Journal of Lie Theory Volume 12 (2002) 289–299 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

A Classification of Reductive Linear Groups

with Spherical Orbits

Ivan Arzhantsev^{∗}

Communicated by E. B. Vinberg

Abstract. We classify finite dimensional G-modules V of an algebraic reductive group G such that any G-orbit in V is spherical. It is shown that any module with this property can be realized as a spherical module after an extension of the group by a central torus. Key words: reductive groups, spherical modules, algebras of invariants. AMS 2000 Subject Classification: primary 20G05, 17B10; secondary 14M17, 14R20.

1. Introduction

Let G be a connected reductive algebraic group over an algebraically closed field K of zero characteristic, and G^{s} be the maximal connected semisimple subgroup of G. Let us recall that an irreducible G-variety Y is called spherical if the induced action B : Y of a Borel subgroup B ⊆ G has an open orbit. By Rosenlicht’s theorem, an action G : Y is spherical iff K(Y)^{B} = K, where K(Y)^{B} is the field of rational B -invariants. An algebraic subgroup H ⊂ G is said to be spherical if the homogeneous space G/H is a spherical G-variety.

Definition 1.1. Let X be an irreducible algebraic variety. We shall say that an action G : X is an action with spherical orbits if there exists an open subset X_{0} ⊂ X such that for any x ∈ X_{0} the orbit Gx is spherical.

Below we list some basic facts about actions with spherical orbits.

(1) Any trivial G-action is an action with spherical orbits.

(2) Suppose that for an action G : X a generic isotropy subgroup exists, see [9, sec. 7.3]. (This is always the case for linear actions.) Denote this subgroup by H. The action G : X is an action with spherical orbits iff H is a spherical subgroup of G.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

∗ The work was supported by INTAS-OPEN-97-1570, CRDF RM1-2088 and RFBR-98-01-00598.

(3) Rosenlicht’s theorem implies that an action G : X is an action with spherical orbits iff K(X)^{G} = K(X)^{B}.

(4) It is shown in [1, Corollary 1] that for an action with spherical orbits any G-orbit is spherical.

(5) Let G_{1} : X_{1} and G_{2} : X_{2} be actions with spherical orbits. Then the action (G_{1} × G_{2}) : (X_{1} × X_{2}) is an action with spherical orbits.

In this paper we classify G-modules with spherical orbits. Let V be a finite dimensional G-module and V = V_{1} ⊕ … ⊕ V_{k} be a decomposition into simple G-submodules. Denote by T the k-dimensional algebraic torus acting by dilatations on any V_{i}. Then \overline{G} = T G is a linear group acting on V . It is shown (Proposition 3.3) that the \overline{G}-module V is spherical. This result and a classification of spherical modules from [8] allow us to list all ”minimal” indecomposable linear actions with spherical orbits (Tables 1-3). In Theorem 6.2 a characterization of arbitrary G-modules with spherical orbits is obtained. It is proved that for any G-module with spherical orbits the algebras of U - and G^{s}-invariants are free. (Here U is a maximal unipotent subgroup of G.) Finally we get a classification of G-actions with spherical orbits on projective spaces ℙ(V).

The author is grateful to E. B. Vinberg, A. G. Khovanskii and K. Kaveh for stimulating questions and discussions.

2. Indecomposable modules

Definition 2.1. A G-module V is said to be a module with spherical orbits if the linear action G : V is an action with spherical orbits.

Definition 2.2. A G-module V is indecomposable if there exist no proper decompositions G^{s} = G_{1}^{s} × G_{2}^{s} and V = V_{1} ⊕ V_{2} such that (g_{1}, g_{2})(v_{1}, v_{2}) = (g_{1}v_{1}, g_{2}v_{2}) for any g = (g_{1}, g_{2}) ∈ G^{s} and any v = (v_{1}, v_{2}) ∈ V .

Definition 2.3. We say that a G′-module V is obtained from a G-module V by a torus extension if there exists a torus T acting on V such that T - and G-actions commute and G′ = T G.

It is clear that any G-module V is obtained by a torus extension from the G^{s}-module V .

Lemma 2.4. Suppose that V is a G-module with spherical orbits and a G′- module V is obtained from this module by a torus extension. Then V is a G′- module with spherical orbits.

Proof. Let H be the generic isotropy subgroup for the action G : V . By assumption, H is spherical in G. Then any subgroup of G′ containing H is spherical in G′. Hence a generic isotropy subgroup for the G′-module V is spherical.

