Journal of Lie Theory Volume 11 (2001) 381–413 ◯^{C}2001 Heldermann Verlag

The Closure Diagrams for Nilpotent Orbits of Real Forms of E_{6}

Dragomir Ž. D – oković^{∗}

Communicated by K. H. Hofmann

Abstract. Let O_{1} and O_{2} be adjoint nilpotent orbits in a real semisimple Lie algebra. Write O_{1} ≥ O_{2} if O_{2} is contained in the closure of O_{1}. This gives a partial order on the set of such orbits, which is known as the closure ordering. We determine this ordering for the adjoint nilpotent orbits of the four noncompact real forms of the simple complex Lie algebra E_{6}.

1. Introduction

In this paper 𝔤 denotes a simple complex Lie algebra of type E_{6} and 𝔤_{0} one of its noncompact real forms. Let G be the adjoint group of 𝔤 and σ the conjugation of 𝔤 with respect to 𝔤_{0}. There is a unique anti-holomorphic involutory automorphism of G whose differential is σ. We denote it also by σ. The adjoint group G_{0} of 𝔤_{0} is the connected Lie subgroup of G corresponding to 𝔤_{0}. According to Matsumoto [14], G_{0} = G_{R} where G_{R} = {a ∈ G : σ(a) = a} is the group of real points of G.

Fix a Cartan decomposition 𝔤_{0} = 𝔨_{0} ⊕ 𝔭_{0} of 𝔤_{0} and denote by θ the corresponding Cartan involution of 𝔤_{0}: θ(X) = X for X ∈ 𝔨_{0} and θ(X) = −X for X ∈ 𝔭_{0}. Let 𝔨 (resp. 𝔭) be the complexification of 𝔨_{0} (resp. 𝔭_{0}). We extend θ to a complex linear map of 𝔤 and use the same letter θ to denote this extension. Furthermore we denote by θ also the corresponding involutory automorphism of G. Let K (resp. K_{0}) be the connected Lie subgroup of G with Lie algebra 𝔨 (resp. 𝔨_{0}). Then K_{0} is a maximal compact subgroup of G_{0}, and K = {a ∈ G : θ(a) = a}.

Let N be the nilpotent variety of 𝔤 and N /G the orbit space for the adjoint action of G, equipped with the quotient topology. The adjoint nilpotent orbits in 𝔤 were enumerated a long time ago by Dynkin. There are 21 such orbits (including the trivial one). Nowadays one uses the Bala–Carter symbols to label these orbits [3]. For any orbit O ∈ N /G we denote by \overline{O} its closure in 𝔤. It is a union of O and some orbits of smaller dimension.

If O_{1}, O_{2} ∈ N /G and O_{2} ⊂ \overline{O}_{1}, then we write O_{1} ≥ O_{2}. If O_{1} ≥ O_{2} and O_{1} ≠ O_{2}, then we write O_{1} > O_{2}. If O_{1} > O_{2} and there is no G-orbit O such that

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag O_{1} > O > O_{2}, then we write O_{1} → O_{2}. The topology of N /G can be represented by the so-called closure diagram. Each orbit O ∈ N /G is represented by a dot and if O_{1} → O_{2} then the dots corresponding to these two orbits are joined by a line. The dot for O_{1} is placed higher than the one for O_{2}.

∗ Supported in part by the NSERC Grant A-5285.

Since the closure diagram for N /G plays an essential role in the paper, we have reproduced it from [3] in Figure 1. Near each dot, the Bala–Carter symbol is displayed for the corresponding orbit O, which may be followed by the Cartan symbols of other regular semisimple subalgebras of 𝔤 whose principal nilpotent orbit is contained in O. On the left hand side of the diagram we indicate the complex dimensions of the orbits on each level.

Let N_{R} = N ∩ 𝔤_{0} be the nilpotent variety of 𝔤_{0} and N_{1} = N ∩ 𝔭 that of 𝔭. The corresponding orbit spaces N_{R}/G_{0} and N_{1}/K are also equipped with their quotient topologies. Both of these spaces are finite. The Kostant–Sekiguchi correspondence establishes a bijection N_{R}/G_{0} → N_{1}/K. For more details about this correspondence we refer the reader to the book [5]. Barbasch and Sepanski [1] have shown recently that this bijection is a homeomorphism.

The K -orbits in N_{1} were enumerated in our papers [6, 7]. For the sake of consistency, we use here the same enumeration. Our original enumeration is reproduced in [5] where the trivial orbit {0} has been given the number 0. Up to G-conjugacy, 𝔤 has four noncompact real forms:

E I = E_{6(6)}, E II = E_{6(2)}, E III = E_{6(−14)}, E IV = E_{6(−26)}

where the subscript k inside the parentheses is the so-called Cartan index

k = dim(𝔭_{0}) − dim(𝔨_{0}).

Our main result is an explicit description of the topology of N_{R}/G_{0} (or, equivalently, N_{1}/K): The closure diagrams for N_{R}/G_{0} (or N_{1}/K) are given by Figures 2 (p. 389), 3 (p. 397), and 5 (p.407). In the case E I we work directly with N_{R}/G_{0} since this real form is of outer type. In the cases E II and E III, which are of inner type, it is more convenient to work with N_{1}/K. The case E IV is rather trivial as it has only two nonzero nilpotent G_{0}-orbits.

The representatives of nilpotent G_{0}-orbits given in [8] are of special kind because they are embedded in real Cayley triples. Dropping that restriction, one can find simpler representatives. Table 3 below (p. 407) gives such representatives for the case E I.

Let us describe briefly the action of Aut(𝔤_{0}) on N_{R}/G_{0}. We recall that Aut(𝔤_{0})/G_{0} = Z_{2}. (By Z_{k} we denote a cyclic group of order k.) If 𝔤_{0} is of outer type (E I or E IV), then Z_{2} acts trivially on N_{R}/G_{0}. If 𝔤_{0} is of type E II, the generator of Z_{2} interchanges the orbits 9 and 10, 12 and 13, 28 and 27, and 29 and 30. Finally, if 𝔤_{0} is of type E III, the generator of Z_{2} acts as the reflection in the vertical axis of symmetry of the E III diagram in Figure 5.

A few words are in order concerning the use of the computer. First of all we used it to compile most of our tables. Several of these tables can be easily verified by hand. Secondly, we often use the computer to determine the dimensions of various orbits and to analyze the orbit structure of some important prehomogeneous vector spaces.

A typical problem that we encounter is the following: Given a nilpotent element X ∈ 𝔤_{0}, decide to which G_{0}-orbit it belongs. As a rule, by using a computer, it is easy to find the dimension of the orbit G_{0} · X. If there is only one nilpotent G_{0}-orbit of that dimension, then the job is finished. Otherwise, for special types of elements X, we may be able to use the method developed in our paper [8]. On several occasions we had to resort to additional ad hoc arguments to finish the job. Two such cases occur in our justification of Table 3.

In addition to our own programs, we used extensively the software packages Maple and LiE (see [4, 16]).

It is a pleasure to thank a referee for very thorough reading of the manuscript and useful comments.

[figure: Figure 1: Closure diagram for E_{6}; 0; 0; 22; A_{1}; 32; 2A_{1}; 40; 3A_{1}; 42; A_{2}, 4A_{1}; 46; A_{2} + A_{1}; 48; 2A_{2}; 50; A_{2} + 2A_{1}; 52; A_{3}; 54; 2A_{2} + A_{1}; 56; A_{3} + A_{1}; 58; D_{4}(a_{1}), A_{3} + 2A_{1}, 3A_{2}; 60; D_{4}; A_{4}; 62; A_{4} + A_{1}; 64; D_{5}(a_{1}); A_{5}; 66; E_{6}(a_{3}), A_{5} + A_{1}; 68; D_{5}; 70; E_{6}(a_{1}); 72; E_{6}]

2. Preliminaries

Fix a σ-stable Cartan subalgebra 𝔥 of 𝔤. Denote by R the root system of (𝔤, 𝔥). Then 𝔥_{0} = 𝔤_{0} ∩ 𝔥 is a Cartan subalgebra of 𝔤_{0}. Choose a system of positive roots R^{+} ⊂ R. Let α_{i}, 1 ≤ i ≤ 36, be the enumeration of R^{+} used in [8]. It is reproduced in the Appendix. In particular, {α_{1}, …, α_{6}} is a system of fundamental roots as in [2]. The negative root −α_{i} is also written as α_{−i}. Let H_{i} ∈ 𝔥 be the coroot corresponding to α_{i}. Then H_{−i} = −H_{i}. For α ∈ R, 𝔤^{α} is the root space of α.

