ_{i_{1}}, …, λ_{i_{s}}})) ∩ int(cone({λ_{j_{1}}^{∗}, …, λ_{j_{t}}^{∗}})) ≠ ∅. (63)

Proof. (1) Discussion in Section 5 (see formula (16)) implies that k[X_{λ_{1},...,λ_{d}}]_{(n_{1},...,n_{d})} is a simple G^{d}-module with highest weight (n_{1}λ_{1}^{∗}, …, n_{d}λ_{d}^{∗}). This and (15) imply that

S(G^{d}, X_{λ_{1},...,λ_{d}}) = ⟨λ_{1}^{∗}, …, λ_{d}^{∗}⟩.  (64)

By [V, Theorem 10] the action of G on X_{λ_{1},...,λ_{d}} is stable if S(G^{d}, X_{λ_{1},...,λ_{d}})−S(G^{d}, G^{d}/G) is a group. But Γ(G, d)^{∗} = Γ(G, d) by (19) and (60). Hence (64) and Theorem 12 imply that the action of G on X_{λ_{1},...,λ_{d}} is stable if (i) holds.

(2) The variety X_{λ_{1},...,λ_{d}} is G-isomorphic to Y × Z, where Y := X_{λ_{1},...,λ̂_{i},...,λ_{d}} and Z := X_{λ_{i}}. Discussion in Section 5 shows that S(G, Y)^{∗} ∋ λ_{1}, …, λ̂_{i}, …, λ_{d} and S(G, Z) = ℤ_{⩾0}λ_{i}^{∗}. Hence

cone(S(G, Y)^{∗}) ⊇ cone({λ_{1}, …, λ̂_{i}, …, λ_{d}}) and cone(S(G, Z)) = ℚ_{⩾0}λ_{i}^{∗}. (65)

If (ii) holds, we deduce from (62) and (65) that

int(cone(S(G, Y)^{∗})) ∩ int(cone(S(G, Z))) ≠ ∅.  (66)

By [V, Theorem 9] inequality (66) implies that the action of G on Y × Z is stable.

(3) Assume now that (iii) holds. The variety X_{λ_{1},...,λ_{d}} is isomorphic to Y × Z, where Y := X_{λ_{i_{1}},...,λ_{is}} and Z := X_{λ_{j_{1}},...,λ_{j_{t}}}. Hence

cone(S(G, Y)^{∗}) ⊇ cone({λ_{i_{1}}, …, λ_{i_{s}}}),

cone(S(G, Z)) ⊇ cone({λ_{j_{1}}^{∗}, …, λ_{j_{t}}^{∗}}). (67)

It follows from (63) and (67) that, as above, (66) holds and hence the action of G on Y × Z is stable. □

16. Case of SL_{n} Let G = SL_{n}. In this case, combining the above results with that of the representation

theory of quivers (we refer to [K_{1}], [K_{2}], [DW], [Sc] for the notions of this theory) leads to a characterization of primitive d-tuples of fundamental weights in terms of canonical decomposition of dimension vectors of representations of some graphs and to an algorithmic way of solving, for every such d-tuple (ϖ_{i_{1}}, …, ϖ_{i_{d}}), whether it is primitive or not.

Namely, in this case, G/P_{ϖ_{i}} is the Grassmannian variety of i-dimensional linear subspaces on k^{n}, and the existence of an open G-orbit in G/P_{ϖ_{i_{1}}} × … × G/P_{ϖ_{i_{d}}} admits the following reformulation in terms of the representation theory of quivers. Let V_{d} be the quiver with d + 1 vertices, d outside, one inside, and the arrows from each vertex outside to a vertex inside (the vertices are enumerated by 1, …, d + 1 so that the inside vertex is enumerated by 1):

Given a vector

α := (a_{1}, …, a_{d+1}) ∈ ℤ_{⩾0}^{d+1},

put GL_{α} := GL_{a_{1}} × … × GL_{a_{d+1}} (we set GL_{0} := {e}). Let

Rep(V_{d}, α) := Mat_{a_{1}×a_{2}} ⊕ … ⊕ Mat_{a_{1}×a_{d+1}}

be the space of α-dimensional representations of V_{d} endowed with the natural GL_{α}- action. For V_{d}, the Euler inner product ⟨ | ⟩ on ℤ^{d+1} is given by

⟨(x_{1}, … x_{d+1}) | (y_{1}, …, y_{d+1})⟩ = (x_{1}y_{1} + … + x_{d+1}y_{d+1}) − y_{1}(x_{2} + … + x_{d+1}). (68)

It then follows from the basic definitions that the following properties are equivalent:

(a) the multiple flag variety G/P_{ϖ_{i_{1}}} × … × G/P_{ϖ_{i_{d}}} contains an open G-orbit;

(b) the space Rep(V_{d}, γ), where

γ := (n, i_{1}, …, i_{d}),

contains an open GL_{γ}-orbit.

Theorem 14. Let G = SL_{n}. The following properties are equivalent:

(i) (ϖ_{i_{1}}, …, ϖ_{i_{d}}) is primitive; (ii) all the roots β_{i} appearing in the canonical decomposition of γ,

γ = β_{1} + … + β_{s},  (69)

are real, i.e., ⟨β_{i} | β_{i}⟩ = 1.

Proof. By Theorems 1 and 4 properties (i) and (a) are equivalent. On the other hand, properties (ii) and (b) are equivalent by [K_{2}, Cor. 1 of Prop. 4]. □

Note that there are combinatorial algorithms for finding decomposition (69) (see [Sc], [DW]; the algorithm in [DW] is fast). Hence they, Theorem 14, and formula (68) yield algorithms verifying, for every concrete d-tuple (ϖ_{i_{1}}, …, ϖ_{i_{d}}), whether it is primitive or not.

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Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, Moscow, 119991, Russia

   E-mail address: popovvl@orc.ru