Theorem 2.5. All indecomposable G-modules with spherical orbits are either indicated in Tables 1-3 or are obtained from the indicated modules by a torus extension.

Table 1

G  weights  dim V  H  codim

0  {e}  0  1  0  1

1  SL(n)  ϕ_{1}  n  A_{n−2} + R_{n−1}  0

2  Λ^{2}SL(2n)  ϕ_{2}  2n^{2} − n  C_{n}  1

3  Λ^{2}SL(2n + 1)  ϕ_{2}  2n^{2} + n  C_{n} + R_{2n}  0

4  S^{2}SL(2n)  2ϕ_{1}  2n^{2} + n  D_{n}  1

5  S^{2}SL(2n + 1)  2ϕ_{1}  2n^{2} + 3n + 1  B_{n}  1

6  SO(2n)  ϕ_{1}  2n  B_{n−1}  1

7  SO(2n + 1)  ϕ_{1}  2n + 1  D_{n}  1

8  Spin(7)  ϕ_{3}  8  G_{2}  1

9  Spin(9)  ϕ_{4}  16  B_{3}  1

10  Spin(10)  ϕ_{4}  16  B_{3} + R_{8}  0

11  Sp(2n)  ϕ_{1}  2n  C_{n−1} + R_{2n−1}  0

12  G_{2}  ϕ_{1}  7  A_{2}  1

13  E_{6}  ϕ_{1}  27  F_{4}  1

Comments to the Tables. The column ”G” contains a reductive group G. In Table 1 the linear group Λ^{2}SL(n) is the image of SL(n) under the action in the second exterior power of the tautological representation, and S^{2}SL(n) is the same thing with respect to the second symmetric power.

In the column ”weights” the highest weights of the G-module are indicated.

Table 2

G  weights  dim V  H  codim
14  SL(2) × K^{∗}  ϕ_{1} ⊗ ϵ + ϕ_{1} ⊗ ϵ^{−1}  4  t_{1}  1
15  SL(n) × K^{∗}, n > 2  ϕ_{1} ⊗ ϵ^{a} + ϕ_{1} ⊗ ϵ^{b}, a ≠ b  2n  A_{n−3} + t_{1} + R_{2(n−2)}  0
16  SL(n), n > 2  ϕ_{1} + ϕ_{n−1}  2n  A_{n−2}  1
17  SL(2n + 1)  ϕ_{1} + ϕ_{2}  (2n + 1)(n + 1)  C_{n}  1
18  SL(2n + 1) × K^{∗}  ϕ_{1} ⊗ ϵ^{a} + ϕ_{2n−1} ⊗ ϵ^{b}, a ≠ nb (2n + 1)(n + 1)  C_{n−1} + t_{1} + R_{2(2n−1)}  0
19  SL(2n)  ϕ_{1} + ϕ_{2}  n(2n + 1)  C_{n−1} + R_{2n−1}  1

ϕ_{1} + ϕ_{2n−2}

20  SO(8)  ϕ_{1} + ϕ_{3}  16  G_{2}  2
21  Sp(2n) × K^{∗}  ϕ_{1} ⊗ ϵ + ϕ_{1} ⊗ ϵ^{−1}  4n  C_{n−1} + t_{1}  1
22  SL(n) × SL(m), n > m  ϕ_{1} ⊗ ϕ_{1}  nm  A_{n−m−1} + A_{m−1} + R_{nm−m^{2}}  0
23  SL(n) × SL(n)  ϕ_{1} ⊗ ϕ_{1}  n^{2}  A_{n−1}  1
24  SL(2) × Sp(2n)  ϕ_{1} ⊗ ϕ_{1}  4n  C_{n−1} + A_{1}  1
25 SL(3) × Sp(2n) × K^{∗}, n > 1  ϕ_{1} ⊗ ϕ_{1} ⊗ ϵ  6n  C_{n−2} + A_{1} + t_{1} + R_{2n−1}  0
26  SL(4) × Sp(4)  ϕ_{1} ⊗ ϕ_{1}  16  C_{2}  1
27  SL(n) × Sp(4), n > 4  ϕ_{1} ⊗ ϕ_{1}  4n  A_{n−5} + C_{2} + R_{4(n−4)}  0

For the group G_{1}×G_{2} the weight ϕ⊗ψ corresponds to the tensor product of simple G_{1}- and G_{2}-modules with highest weights ϕ and ψ respectively. The symbol + denotes a direct sum of modules. If G^{s} is the product of several simple groups, then their fundamental weights are denoted successively by letters ϕ_{i}, ψ_{i} and τ_{i}. The fundamental weight of the central torus is denoted by ϵ (for a two-dimensional torus – by ϵ_{1} and ϵ_{2}).