Let X_{i} ∈ 𝔤^{α_{i}} for ±i ∈ {1, …, 36} be the root vectors chosen so that together with the H_{i}, 1 ≤ i ≤ 6, they form a Chevalley basis of 𝔤. For the convenience of the reader, the Appendix includes the structure constants of 𝔤 from our paper [8, Table 13]. They are given here in a more extensive form and user-friendly way.

An ordered triple (E, H, F) of nonzero elements of 𝔤 is a standard triple if

[H, E] = 2E, [H, F] = −2F, [F, E] = H.

For instance, (X_{i}, H_{i}, X_{−i}) are standard triples for all i. A standard triple (E, H, F) is a normal triple if H ∈ 𝔨 and E, F ∈ 𝔭.

We enumerate the nonzero G-orbits in N as O^{k}, 1 ≤ k ≤ 20. We can choose a standard triple (E^{k}, H^{k}, F^{k}) with E^{k} ∈ O^{k}, H^{k} ∈ 𝔥, and such that α_{i}(H^{k}) ≥ 0 for 1 ≤ i ≤ 6. The nonzero G_{0}-orbits in N_{R} will be enumerated as in our papers [6, 7] and in [5], and we denote them by O_{0}^{i} (i = 1, 2, …). The K -orbit in N_{1} that corresponds to O_{0}^{i} under the Kostant–Sekiguchi bijection is denoted by O_{1}^{i}.

If nonempty, 𝔭 ∩ O^{k} is an equidimensional complex algebraic variety [12, Proposition 5] with

dim_{C}(𝔭 ∩ O^{k}) = \frac{1}{2} dim_{C}(O^{k}).

(Recall that dim_{C}(O^{k}) is always even.) Each connected component of 𝔭 ∩ O^{k} is a single K -orbit. Similarly 𝔤_{0} ∩ O^{k} (if nonempty) is an equidimensional real algebraic variety with

dim_{R}(𝔤_{0} ∩ O^{k}) = dim_{C}(O^{k}),

and each connected component of 𝔤_{0} ∩ O^{k} is a single G_{0}-orbit.

Furthermore

O_{0}^{i} ⊂ 𝔤_{0} ∩ O^{k} ⟺ O_{1}^{i} ⊂ 𝔭 ∩ O^{k}.

Hence 𝔤_{0} ∩ O^{k} and 𝔭 ∩ O^{k} have the same number of connected components. In particular

𝔤_{0} ∩ O^{k} ≠ Ø ⟺ 𝔭 ∩ O^{k} ≠ Ø.

All topological notions (such as closure, connectedness, etc.) refer to the Euclidean topology in 𝔤 and the Lie group topology in G.

In Table 1 we enumerate the nonzero G-orbits O^{k} ⊂ N . Column 2 contains the Bala–Carter label of O^{k}. In Column 3 we list the integers α_{j}(H^{k}), 1 ≤ j ≤ 6. The next four columns show which orbits O_{0}^{i} (or, equivalently, O_{1}^{i}) are contained in O^{k}. This depends on the type of the real form 𝔤_{0}. For instance,

E I :  𝔤_{0} ∩ O^{4} = O_{0}^{4} ∪ O_{0}^{5},  𝔭 ∩ O^{4} = O_{1}^{4} ∪ O_{1}^{5};
E II :  𝔤_{0} ∩ O^{4} = O_{0}^{6} ∪ O_{0}^{7} ∪ O_{0}^{8},  𝔭 ∩ O^{4} = O_{1}^{6} ∪ O_{1}^{7} ∪ O_{1}^{8};
E III :  𝔤_{0} ∩ O^{4} = O_{0}^{6},  𝔭 ∩ O^{4} = O_{1}^{6};
E IV :  𝔤_{0} ∩ O^{4} = Ø;  𝔭 ∩ O^{4} = Ø.

The last column of this table records the complex dimension of O^{k}.

Table 1: Nonzero nilpotent orbits in E_{6} and its real forms

k Bala-Carter α_{j}(H^{k})  E I  E II  E III E IV dim_{C}(O^{k})
1 A_{1}  0 1 0 0 0 0 1  1  1,2  22
2 2A_{1}  1 0 0 0 0 1 2  2,3  3,4,5 1  32
3 3A_{1}  0 0 0 1 0 0 3  4,5  40
4 A_{2}  0 2 0 0 0 0 4,5  6,7,8  6  42
5 A_{2} + A_{1}  1 1 0 0 0 1 8  9,10  7,8  46
6 2A_{2}  2 0 0 0 0 2 6  11  9  2  48
7 A_{2} + 2A_{1}  0 0 1 0 1 0 10 12,13,14  50
8 A_{3}  1 2 0 0 0 1 7  15,16 10,11  52
9 2A_{2} + A_{1}  1 0 0 1 0 1 11  17  54
10 A_{3} + A_{1}  0 1 1 0 1 0 15  18,19  56
11 D_{4}(a_{1})  0 0 0 2 0 0 12,23 20,21,22  58
12 A_{4}  2 2 0 0 0 2 9  25,26  12  60
13 D_{4}  0 2 0 2 0 0 13  23,24  60
14 A_{4} + A_{1}  1 1 1 0 1 1 16  27,28  62
15 D_{5}(a_{1})  1 2 1 0 1 1 17  29,30  64
16 A_{5}  2 1 1 0 1 2 14  31  64
17 E_{6}(a_{3})  2 0 0 2 0 2 19,22 32,33  66
18 D_{5}  2 2 0 2 0 2 21  34,35  68
19 E_{6}(a_{1})  2 2 2 0 2 2 18  36  70
20 E_{6}  2 2 2 2 2 2 20  37  72

For 1 ≤ k ≤ 20 and any integer j set

𝔤(j, k) = {X ∈ 𝔤 : [H^{k}, X] = jX}, R(j, k) = {α ∈ R : α(H^{k}) = j}.

Then

𝔤(0, k) = 𝔥 + ∑_{α∈R(0,k)} 𝔤^{α}; 𝔤(j, k) = ∑_{α∈R(j,k)} 𝔤^{α}, j ≠ 0.

Introduce the subalgebras

𝔮(i, k) = ∑_{j≥i} 𝔤(j, k), i ≥ 0,

and let Q(k) be the parabolic subgroup of G corresponding to 𝔮(0, k). The centralizer, L(k), of H^{k} in G is a Levi factor of Q(k) with Lie algebra 𝔤(0, k).

The following theorem, due to Kostant (see [11, Theorem 4.3] or [5, Lemma 4.1.4]), is valid for arbitrary complex semisimple Lie algebras. For the last assertion of the theorem see [13, Satz 2, pp. 182–184].

Theorem 2.1. Let (E^{k}, H^{k}, F^{k}) be a standard triple, as above, with E^{k} ∈ O^{k}. Then

O^{k} ∩ 𝔤(2, k) = L(k) · E^{k}

is a dense open subset of 𝔤(2, k), and

O^{k} ∩ 𝔮(2, k) = L(k) · E^{k} + 𝔮(3, k) = Q(k) · E^{k}

is a dense open subset of 𝔮(2, k). Moreover \overline{O^{k}} = G · 𝔮(2, k).

Table 2 lists the indices i of the roots α_{i} that belong to R(j, k) for j ≥ 2. Those for R(2, k) are listed first and separated from the other (if any) by a semicolon.

Table 2: Indices of roots in R(j, k), j ≥ 2.

k  R(2, k); R(j, k), j > 2
1 36;
2 23 27 30 32 33 34 35 36;
3 24 26 28 29 30 31 32 33 34 ; 35 36
4 2 8 13 14 17 19 20 22 24 25 26 27 28 29 30 31 32 33 34 35 ; 36
5 17 20 22 23 25 26 28 29 31 ; 27 30 32 33 34 35 36
6 1 6 7 11 12 16 17 18 20 21 22 25 26 28 29 31 ; 23 27 30 32 33 34 35 36
7 15 18 19 21 22 23 24 25 26 27 28 30 ; 29 31 32 33 34 35 36
8 2 8 13 14 19 23 24 ; 17 20 22 25 26 27 28 29 30 31 32 33 34 35 36
9 12 16 17 18 20 21 22 24 25 ; 23 26 27 28 29 30 31 32 33 34 35 36
10 13 14 15 17 18 20 21 23 ; 19 22 24 25 26 27 28 29 30 31 32 33 34 35 36
11 4 8 9 10 12 13 14 15 16 17 18 19 20 21 22 23 25 27 ;

24 26 28 29 30 31 32 33 34 35 36

12 1 2 6 7 8 11 12 13 14 16 18 19 21 24 ;

17 20 22 23 25 26 27 28 29 30 31 32 33 34 35 36

13 2 4 9 10 12 15 16 18 21 23 ;

8 13 14 17 19 20 22 24 25 26 27 28 29 30 31 32 33 34 35 36

14 7 11 12 13 14 15 16 ;

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

15 2 7 8 11 12 15 16 ;

13 14 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

16 1 6 13 14 15 ;

7 11 12 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

17 1 4 6 7 8 9 10 11 13 14 15 19 ;

12 16 17 18 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

18 1 2 4 6 7 9 10 11 15 ; 8 12 13 14 16 17 18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35 36

19 1 2 3 5 6 8 9 10 ; 7 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

26 27 28 29 30 31 32 33 34 35 36

20 1 2 3 4 5 6 ; 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

26 27 28 29 30 31 32 33 34 35 36

3. Type E I

In this section 𝔤_{0} = E I is the split real form of 𝔤 = E_{6}. Hence 𝔨 is of type C_{4} and K = Sp_{8}/Z_{2}. Let 𝔥′ be a Cartan subalgebra of 𝔨 and {β_{1}, β_{2}, β_{3}, β_{4}} a base of the root system of (𝔨, 𝔥′) as in [2].

In Table 3 we list the nonzero G_{0}-orbits O_{0}^{i} ⊂ N_{R}, 1 ≤ i ≤ 23. In the first column of this table we give the unique integer k such that O_{0}^{i} ⊂ O^{k} (see Table 1). We can choose a normal triple (E′, H′, F′) such that E′ ∈ O_{1}^{i}, H′ ∈ 𝔥′, and β_{j}(H′) ≥ 0 for 1 ≤ j ≤ 4. The integers β_{j}(H′) are listed in the third column. They uniquely determine the orbit O_{1}^{i} (or O_{0}^{i}).

Table 3: Nonzero nilpotent orbits in E I

k i β_{j}(H′)  Representative E ∈ O_{0}^{i}  Type of E
1 1 0 0 0 1 X_{36}  A_{1}
2 2 0 1 0 0 (X_{23}) + (X_{36})  2A_{1}
3 3 1 0 0 1 (X_{24}) + (X_{30}) + (X_{34})  3A_{1}
4 4 0 0 0 2 X_{2} + X_{35}  A_{2}
(X_{2}) + (X_{24}) + (X_{30}) + (X_{34})  4A_{1}
4 5 2 0 0 0 (X_{2}) + (X_{24}) + (X_{30}) + (−X_{34})  4A_{1}
5 8 0 1 0 1 (X_{17} + X_{31}) + (X_{23})  A_{2} + A_{1}
6 6 0 2 0 0 (X_{1} + X_{31}) + (X_{6} + X_{29})  2A_{2}
7 10 1 0 1 0 (X_{22} + X_{28}) + (X_{15}) + (X_{23})  A_{2} + 2A_{1}
8 7 0 1 0 2 X_{2} + X_{23} + X_{24}  A_{3}
9 11 1 1 0 1 (X_{12} + X_{25}) + (X_{16} + X_{22}) + (X_{24})  2A_{2} + A_{1}
10 15 1 0 1 1 (X_{13} + X_{23} + X_{14}) + (X_{15})  A_{3} + A_{1}
11 12 2 0 0 2 (X_{13} + X_{23} + X_{14}) + (X_{4}) + (X_{15})  A_{3} + 2A_{1}
(X_{4} + X_{27}) + (X_{13} + X_{18}) + (X_{14} + X_{21}) 3A_{2}
11 23 0 0 2 0 (X_{13} + X_{23} + X_{14}) + (X_{4}) + (−X_{15})  A_{3} + 2A_{1}
X_{4} + X_{19} + X_{27} + X_{15}  D_{4}(a_{1})
12 9 0 2 0 2 X_{2} + X_{21} + X_{1} + X_{24}  A_{4}
13 13 2 0 0 4 X_{4} + X_{2} + X_{15} + X_{23}  D_{4}
14 16 1 1 1 1 (X_{13} + X_{16} + X_{7} + X_{14}) + (X_{15})  A_{4} + A_{1}
15 17 1 1 1 2 X_{15} + X_{8} + X_{7} + X_{11} + X_{12}  D_{5}(a_{1})
16 14 1 2 1 1 X_{13} + X_{1} + X_{15} + X_{6} + X_{14}  A_{5}
17 19 2 2 0 2 (X_{13} + X_{1} + X_{15} + X_{6} + X_{14}) + (X_{4})  A_{5} + A_{1}
17 22 0 2 2 0 (X_{13} + X_{1} + X_{15} + X_{6} + X_{14}) + (−X_{4}) A_{5} + A_{1}
X_{1} + X_{4} + X_{6} + X_{7} + X_{11} + X_{19}  E_{6}(a_{3})
18 21 2 2 0 4 X_{4} + X_{2} + X_{15} + X_{1} + X_{6}  D_{5}
19 18 2 2 2 2 X_{1} + X_{2} + X_{5} + X_{6} + X_{8} + X_{9}  E_{6}(a_{1})
20 20 4 2 2 4 X_{1} + X_{2} + X_{3} + X_{4} + X_{5} + X_{6}  E_{6}

The fourth column gives a representative E ∈ O_{0}^{i}. In most cases these representatives have been extracted (with some rescaling using the action of the identity component of the maximal torus of G_{0}) from our paper [8]. Only in the cases i = 17, 18 we are using new simpler representatives. Furthermore for i ∈ {4, 12, 22, 23} we have included an additional representative of different type.

The representative of type E_{6}(a_{3}) belongs to either O_{0}^{22} or O_{0}^{19} (see Table 1). By using the method of our paper [8] one can easily verify that the element

E = \sqrt{10}X_{19} + \sqrt{6}(X_{1} + X_{4} + X_{6}) + \sqrt{2}(X_{7} − X_{9} − X_{10} + X_{11})

belongs to O_{0}^{22}. Consequently the same is true for the element

E_{1} = X_{19} + X_{1} + X_{4} + X_{6} + X_{7} − X_{9} − X_{10} + X_{11}.

By using Table 14 from the Appendix it is easy to check that:

exp(ad(−X_{3} − X_{5} + X_{−2}))(E_{1}) = X_{1} + X_{4} + X_{6} + 2X_{7} + 2X_{11} + X_{19}.

We conclude that the representative of type E_{6}(a_{3}) indeed belongs to O_{0}^{22}.

The last column gives the type of E. For instance, if i = 23 the first representative is (X_{13} + X_{23} + X_{14}) + (X_{4}) + (−X_{15}) and its type is A_{3} + 2A_{1}. This means that {α_{13}, α_{23}, α_{14}, α_{4}, α_{15}} is a base for a closed root subsystem of type A_{3} + 2A_{1} with {α_{13}, α_{23}, α_{14}} being a base for the A_{3} component.

The second representative of O_{0}^{23} is E = X_{4} + X_{19} + X_{27} + X_{15} of type D_{4}(a_{1}), i.e., it is a subregular nilpotent element in a regular subalgebra of type D_{4}. Indeed {α_{4}, α_{2}, α_{15}, α_{23}} is a base of a closed root subsystem of type D_{4} with α_{2}+α_{15} = α_{19} and α_{2}+α_{23} = α_{27}. Consequently E ∈ 𝔤_{0}∩O^{11} = O_{0}^{22}∪O_{0}^{23}. A more delicate argument is needed to show that in fact E ∈ O_{0}^{23}. By using the method of my paper [8], one can check that the element 2X_{4} + X_{15} − X_{23} + \sqrt{3}(X_{19} + X_{27}) belongs to O_{0}^{23}. Hence the same is true for E_{1} = X_{4} + X_{15} − X_{23} + X_{19} + X_{27}. By using Table 4 from the Appendix, we find that

exp(−ad X_{−2})(E_{1}) = X_{4} + X_{19} + X_{27} + 2X_{15}.

This implies that also E ∈ O_{0}^{23}.

Note that our enumeration of the orbits O_{0}^{i} has two obvious flaws: The dimensions of O_{0}^{i} do not increase with i, and there are two pairs O_{0}^{i}, O_{0}^{j} ⊂ O^{k} with |i − j| > 1 (for k = 11 and 17).

We now proceed to the proof of the main result of this section.

Theorem 3.1. Let 𝔤_{0} be of type E I. The closure diagram of the orbit space N_{R}/G_{0} is as given in Figure 2. (The dotted horizontal lines in this diagram connect two G_{0}-orbits that are contained in the same G-orbit.)