In the column ” dim V ” the dimension of the module is shown.

In Tables 1 and 2 the column ”H” contains the type of the tangent algebra H of the generic isotropy subgroup H for our module. Here t_{1} is the tangent algebra of the one-dimensional central torus in H, and R_{k} is the tangent algebra of the k-dimensional unipotent radical of H. The information of this column is taken from Elashvili’s tables [4], [5].

Table 3

G  weights  dim V  codim

28  SL(n) × SL(n) × K^{∗}  ϕ_{1} ⊗ ϵ + ϕ_{1} ⊗ ψ_{1}  n(n + 1)  1
ϕ_{1} ⊗ ϵ + ϕ_{n−1} ⊗ ψ_{n−1}

29  SL(n + 1) × SL(n) × K^{∗}  ϕ_{1} ⊗ ϵ^{n} + ϕ_{1} ⊗ ψ_{1} ⊗ ϵ^{−1}  (n + 1)^{2}  1

30 SL(n + 1) × SL(n) × K^{∗} × K^{∗}, n > 1  ϕ_{1} ⊗ ϵ_{1} + ϕ_{n} ⊗ ψ_{n−1} ⊗ ϵ_{2}  (n + 1)^{2}  0

31  SL(n) × SL(m) × K^{∗}, n > m + 1  ϕ_{1} ⊗ ϵ^{a} + ϕ_{1} ⊗ ψ_{1} ⊗ ϵ^{b}, a ≠ b  n(m + 1)  0

32 SL(n) × SL(m) × K^{∗}, n > m + 1 > 2 ϕ_{1} ⊗ ϵ^{a} + ϕ_{n−1} ⊗ ψ_{m−1} ⊗ ϵ^{b}, a ≠ −b n(m + 1)  0

33  SL(n) × SL(m) × K^{∗}, n < m  ϕ_{1} ⊗ ϵ^{a} + ϕ_{1} ⊗ ψ_{1} ⊗ ϵ^{b}, a ≠ 0  n(m + 1)  0
ϕ_{1} ⊗ ϵ^{a} + ϕ_{n−1} ⊗ ψ_{m−1} ⊗ ϵ^{b}, a ≠ 0

34 SL(n) × SL(2) × SL(m), n > 2, m > 2  ϕ_{1} ⊗ ψ_{1} + ψ_{1} ⊗ τ_{1}  2(n + m)  0

35 SL(n) × SL(2) × Sp(2m), n > 2, m ≥ 1  ϕ_{1} ⊗ ψ_{1} + ψ_{1} ⊗ τ_{1}  2(n + 2m)  1

36 Sp(2n) × SL(2) × Sp(2m), n, m ≥ 1  ϕ_{1} ⊗ ψ_{1} + ψ_{1} ⊗ τ_{1}  4(m + n)  2

37  SL(2) × Sp(2n) × K^{∗}  ϕ_{1} ⊗ ϵ + ϕ_{1} ⊗ ψ_{1}  2(2n + 1)  1

In the last column the codimension of a generic G-orbit in V is shown.

3. Spherical modules

It is natural to say that a G-module V is spherical if V is a spherical Gvariety. (In particular, G acts on V with an open orbit.) Such modules are also known as multiplicity free modules. The motivation for this term is the following characterization of spherical G-modules: any simple G-module appears in the decomposition of the G-module K[V] of polynomial functions on V with multiplicity ≤ 1 [7]. A classification of spherical modules was obtained in [6] (simple modules), and in [2], [3], [8] (non-simple modules).

Let V = V_{1} ⊕ … ⊕ V_{k} be a decomposition of a G-module as a sum of simple submodules, and T = T_{1} × … × T_{k} be a torus acting on V , where T_{i} is a one-dimensional torus acting on V_{i} by dilatations and trivially on the other components. We shall say that T is the saturating torus for the G-module V , and set \overline{G} = GT.

Definition 3.1. A G-module V is said to be saturated if G = \overline{G}.

We need the following well-known lemma.