Proof. We claim that if i and j are two vertices in Figure 2, with i higher than j, which are connected by a solid line, then O_{0}^{i} > O_{0}^{j}. Let k be such that O_{0}^{i} ⊂ O^{k}.

Assume first that k ≠ 4, 11, 17 or, equivalently, that 𝔤_{0} ∩ O^{k} = O_{0}^{i}. Then Theorem 2.1 implies that the intersection of O_{0}^{i} with

𝔮_{0}(2, k) = 𝔤_{0} ∩ 𝔮(2, k)

is an open dense subset of 𝔮_{0}(2, k). Hence in order to prove that O_{0}^{i} > O_{0}^{j} it suffices to exhibit an element E ∈ 𝔮_{0}(2, k) ∩ O_{0}^{j}. Table 4 provides the list of such elements E for each pair i, j as above.

[figure: Figure 2: Closure diagram for E I; 0; 0; 22; 1; 32; 2; 40; 3; 42; 5; .; .; .; .; .; .; 4; 46; 8; 48; 6; 50; 10; 52; 7; 54; 11; 56; 15; 58; 23; .; .; .; .; .; .; 12; 60; 13; 9; 62; 16; 64; 17; 14; 66; 22; .; .; .; .; .; .; 19; 68; 21; 70; 18; 72; 20]

For instance, let i = 15 and j = 11. Then Table 4 gives the element

E = (X_{13} + X_{18}) + (X_{14} + X_{21}) + (X_{27})

of type 2A_{2} + A_{1}. Consulting Table 1 we see that E ∈ O^{9} (Col. 2) and E ∈ O_{0}^{11} (Col. 4). Since k = 10 and the roots α_{13}, α_{18}, α_{14}, α_{21}, α_{27} belong to the union of R(s, 10) with s ≥ 2 (see Table 2), we indeed have E ∈ 𝔮_{0}(2, 10). It follows that O_{0}^{15} > O_{0}^{11}.

Table 4: Elements E ∈ 𝔮_{0}(2, k) ∩ \overline{O_{0}^{i}} ∩ O_{0}^{j}

\begin{matrix}i & j & E & Type \\ 20 & 18 & X_{1} + X_{2} + X_{5} + X_{6} + X_{8} + X_{9} & E_{6}(a_{1}) \\ 18 & 21 & X_{10} + X_{2} + X_{9} + X_{1} + X_{11} & D_{5} \\ 21 & 22 & (X_{13} + X_{1} − X_{15} + X_{6} + X_{14}) + (X_{4}) & A_{5} + A_{1} \\ 21 & 19 & (X_{13} + X_{1} + X_{15} + X_{6} + X_{14}) + (X_{4}) & A_{5} + A_{1} \\ 22 & 14 & X_{13} + X_{1} − X_{15} + X_{6} + X_{14} & A_{5} \\ 19 & 14 & X_{13} + X_{1} + X_{15} + X_{6} + X_{14} & A_{5} \\ 22 & 17 & X_{15} + X_{8} + X_{7} + X_{11} + X_{12} & D_{5}(a_{1}) \\ 14,17 & 16 & (X_{13} + X_{16} + X_{7} + X_{14}) + (X_{15}) & A_{4} + A_{1} \\ 17 & 13 & X_{15} + X_{8} + X_{7} + X_{11} & D_{4} \\ 16 & 9 & X_{13} + X_{16} + X_{7} + X_{14} & A_{4} \\ 9 & 12 & (X_{1} + X_{19} + X_{16}) + (X_{8}) + (X_{32}) & A_{3} + 2A_{1} \\ 9 & 23 & (X_{1} + X_{19} + X_{16}) + (−X_{8}) + (X_{32}) & A_{3} + 2A_{1} \\ 13 & 12 & (X_{13} + X_{23} + X_{14}) + (X_{4}) + (X_{15}) & A_{3} + 2A_{1} \\ 13 & 23 & (X_{13} + X_{23} + X_{14}) + (X_{4}) + (−X_{15}) & A_{3} + 2A_{1} \\ 12,23 & 15 & (X_{13} + X_{23} + X_{14}) + (X_{4}) & A_{3} + A_{1} \\ 15 & 11 & (X_{13} + X_{18}) + (X_{14} + X_{21}) + (X_{27}) & 2A_{2} + A_{1} \\ 15 & 7 & X_{13} + X_{23} + X_{14} & A_{3} \\ 7 & 10 & (X_{2} + X_{23}) + (X_{26}) + (X_{28}) & A_{2} + 2A_{1} \\ 11 & 10 & (X_{12} + X_{25}) + (X_{16}) + (X_{24}) & A_{2} + 2A_{1} \\ 11 & 6 & (X_{12} + X_{25}) + (X_{16} + X_{22}) & 2A_{2} \\ 10 & 8 & (X_{22} + X_{28}) + (X_{15}) & A_{2} + A_{1} \\ 6 & 8 & (X_{1} + X_{31}) + (X_{6}) & A_{2} + A_{1} \\ 10 & 5 & (−X_{15}) + (X_{22}) + (X_{23}) + (X_{25}) & 4A_{1} \\ 8 & 4 & X_{17} + X_{31} & A_{2} \\ 4,5 & 3 & (X_{2}) + (X_{24}) + (X_{30}) & 3A_{1} \\ 3 & 2 & (X_{24}) + (X_{30}) & 2A_{1} \\ 2 & 1 & X_{36} & A_{1} \\ 1 & 0 & 0 & 0 \end{matrix}

Now let k = 4. Then from Table 3 (p. 387) we see that i = 4 or 5 and j = 3. Since

X_{2} + X_{24} + X_{30} + X_{34} ∈ O_{0}^{4},
X_{2} + X_{24} + X_{30} − X_{34} ∈ O_{0}^{5},
X_{2} + X_{24} + X_{30} ∈ O_{0}^{3},

it is clear that O_{0}^{4}, O_{0}^{5} > O_{0}^{3}. A similar argument shows that O_{0}^{12}, O_{0}^{23} > O_{0}^{15} (when k = 11) and O_{0}^{19}, O_{0}^{22} > O_{0}^{14} (when k = 17).

Now let i = 22 and j = 17. Then k = 17 and Table 3 gives the representative

E = X_{1} + X_{4} + X_{6} + X_{7} + X_{11} + X_{19} ∈ O_{0}^{22}

of type E_{6}(a_{3}). Since the roots α_{1}, α_{4}, α_{6}, α_{7}, α_{11}, and α_{19} are linearly independent, it is clear that the element

E_{1} = X_{1} + X_{4} + X_{6} + X_{7} + X_{19}

belongs to the closure of O_{0}^{22}. Since E_{1} is of type D_{5}(a_{1}), it must belong to O_{0}^{17}, cf. Table 1. Hence O_{0}^{22} → O_{0}^{17}. This completes the proof of our claim that if i and j are two vertices which are connected in Figure 2, with i above j, then O_{0}^{i} > O_{0}^{j}.

Now we are going to prove that there are no edges missing in Figure 2. Analyzing the graph in Figure 2, it turns out that it suffices to show

O_{0}^{6} ⊄ \overline{O_{0}^{7}}, O_{0}^{13} ⊄ \overline{O_{0}^{19}}, O_{0}^{5} ⊄ \overline{O_{0}^{6}}.

Since O_{0}^{7} ⊂ O^{8}, O_{0}^{6} ⊂ O^{6}, and O^{6} ⊄ \overline{O^{8}} (see Table 1 and Figure 1), it follows that O_{0}^{6} ⊄ \overline{O_{0}^{7}}. It remains to prove that O_{0}^{13} ⊄ \overline{O_{0}^{19}} and O_{0}^{5} ⊄ \overline{O_{0}^{6}}.

Let k = 17. From Table 1 we see that

𝔤_{0} ∩ O^{k} = O_{0}^{19} ∪ O_{0}^{22}.

We examine in more details the prehomogeneous vector spaces (L(k), 𝔤(2, k)) and (Q(k), 𝔮(2, k)). We have L(k) = (GL_{2})^{3}/⟨(1, ζ, ζ^{2})⟩ where the positive roots corresponding to the GL_{2} factors are α_{2}, α_{3}, and α_{5} and ζ = e^{2πi/3}. As an L(k)-module, 𝔤(2, k) is a direct sum of three simple modules V_{1}, V_{2}, V_{3} whose bases are:

V_{1} : {X_{13}, X_{8}, X_{4}, X_{9}, X_{19}, X_{14}, X_{10}, X_{15}},
V_{2} : {X_{1}, X_{7}},
V_{3} : {X_{6}, X_{11}}.

The action of L(k) on 𝔤(2, k) lifts to (GL_{2})^{3}. As a module for (SL_{2})^{3}, V_{1} is the tensor product of the 2-dimensional simple modules for each of the three factors SL_{2}, while V_{2} (resp. V_{3}) is the 2-dimensional simple module for the second (resp. third) factor SL_{2} on which the other two factors SL_{2} act trivially. The elements (t_{1}, t_{2}, t_{3}) of the central 3-dimensional torus (GL_{1})^{3} of (GL_{2})^{3} act on each V_{i} by scalar multiplications: as (t_{1}t_{2}t_{3})^{−1} on V_{1}, as t_{1}^{2}t_{2}^{3} on V_{2}, and t_{1}^{2}t_{3}^{3} on V_{3}. Note that the element (1, ζ, ζ^{2}) indeed acts trivially. We mention that the prehomogeneous vector space (L(k), 𝔤(2, k)) has only finitely many orbits [10, Theorem 5.21].

Write an arbitrary X ∈ 𝔮(2, k) as X = X^{(1)} + X^{(2)} + X^{(3)} + X′ where X′ ∈ 𝔮(3, k) and the X^{(s)} ∈ V_{s} are written as:

X^{(1)} = x_{1}X_{13} + x_{2}X_{8} + x_{3}X_{4} + x_{4}X_{9} + y_{1}X_{19} + y_{2}X_{14} + y_{3}X_{10} + y_{4}X_{15},
X^{(2)} = z_{1}X_{1} + z_{2}X_{7},
X^{(3)} = w_{1}X_{6} + w_{2}X_{11}.

Define the homogeneous polynomials f_{i} : 𝔮(2, k) → C (i = 1, 2, 3) by

f_{1}(X) = 4(x_{1}x_{3} + x_{2}x_{4})(−y_{1}y_{3} + y_{2}y_{4}) − (−x_{1}y_{3} + x_{2}y_{4} + x_{3}y_{1} + x_{4}y_{2})^{2},
f_{2}(X) = (z_{1}x_{4} − z_{2}x_{3})(z_{1}y_{1} + z_{2}y_{2}) − (z_{1}x_{1} + z_{2}x_{2})(z_{1}y_{4} + z_{2}y_{3}),
f_{3}(X) = (w_{1}y_{3} − w_{2}x_{3})(w_{1}y_{1} + w_{2}x_{1}) − (w_{1}y_{2} + w_{2}x_{2})(w_{1}y_{4} + w_{2}x_{4}).

It is tedious but straightforward to check that each f_{i} is a relative invariant for the action of Q(k) on 𝔮(2, k). The singular set S of this prehomogeneous vector space has three irreducible components: The three hypersurfaces S_{i} defined by the equations f_{i}(X) = 0, respectively. The generic Q(k)-orbit in 𝔮(2, k) is

Ω = 𝔮(2, k) \ S = 𝔮(2, k) ∩ O^{17}.

By computing the dimension of the orbit Q(k) · E, where E is the representative of O_{0}^{14} from Table 3, we conclude that S_{1} ∩ O^{16} is a dense open subset in S_{1}. Hence S_{1} ⊂ \overline{O^{16}}.

Let Ω_{0} = Ω ∩ 𝔮_{0}(2, k) and set

Γ^{+} = {X ∈ 𝔮_{0}(2, k) : f_{1}(X) > 0}, Γ^{−} = {X ∈ 𝔮_{0}(2, k) : f_{1}(X) < 0}.

By analyzing the action of Q(k)_{R} on 𝔮_{0}(2, k), we find that Ω_{0} is the union of two Q(k)_{R}-orbits Ω′_{0} and Ω′′_{0}. As representatives of these orbits we can take the following elements, cf. Table 3:

E′_{1} = (X_{13} + X_{1} + X_{15} + X_{6} + X_{14}) + (X_{4}) ∈ Ω′_{0},
E′′_{1} = (X_{13} + X_{1} + X_{15} + X_{6} + X_{14}) + (−X_{4}) ∈ Ω′′_{0}.

One can show that L(k)_{R} has exactly 8 connected components. It is not hard to exhibit an element from a non-identity component of L(k)_{R} that fixes E′_{1}, and similarly for E′′_{1}. Hence each of the orbits Ω′_{0} and Ω′′_{0} has at most 4 connected components. Now consider the elements:

E′ = x_{1}X_{13} + y_{2}X_{14} + y_{4}X_{15} + x_{3}X_{4} + X_{1} + X_{6}

where the coefficients x_{1}, y_{2}, y_{4}, x_{3} are ±1. All of them belong to Ω_{0}. We choose 8 of them by indicating the signs of the coefficients and compute the signs of the nonzero real numbers f_{i}(E′):

\begin{matrix}x_{1} & y_{2} & y_{4} & x_{3} & f_{1} & f_{2} & f_{3} \\ + & + & + & + & + & − & − \\ − & − & + & + & + & + & + \\ − & + & − & + & + & − & + \\ − & + & + & − & + & + & − \\ − & + & + & + & − & + & − \\ + & − & + & + & − & − & + \\ + & + & − & + & − & + & + \\ + & + & + & − & − & − & − \end{matrix}

Since the sign patterns in the last 3 colums of the above table are all different, these 8 elements belong to different connected components of Ω_{0}. We conclude that Ω′_{0} and Ω′′_{0} have each exactly 4 connected components.

A computation shows that the representatives E′ with f_{1}(E′) > 0 belong to O_{0}^{19} and those with f_{1}(E′) < 0 belong to O_{0}^{22}. We obtain that

𝔮_{0}(2, k) ∩ O_{0}^{19} = Ω′_{0} ⊂ Γ^{+}, 𝔮_{0}(2, k) ∩ O_{0}^{22} = Ω′′_{0} ⊂ Γ^{−}.

We claim that if X ∈ Γ^{+} ∩ S_{2}, then X^{(2)} = 0, i.e., z_{1} = z_{2} = 0. Observe that f_{2}(X) is a quadratic form in the variables z_{1} and z_{2} and that its discriminant is −f_{1}(X). As we assume that f_{1}(X) > 0 and f_{2}(X) = 0, it follows that z_{1} = z_{2} = 0 as claimed. Another computation shows that the intersection of O^{16} with the codimension 2 subspace defined by z_{1} = z_{2} = 0 is dense in this subspace. Consequently, Γ^{+} ∩ S_{2} ⊂ \overline{O^{16}}. A similar argument shows that Γ^{+} ∩ S_{3} ⊂ \overline{O^{16}}. Since we have already shown that S_{1} ⊂ \overline{O^{16}}, it follows that the boundary, say Δ, of Ω′_{0} in 𝔮_{0}(2, k) is contained in \overline{O^{16}}. Consequently,

G_{0} · Δ ⊂ \overline{O^{16}}.

As O^{13} ⊄ \overline{O^{16}} (see Figure 1 and Table 1) and O_{0}^{13} ⊂ O^{13}, we deduce that O_{0}^{13} ⊄ G_{0} · Δ. Since Q(k)_{R} is a parabolic subgroup of G_{0}, the homogeneous space G_{0}/Q(k)_{R} is compact. It follows that the set G_{0} · Δ is closed. Hence

\overline{O_{0}^{19}} = O_{0}^{19} ∪ G_{0} · Δ

and so O_{0}^{13} ⊄ \overline{O_{0}^{19}}.

Finally we show that O_{0}^{5} ⊄ \overline{O_{0}^{6}}, so let k = 6. Then 𝔤_{0} ∩ O^{k} = O_{0}^{6}, and L(k) = Spin_{8}·T_{2}. As an L(k)-module, 𝔤(2, k) is a direct sum of two simple modules V_{1} and V_{2} with bases:

V_{1} :  {X_{1}, X_{7}, X_{12}, X_{17}, X_{18}, X_{22}, X_{26}, X_{29}},
V_{2} :  {X_{6}, X_{11}, X_{16}, X_{20}, X_{21}, X_{25}, X_{28}, X_{31}}.

The space V_{3} = 𝔤(4, k) is also a simple L(k)-module. The three modules V_{1}, V_{2}, V_{3} are pairwise non-isomorphic and 𝔮(2, k) = V_{1} ⊕ V_{2} ⊕ V_{3}.

Write any X ∈ 𝔮(2, k) as X = X^{(1)} + X^{(2)} + X^{(3)} with X^{(s)} ∈ V_{s} and

X^{(1)} = x_{1}X_{1} + x_{2}X_{7} + x_{3}X_{12} + x_{4}X_{17} + x_{5}X_{18} + x_{6}X_{22} + x_{7}X_{26} + x_{8}X_{29},
X^{(2)} = y_{1}X_{6} + y_{2}X_{11} + y_{3}X_{16} + y_{4}X_{20} + y_{5}X_{21} + y_{6}X_{25} + y_{7}X_{28} + y_{8}X_{31}.