Lemma 3.2. ( [6, Lemma 3.11]) If V is a simple G-module and H is a spherical subgroup of G, then for the subspace V^{H} of H -fixed points one has dim V^{H} ≤ 1.

Proposition 3.3. If V is a G-module with spherical orbits, then V is a spherical \overline{G}-module.

Proof. Let H be the generic isotropy subgroup for V . By assumption, H is a spherical subgroup of G. Hence dim V_{i}^{H} ≤ 1 and dim V^{H} ≤ k. This implies that the torus T acts on V^{H} with an open orbit, and the \overline{G}-orbit of a point of the open T -orbit in V^{H} is open in V . On the other hand, this orbit is spherical by Lemma 2.4.

4. The algebra of U -invariants

Let U be a maximal unipotent subgroup of G. For any G-module V the algebra K[V]^{U} of polynomial U -invariants is the linear span of the highest weight vectors of the G-module K[V]. The following proposition can be found, for example, in [8, Theorem 1.2].

Proposition 4.1. A G-module V is spherical iff K[V]^{U} = K[g_{1}, …, g_{m}], where the g′_{i}s are algebraically independent and the weights of g_{i} are linearly independent.

In particular, the algebra K[V]^{U} is free for any spherical G-module V .

Proposition 4.2. For any G-module V with spherical orbits the algebras K[V]^{U} and K[V]^{G^{s}} are free.

Proof. The subgroup U is a maximal unipotent subgroup of the saturated linear group \overline{G}. Thus the first statement follows from Proposition 3.3 and Proposition 4.1. For a semisimple group the semigroup of dominant weights generates (over ℚ) a strictly convex cone. Hence the algebra K[V]^{G^{s}} is generated by G^{s}- invariant free generators g_{i} of the algebra K[V]^{U}.

Finally we get an analog of Proposition 4.1 for modules with spherical orbits.

Proposition 4.3. Let G be a connected semisimple group. The following conditions are equivalent:

(1) V is a G-module with spherical orbits;

(2) K[V]^{U} = K[f_{1}, …, f_{h}, g_{1}, …, g_{m}], where the f′_{i}s are generators of K[V]^{G}, the g′_{i}s are algebraically independent and their weights are linearly independent.

Proof. (1) ⇒ (2) We may suppose that f_{i}, g_{j} are free generators of K[V]^{U}. Denote by ω_{i} the weight of g_{i}. If ω_{i} are linearly dependent, then (after reindexing) one has b_{1}ω_{1} + … + b_{p}ω_{p} = c_{p+1}ω_{p+1} + … + c_{m}ω_{m}, where b_{i}, c_{j} are non-negative integers. Hence

\frac{g_{1}^{b_{1}} … g_{p}^{b_{p}}}{g_{p+1}^{c_{p+1}} … g_{m}^{c_{m}}} ∈ K(V)^{B} = K(V)^{G}.

But for a semisimple group any rational G-invariant is the ratio of two polynomial G-invariants [9, sec. 3.2]. Therefore

g_{1}^{b_{1}} … g_{p}^{b_{p}}F_{1}(f_{1}, …, f_{h}) = g_{p+1}^{c_{p+1}} … g_{m}^{c_{m}}F_{2}(f_{1}, …, f_{h}).

This contradiction concludes the proof.

(2) ⇒ (1) Any rational B -invariant is the ratio of two polynomial B - semi-invariants of the same weight [9, sec. 3.2]. Thus any element of K(V)^{B} is the ratio of g_{1}^{d_{1}} … g_{m}^{d_{m}}F_{1}(f_{1}, …, f_{h}) and g_{1}^{d_{1}} … g_{m}^{d_{m}}F_{2}(f_{1}, …, f_{h}). This shows that K(V)^{B} = K(V)^{G}.

5. A technical lemma

Let T be an algebraic torus and X(T) be the character group of T. Consider a field K_{1} = K(a_{1}, …, a_{n}) with a T -action (by automorphisms) such that t∗a_{i} = λ_{i}(t)a_{i} for some λ_{i} ∈ X(T). The subfield of T -fixed elements of K_{1} is denoted by K_{1}^{T}. By tr.deg F we denote the transcendency degree of a field F over K. Consider the subfield K_{2} = K(a_{1}, …, a_{q}), q ≤ n. Denote by S_{1} =< λ_{1}, …, λ_{n} > and S_{2} =< λ_{1}, …, λ_{q} > the corresponding sublattices in X(T). For any sublattice L ⊂ X(T) define L_{ℚ} as the vector space L ⊗_{ℤ} ℚ.