The quadratic forms f_{1}, f_{2} : 𝔮(2, k) → C defined by

f_{1}(X) = x_{1}x_{8} + x_{2}x_{7} − x_{3}x_{6} + x_{4}x_{5},
f_{2}(X) = y_{1}y_{8} + y_{2}y_{7} − y_{3}y_{6} + y_{4}y_{5},

are relative invariants for the action of Q(k) on 𝔮(2, k). Let S_{i} ⊂ 𝔮_{0}(2, k) be the hypersurface defined by f_{i}(X) = 0 (i = 1, 2). Let S = S_{1} ∪ S_{2} and let Ω_{0} = 𝔮_{0}(2, k)\S. Then Ω_{0} = 𝔮_{0}(2, k) ∩ O_{0}^{6} and 𝔮_{0}(2, k) ∩ O_{0}^{8} is a dense open subset of S. Consequently

\overline{O_{0}^{6}} = O_{0}^{6} ∪ \overline{O_{0}^{8}}.

By applying [9, Theorem 4.1] to the adjoint module 𝔤 = 𝔨 ⊕ 𝔭, one can show easily that O_{1}^{8} ≯ O_{1}^{5}. Indeed we have d_{1}(2, 8) = 29 while d_{1}(2, 5) = 28. (For the definition of the integers d_{i}(j, k) see the next section.) It follows that O_{0}^{8} ≯ O_{0}^{5} which entails that O_{0}^{6} ≯ O_{0}^{5}.

394 D – oković

Table 5: Nonzero nilpotent orbits in 𝔭 (𝔤_{0} = E II)

k i β_{j}(H^{i}) E^{i} Type of E^{i}
1 1 00100 1 X_{−2}  A_{1}
2 2 10001 2 (X_{−13}) + (X_{−14})  2A_{1}
2 3 01010 0 (X_{35}) + (X_{−8})  2A_{1}
3 4 00100 3 (X_{−8}) + (X_{−22}) + (X_{−25})  3A_{1}
3 5 10101 1 (X_{35}) + (X_{−8}) + (X_{−19})  3A_{1}
4 6 00000 4 X_{−24} + X_{−27}  A_{2}
(X_{−2}) + (X_{−24}) + (X_{−30}) + (X_{−34})  4A_{1}
4 7 20002 0 X_{34} + X_{−19}  A_{2}
(X_{27}) + (X_{35}) + (X_{−8}) + (X_{−19})  4A_{1}
4 8 00200 2 (X_{35}) + (X_{−17}) + (X_{−19}) + X_{−20})  4A_{1}
5 9 21001 1 (X_{32} + X_{−13}) + (X_{−20})  A_{2} + A_{1}
5 10 10012 1 (X_{33} + X_{−14}) + (X_{−17})  A_{2} + A_{1}
6 11 02020 0 (X_{31} + X_{−20}) + (X_{32} + X_{−13})  2A_{2}
7 12 30100 0 (X_{33} + X_{−20}) + (X_{32}) + (X_{−19})  A_{2} + 2A_{1}
7 13 00103 0 (X_{32} + X_{−17}) + (X_{33}) + (X_{−19)})  A_{2} + 2A_{1}
7 14 11011 2 (X_{34} + X_{−19}) + (X_{−17}) + (X_{−20})  A_{2} + 2A_{1}
8 15 10201 4 X_{35} + X_{−24} + X_{−27}  A_{3}
8 16 01210 2 X_{−22} + X_{34} + X_{−25}  A_{3}
9 17 11111 1 (X_{32} + X_{−17}) + (X_{33} + X_{−20}) + X_{−19}  2A_{2} + A_{1}
10 18 10301 1 (X_{−22} + X_{34} + X_{−25}) + (X_{30})  A_{3} + A_{1}
10 19 11111 3 (X_{−22} + X_{34} + X_{−25}) + (X_{−24})  A_{3} + A_{1}
11 20 00400 0 X_{29} + X_{30} + X_{−17} + X_{−19}  D_{4}(a_{1})
(X_{32} + X_{−27} + X_{33}) + (X_{−8}) + (X_{−19})  A_{3} + 2A_{1}
(X_{24} + X_{−13}) + (X_{30} + X_{−20}) + (X_{34} + X_{−22}) 3A_{2}
11 21 02020 4 X_{35} + X_{−19} + X_{−24} + X_{−30}  D_{4}(a_{1})
(X_{−26} + X_{35} + X_{−28}) + (X_{−19}) + (X_{−27})  A_{3} + 2A_{1}
11 22 20202 2 (X_{−20} + X_{33} + X_{−22}) + (X_{32}) + (X_{−24})  A_{3} + 2A_{1}
12 25 40004 4 X_{32} + X_{−25} + X_{−26} + X_{33}  A_{4}
12 26 22022 0 X_{30} + X_{−17} + X_{29} + X_{−24}  A_{4}
13 23 00400 8 X_{−29} + X_{35} + X_{−30} + X_{−31}  D_{4}
13 24 20402 4 X_{32} + X_{−27} + X_{33} + X_{−24}  D_{4}
14 27 12113 1 (X_{30} + X_{−20} + X_{31} + X_{−24}) + (X_{−22})  A_{4} + A_{1}
14 28 31121 1 (X_{30} + X_{−17} + X_{29} + X_{−24}) + (X_{−25})  A_{4} + A_{1}
15 29 31310 4 X_{33} + X_{−27} + X_{32} + X_{−24} + X_{29}  D_{5}(a_{1})
15 30 01313 4 X_{32} + X_{−25} + X_{−26} + X_{31} + X_{−24}  D_{5}(a_{1})
16 31 13131 3 X_{−22} + X_{29} + X_{−24} + X_{31} + X_{−25}  A_{5}
17 32 22222 2 (X_{−22} + X_{29} + X_{−24} + X_{31} + X_{−25}) + (X_{30}) A_{5} + A_{1}
17 33 04040 4 X_{29} + X_{31} + X_{32} + X_{33} + X_{−24} + X_{−27}  E_{6}(a_{3})
(X_{−22} + X_{29} + X_{−24} + X_{31} + X_{−25}) + (X_{−30}) A_{5} + A_{1}
18 34 22422 4 X_{30} + X_{−27} + X_{−24} + X_{29} + X_{31}  D_{5}
18 35 40404 8 X_{34} + X_{−27} + X_{30} + X_{−26} + X_{−28}  D_{5}
19 36 44044 4 X_{27} + X_{29} + X_{30} + X_{31} + X_{−22} + X_{−28}  E_{6}(a_{1})
20 37 44444 8 X_{29} + X_{−27} + X_{−26} + X_{30} + X_{−28} + X_{31}  E_{6}

4. Type E II

In this section 𝔤_{0} = E II and so 𝔨 is of type A_{5} + A_{1} and K = (SL_{6}/Z_{3} × SL_{2})/Z_{2}. As 𝔤_{0} is of inner type, we may assume that 𝔥 ⊂ 𝔨. The roots

β_{1} = α_{1}, β_{2} = α_{3}, β_{3} = α_{4}, β_{4} = α_{5}, β_{5} = α_{6}, β_{6} = −α_{36}

form a base for the root system of (𝔨, 𝔥).

In Table 5 we list the nonzero K -orbits O_{1}^{i} ⊂ N_{1}, 1 ≤ i ≤ 37. In the first column we give the integer k such that O_{1}^{i} ⊂ O^{k}. We choose a normal triple (E^{i}, H^{i}, F^{i}) such that E^{i} ∈ O_{1}^{i}, H^{i} ∈ 𝔥, and β_{j}(H^{i}) ≥ 0 for 1 ≤ j ≤ 6. The integers β_{j}(H^{i}) are listed in the third column. They uniquely determine the orbit O_{1}^{i}. The fourth column gives the representative E^{i} ∈ O_{1}^{i} (in some cases several representatives are listed). The last column gives the type of E^{i}.

As in [9] we set

𝔤_{H^{i}}(0, j) = 𝔨 ∩ 𝔤(j, H^{i}), 𝔤_{H^{i}}(1, j) = 𝔭 ∩ 𝔤(j, H^{i}),

and

𝔭_{s}(H^{i}) = ∑_{j≥s} 𝔤_{H^{i}}(1, j).

By Q_{H^{i}} we denote the parabolic subgroup of K with Lie algebra.

𝔮_{H^{i}} = ∑_{j≥0} 𝔤_{H^{i}}(0, j).

In Table 6 we list, for each i, the indices k of the roots α_{k} whose root space is contained in 𝔭_{2}(H^{i}). We first list those k for which 𝔤^{α_{k}} is contained in 𝔤_{H^{i}}(1, 2) and separate them by a semi-colon from the other indices (if any).

Table 6: Root spaces in 𝔤_{H^{i}}(1, 2) and 𝔭_{3}(H^{i})
i  𝔭_{2}(H^{i})
1 −2;
2 −2, −8, −13, −14, −19, −24;
3 −2, −8, 34, 35;
4 −8, −13, −14, −17, −19, −20, −22, −25, −27; −2
5 −8, −13, −14, −19, 35; −2
6 −2, −8, −13, −14, −17, −19, −20, −22, −24, −25, −26, −27, −28, −29, −30,
−31, −32, −33, −34, −35;
7 −2, −8, −13, −14, −19, −24, 27, 30, 32, 33, 34, 35;
8 −8, −13, −14, −17, −19, −20, −22, −25, −27, 35; −2
9 −13, −19, −20, −24, 32, 34, 35; −2, −8, −14
10 −14, −17, −19, −24, 33, 34, 35; −2, −8, −13
11 −13, −14, −17, −20, 29, 31, 32, 33; −2, −8, 34, 35

Table 6: (continued)

i  𝔭_{2}(H^{i})
12 −8, −13, −14, −19, −20, −25, 26, 29, 30, 32, 33, 34; −2, 35
13 −8, −13, −14, −17, −19, −22, 28, 30, 31, 32, 33, 34; −2, 35
14 −17, −19, −20, −24, 34, 35; −13, −14, −2, −8
15 −24, −27, 35; −17, −20, −22, −25, −8, −13, −14, −19, −2
16 −19, −22, −25, −27, 34; −13, −14, −17, −20, −8, 35, −2
17 −17, −19, −20, 32, 33; −13, −14, 34, −8, 35, −2
18 −17, −20, −22, −25, 30, 32, 33, 34; −8, −13, −14, −19, 35, −2
19 −22, −24, −25, 34; −17, −19, −20, 35, −13, −14, −8, −2
20 −8, −13, −14, −17, −19, −20, −22, 24, −25, 26, −27, 28, 29, 30, 31, 32, 33, 34;
−2, 35
21 −19, −22, −24, −25, −26, −27, −28, −30, 34, 35;
−13, −14, −17, −20, −2, −8
22 −17, −20, −22, −24, −25, 30, 32, 33, 34; −8, −13, −14, −19, 35, −2
23 −24, −26, −28, −29, −30, −31, −32, −33, −34, 35;
−8, −13, −14, −17, −19, −20, −22, −25, −27, −2
24 −24, −27, 30, 32, 33, 34; −17, −20, −22, −25, −8, −13, −14, −19, 35, −2
25 −17, −20, −22, −25, −26, 27, −28, −29, 30, −31, 32, 33, 34, 35;
−2, −8, −13, −14, −19, −24
26 −17, −19, −20, −24, 27, 29, 30, 31; −13, −14, 32, 33, −2, −8, 34, 35
27 −20, −22, −24, 30, 31; −17, −19, 33, −13, 32, −14, 34, −8, 35, −2
28 −17, −24, −25, 29, 30; −19, −20, 32, −14, 33, −13, 34, −8, 35, −2
29 −22, −24, −27, −28, 29, 32, 33;
−17, 34, −19, −25, −13, −14, −20, 35, −8, −2
30 −24, −25, −26, −27, 31, 32, 33;
−20, 34, −19, −22, −13, −14, −17, 35, −8, −2
31 −22, −24, −25, 29, 31; −19, 32, 33, −17, −20, −13, −14, 34, 35, −8, −2
32 −22, −24, −25, 29, 30, 31; −17, −19, −20, 32, 33, −13, −14, 34, −8, 35, −2
33 −19, −22, −24, −25, −26, −27, −28, 29, −30, 31, 32, 33;
−13, −14, −17, −20, 34, 35, −2, −8
34 −24, −27, 29, 30, 31;
−22, −25, 32, 33, −17, −19, −20, 34, −13, −14, −8, 35, −2
35 −26, −27, −28, −29, 30, −31, 32, 33, 34;
−17, −20, −22, −24, −25, 35, −8, −13, −14, −19, −2
36 −22, −25, −26, 27, −28, 29, 30, 31;
−17, −19, −20, −24, 32, 33, −13, −14, 34, 35, −2, −8
37 −26, −27, −28, 29, 30, 31;
−22, −24, −25, 32, 33, −17, −19, −20, 34, −13, −14, 35, −8, −2

Theorem 4.1. Let 𝔤_{0} be of type E II. The closure diagram of the orbit space N_{1}/K is as given in Figure 3. (The dotted horizontal lines in this diagram join the K -orbits that are contained in the same G-orbit.)

[figure: Figure 3: Closure diagram for E II; 0; 0; 22; 1; 32; 2; .; .; .; .; .; .; 3; 40; 4; .; .; .; .; .; .; 5; 42; 6; .; .; .; 8; .; .; .; .; .; .; .; .; .; .; 7; 46; 9; .; .; .; .; .; .; 10; 48; 11; 50; 14; .; .; .; .; .; .; 12; .; .; .; .; .; .; 13; 52; 15; .; .; .; 16; 54; 17; 56; 19; .; .; .; .; .; .; 18; 58; 21; .; .; .; .; .; .; 22; .; .; .; .; .; .; 20; 60; 23; .; .; .; .; .; .; 24; 25; .; .; .; .; .; .; 26; 62; 28; .; .; .; .; .; .; 27; 64; 29; .; .; .; .; .; .; 30; 31; 66; 33; .; .; .; .; .; .; 32; 68; 35; .; .; .; .; .; .; .; .; .; .; .; .; .; .; 34; 70; 36; 72; 37]

Proof. Let i and j be two vertices in the diagram of Figure 3 which are joined by a solid line with i being above j. In order to prove that O_{1}^{i} > O_{1}^{j} it suffices, by [9, Theorem 3.1], to show that the orbit O_{1}^{j} meets the subspace 𝔭_{2}(H^{i}). For each such pair (with j ≠ 0) we have exhibited in Table 7 an element

E ∈ 𝔭_{2}(H^{i}) ∩ O_{1}^{j}.

The fact that E ∈ 𝔭_{2}(H^{i}) can be verified using Table 6. The fact that E ∈ O_{1}^{j} follows from the observation that E ∈ 𝔤_{H^{j}}(1, 2). There are a few cases where this condition fails. In these cases we exhibit in the last column of the table an element w of the Weyl group W_{0} of (𝔨, 𝔥) such that w(E) ∈ 𝔤_{H^{j}}(1, 2). The element w is expressed as a product of reflections s_{k} (corresponding to the roots α_{k}).