A sublattice L determines the subgroup

T_{L} = {t ∈ T | λ(t) = 1 ∀λ ∈ L}.

Lemma 5.1. Suppose that K_{1}^{T} = K. Then the following conditions are equivalent:

(1) tr.deg K_{1}^{T_{L}} = tr.deg K_{2}^{T_{L}};
(2) L ∩ S_{1} ⊂ (S_{2})_{ℚ}.

Proof. Set A_{1} = K[a_{1}, …, a_{n}] and A_{2} = K[a_{1}, …, a_{q}]. Denote by Spec A_{i} the affine variety corresponding to the algebra A_{i}, i = 1, 2. For the natural T - action one has tr.deg K_{i}^{T_{L}} = dim Spec A_{i}−h_{i}, where h_{i} is the dimension of generic T -orbits in Spec A_{i}. The condition K_{1}^{T} = K implies dim Spec A_{1} = rk S_{1} and dim Spec A_{2} = rk S_{2}. Moreover, h_{i} = dim T_{L} − dim T_{S_{i}+L}. Thus condition (1) is equivalent to rk (S_{1} ∩ L) = rk (S_{2} ∩ L). This is a reformulation of (2).

6. Modules with spherical orbits

Let V be a saturated spherical G-module. Any character of the saturating torus T can be considered as a character of G with trivial restriction to G^{s}. For any sublattice L ⊂ X(T) define

G_{L} = {g ∈ G | λ(g) = 1 ∀λ ∈ L}.

Let B^{s} be a Borel subgroup of G^{s}. Then B = T B^{s} and B_{L} = T_{L}^{0}B^{s} are Borel subgroups of G and G_{L} respectively. Set

S(V) = {λ ∈ X(T) | ∃f ∈ K(V)^{B^{s}} : t ∗ f = λ(t)f ∀t ∈ T }

and

M(V) = {λ ∈ X(T) | ∃f ∈ K(V)^{G^{s}} : t ∗ f = λ(t)f ∀t ∈ T }.

Proposition 6.1. (cf. [8, Theorem 2.6]) The G_{L}-module V is spherical iff L ∩ S(V) = {0}.

Proof. Lemma 5.1 with K_{1} = K(V)^{B^{s}} and K_{2} = K shows that the condition L ∩ S(V) = {0} is equivalent to tr.deg K(V)^{B_{L}} = 0, or K(V)^{B_{L}} = K.

Theorem 6.2. A G-module V is a module with spherical orbits iff the \overline{G}- module V is spherical and G = G_{L} with L ∩ S(V) ⊂ M(V)_{ℚ}.

Proof. Appling Lemma 5.1 with K_{1} = K(V)^{B^{s}} and K_{2} = K(V)^{G^{s}} one shows that the second condition is equivalent to

tr.deg (K(V)^{B^{s}})^{T_{L}} = tr.deg (K(V)^{G_{s}})^{T_{L}},

or K(V)^{B_{L}} = K(V)^{G_{L}}.

In Table 4 we indicate bases of the subspaces S(V)_{ℚ} and M(V)_{ℚ} for all indecomposable saturated spherical G-modules. There is a natural bijection between such modules and items of Tables 1-3, see [8, Theorem 2.5]. Our calculations are based on [8, Table 2].

7. Proof of Theorem 2.5

Suppose that V is a spherical indecomposable saturated \overline{G}-module. By Proposition 3.3 and Lemma 2.4, we need to find all minimal (with respect to inclusion) subgroups G, G^{s} ⊆ G ⊆ \overline{G}, such that V is a G-module with spherical orbits. By Theorem 6.2, these subgroups coincide with G_{L} for a maximal sublattice L ⊂ X(T) satisfying L ∩ S(V) ⊂ M(V)_{ℚ}.

Case 1. S(V) = M(V) = 0. Here V is a spherical G^{s}-module, G = G^{s}, and we obtain items 1, 3, 10, 11, 22, 27, 34.

Table 4

\begin{aligned}items & S(V)ℚ & M(V)ℚ \\ 1, 3, 10, 11, 22, 27, 34 & 0 & 0 \\ 0, 2, 4 − 9, 12, 13, 23, 24, 26 & < ϵ_{1} > & < ϵ_{1} > \\ 19, 35 & < ϵ_{2} > & < ϵ_{2} > \\ 25, 33 & < ϵ_{1} > & 0 \\ 20, 36 & < ϵ_{1}, ϵ_{2} > & < ϵ_{1}, ϵ_{2} > \\ 15, 31 & < ϵ_{1} − ϵ_{2} > & 0 \\ 18 & < ϵ_{1} − nϵ_{2} > & 0 \\ 30 & < ϵ_{1}, ϵ_{2} > & 0 \\ 32 & < ϵ_{1} + ϵ_{2} > & 0 \\ 16 & < ϵ_{1} + ϵ_{2} > & < ϵ_{1} + ϵ_{2} > \\ 14, 21 & < ϵ_{1}, ϵ_{2} > & < ϵ_{1} + ϵ_{2} > \\ 17 & < ϵ_{1} + nϵ_{2} > & < ϵ_{1} + nϵ_{2} > \\ 28, 37 & < ϵ_{1}, ϵ_{2} > & < ϵ_{2} > \\ 29 & < ϵ_{1}, ϵ_{2} > & < ϵ_{1} + nϵ_{2} > \end{aligned}

Case 2. rk S(V) = rk M(V) ≠ 0. Here V is a G^{s}-module with spherical orbits. One gets items 0, 2, 4 – 9, 12, 13, 16, 17, 19, 20, 23, 24, 26, 35, 36.

Case 3. rk S(V) = 1 and M(V) = 0. In this case G can be obtained as an extension of G^{s} by any one-dimensional subtorus of T_{L} with the restriction L ∩ S(V) = {0}. We get items 15, 18, 25, 31, 32, 33.

Case 4. rk S(V) = 2 and M(V) = 0. In this case the \overline{G}-module V is the only module with spherical orbits – item 30.

Case 5. rk S(V) = 2 and rk M(V) = 1. In this case G is obtained as an extension of G^{s} by a one-dimensional subtorus of T_{L} with L_{ℚ} = M(V)_{ℚ} (this is the torus fixing generic G^{s}-orbits in V ). One gets items 14, 21, 28, 29, 37.

This completes the proof of Theorem 2.5.

Remark 7.1. Suppose that there is a decomposable G-module V . We are going to check that all orbits in V are spherical. First we verify that for all indecomposable components V_{i} the G-module V_{i} is a module with spherical orbits (Theorem 2.5). If this is the case we apply Theorem 6.2 and use Table 4.

8. Actions with spherical orbits on projective spaces

Any linear action G : V determines an action G : ℙ(V) on the projective space ℙ(V). Let T′ be a one-dimensional torus acting on V by dilatations and G′ = T′G be the torus extension. It is clear that the action G : ℙ(V) is an action with spherical orbits iff the action G′ : V has this property. Let us recall that T is the saturating torus for the G-module V . Denote by F the sublattice in X(T) corresponding to T′. (In standart coordinates F is given as x_{1}+…+x_{k} = 0.) The following results are immediate consequences of Theorem 6.2 and Theorem 2.5.

Proposition 8.1. An action G : ℙ(V) is an action with spherical orbits iff the \overline{G}-module V is spherical and G = G_{L} with (L ∩ F ∩ S(V)) ⊂ M(V)_{ℚ}.

Proposition 8.2. Suppose that G is semisimple and a G-module V is indecomposable. The action G : ℙ(V) is an action with spherical orbits iff either the G-module V is a module with spherical orbits or V is one of the modules indicated in Table 5.

Table 5

G  weights  dim V

1  SL(3) × Sp(2n), n > 1  ϕ_{1} ⊗ ψ_{1}  6n

2  SL(2n + 1), n > 1  ϕ_{1} + ϕ_{2n−1}  (2n + 1)(n + 1)

3 SL(n) × SL(m), n > m + 1 > 2 ϕ_{1} + ϕ_{n−1} ⊗ ψ_{m−1}  n(m + 1)

4  SL(n) × SL(m), n < m  ϕ_{1} + ϕ_{1} ⊗ ψ_{1}  n(m + 1)

ϕ_{1} + ϕ_{n−1} ⊗ ψ_{m−1}

References

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Ivan Arzhantsev Chair of Algebra Moscow State University Vorobievy Gory, Moscow, 119899, Russia arjantse@mccme.ru

Received March 2, 2001 and in final form April 1, 2001