Table 7: Elements E ∈ 𝔭_{2}(H^{i})∩ O_{1}^{j}
i  j  Type  E  w
37 36 E_{6}(a_{1})  X_{29} + X_{30} + X_{31} + X_{−25} + X_{−26} + X_{−28}
37 35 D_{5}  X_{34} + X_{−27} + X_{30} + X_{−26} + X_{−28}
36 34 D_{5} X_{31} + X_{−28} + X_{−22} + X_{27} + X_{29} s_{5}s_{4}s_{1}
35 33 A_{5} + A_{1} (X_{−26} + X_{33} + X_{−27} + X_{32} + X_{−28})
+(X_{−19})
34 33 E_{6}(a_{3})  X_{29} + X_{31} + X_{−22} + X_{−24} + X_{−25} + X_{−27}
34 32 A_{5} + A_{1} (X_{−22} + X_{29} + X_{−24} + X_{31} + X_{−25}) + (X_{30})
33 30 D_{5}(a_{1})  X_{32} + X_{−25} + X_{33} + X_{−27} + X_{−26}
33 29 D_{5}(a_{1})  X_{33} + X_{−22} + X_{32} + X_{−27} + X_{−28}
32,33 31 A_{5}  X_{−22} + X_{29} + X_{−24} + X_{31} + X_{−25}
29,31 28 A_{4} + A_{1} (X_{33} + X_{−22} + X_{29} + X_{−24}) + (X_{−25})  s_{5}
30,31 27 A_{4} + A_{1} (X_{32} + X_{−25} + X_{31} + X_{−24}) + (X_{−22})  s_{3}
35 23 D_{4} X_{−27} + X_{34} + X_{−29} + X_{−31} s_{4}
29,30 24 D_{4}  X_{32} + X_{−27} + X_{33} + X_{−24}
30 25 A_{4}  X_{32} + X_{−25} + X_{−26} + X_{33}
29 25 A_{4}  X_{32} + X_{−28} + X_{−22} + X_{33}
28 26 A_{4}  X_{30} + X_{−17} + X_{29} + X_{−24}
27 26 A_{4}  X_{30} + X_{−20} + X_{31} + X_{−24}
25 21 D_{4}(a_{1})  X_{34} + X_{−22} + X_{−25} + X_{−28}
24 21 D_{4}(a_{1})  X_{34} + X_{−19} + X_{−24} + X_{−27}
23 21 A_{3} + 2A_{1} (X_{−26} + X_{35} + X_{−28}) + (X_{−19}) + (X_{−27})
24,25,26 22 A_{3} + 2A_{1} (X_{−17} + X_{30} + X_{−20}) + (X_{34}) + (X_{−24})
26 20 D_{4}(a_{1})  X_{29} + X_{30} + X_{−17} + X_{−19}
24 20 D_{4}(a_{1})  X_{30} + X_{34} + X_{−8} + X_{−27}
21,22 19 A_{3} + A_{1} (X_{−22} + X_{34} + X_{−25}) + (X_{−24})
20,22 18 A_{3} + A_{1} (X_{−22} + X_{34} + X_{−25}) + (X_{30})
18 17 2A_{2} + A_{1} (X_{32} + X_{−17}) + (X_{33} + X_{−20}) + (X_{−19)}
18,19 16 A_{3}  X_{−22} + X_{34} + X_{−25}
21 15 A_{3}  X_{35} + X_{−24} + X_{−27}
16,17 14 A_{2} + 2A_{1} (X_{34} + X_{−19}) + (X_{−17}) + (X_{−20})
15 14 A_{2} + 2A_{1} (X_{35} + X_{−24}) + (X_{−17}) + (X_{−20})
17 13 A_{2} + 2A_{1} (X_{32} + X_{−17}) + (X_{33}) + (X_{−19})
17 12 A_{2} + 2A_{1} (X_{33} + X_{−20}) + (X_{32}) + (X_{−19})

Table 7: (continued)

i  j Type  E  w
17 11 2A_{2}  (X_{32} + X_{−17}) + (X_{33} + X_{−20})
13,14 10 A_{2} + A_{1} (X_{34} + X_{−19}) + (X_{−17})
11 10 A_{2} + A_{1} (X_{33} + X_{−14}) + (X_{−17})
12,14 9 A_{2} + A_{1} (X_{34} + X_{−19}) + (X_{−20})
11 9 A_{2} + A_{1} (X_{32} + X_{−13}) + (X_{−20})
14 8 4A_{1}  (X_{35}) + (X_{−17}) + (X_{−19}) + (X_{−20})
15 6 A_{2}  X_{−24} + X_{−27}
9,10 7 A_{2}  X_{34} + X_{−19}
7,8 5 3A_{1}  (X_{35}) + (X_{−8}) + (X_{−19})
6,8 4 3A_{1}  (X_{−8}) + (X_{−22}) + (X_{−25})
5 3 2A_{1}  (X_{35}) + (X_{−8})
4,5 2 2A_{1}  (X_{−13}) + (X_{−14})
2,3 1 A_{1}  X_{−2}

In order to complete the proof of Theorem 4.1 it remains to show that O_{1}^{i} ≯ O_{1}^{j} when (i, j) is one of the following pairs:

(6, 3), (12, 4), (13, 4), (12, 10), (13, 9),
(17, 16), (20, 19), (29, 27), (30, 28),
(32, 6), (23, 11), (23, 12), (23, 13),
(35, 32), (11, 4), (25, 20), (36, 23).

For the first nine pairs the assertion follows immediately from [9, Theorem 4.1] and Table 8 where we list the dimensions d_{i}(j, k) for the Z_{2}-graded 27-dimensional simple 𝔤-module V = V_{0} ⊕ V_{1} (dim V_{0} = 15, dim V_{1} = 12). These dimensions are defined by

d_{i}(j, k) = dim(V_{i} ∩ ker ρ(E)^{j})

where (E, H, F) is a normal triple with E ∈ O_{1}^{k} and ρ is the representation of 𝔤 afforded by the module V. For each of the nine pairs (r, s) we give in Table 9 the reason why O_{1}^{r} ≯ O_{1}^{s}: We have d_{i}(j, r) > d_{i}(j, s) for suitably chosen i and j.

In order to deal with the remaining eight pairs (i, j), we need a more detailed description of the space 𝔭. As a module for SL_{6} × SL_{2}, 𝔭 is isomorphic to V_{0} ⊗ V_{1} where V_{0} is the fundamental 20-dimensional module for SL_{6} (i.e. the third exterior power of the defining representation) and V_{1} is the defining 2-dimensional module for SL_{2}. Let R_{0} (resp. R_{1}) be the set of roots whose root spaces are contained in 𝔨 (resp. 𝔭). Set R_{1}^{+} = R_{1} ∩ R^{+} and R_{1}^{−} = R_{1}\R_{1}^{+}. Each of the sets R_{1}^{+} and R_{1}^{−} consists of 20 roots. The subspaces

𝔭^{+} = ∑_{α∈R_{1}^{+}} 𝔤^{α}, 𝔭^{−} = ∑_{α∈R_{1}^{−}} 𝔤^{α}

are simple SL_{6}-modules isomorphic to V_{0} and 𝔭 = 𝔭^{+} ⊕ 𝔭^{−}. An isomorphism φ : 𝔭^{+} → 𝔭^{−} of SL_{6}-modules is given by φ(X) = [X_{−36}, X]. In Table 10 we give the φ-images of the basis of 𝔭^{+} consisting of root vectors.

400 D – oković

Table 8: The integers d_{i}(j, k) for the module V(ω_{1})

k d_{0}(j, k), j ≥ 1 d_{1}(j, k), j ≥ 1
1 12 15  9 12
2 10 15  7 11 12
3 10 14 15  7 12
4 9 15  6 9 12
5 9 13 15  6 11 12
6 9 15  6 6 12
7 9 11 15  6 10 12
8 9 12 15  6 9 12
9 8 11 14 15  4 10 12
10 7 11 15  5 10 11 12
11 6 9 14 14 15  3 8 11 12
12 8 10 13 15  3 10 12
13 6 10 15  5 10 10 12
14 7 11 14 15  4 9 11 12
15 7 10 12 15  4 6 9 11 12
16 7 9 12 14 15  4 7 9 12
17 6 9 13 14 15  3 8 10 12
18 6 9 12 13 15  3 7 9 12
19 6 9 12 14 15  3 7 9 11 12
20 6 9 12 12 15  3 6 9 12
21 6 9 12 14 15  3 6 9 10 12
22 6 8 12 13 15  3 7 9 11 12
23 6 9 9 12 12 15  3 3 6 6 9 9 12
24 6 7 9 10 12 13 15  3 5 6 8 9 11 12
25 5 7 10 11 14 15  2 5 7 10 11 11 12
26 5 6 10 11 14 14 15  2 6 7 10 11 12
27 4 6 10 11 14 14 15  2 6 7 10 10 12
28 5 6 10 11 13 14 15  1 6 7 10 11 12
29 5 6 9 10 12 13 14 15  1 5 6 8 9 11 12
30 4 6 9 10 12 13 15  2 5 6 8 9 11 11 12
31 4 5 8 9 12 13 14 14 15  1 4 5 8 9 11 11 12
32 4 5 8 9 12 12 14 14 15  1 4 5 8 9 11 11 12
33 4 5 8 9 12 13 14 14 15  1 4 5 8 9 10 11 12
34 4 5 7 8 10 10 12 12 14 14 15  1 3 4 6 7 9 9 11 11 12
35 4 5 7 8 10 11 12 13 14 15  1 3 4 6 7 8 9 10 11 11 12
36 3 3 6 6 9 9 11 11 13 13 14 14 15 0 3 3 6 6 8 8 10 10 11 11 12
37 3 3 5 5 7 7 9 9 11 11 12 12 13 13 0 2 2 4 4 6 6 8 8 9 9 10 10
14 14 15  11 11 12

We shall also need the subspaces

𝔭_{2}^{±}(H^{i}) = 𝔭^{±} ∩ 𝔭_{2}(H^{i}).

Note that

𝔭_{2}(H^{i}) = 𝔭_{2}^{+}(H^{i}) ⊕ 𝔭_{2}^{−}(H^{i}).

When viewed as a K -module, all the weights of 𝔭 are simple. The weight diagram of this module is exhibited in Figure 4 where a vertex with label i designates the root space of α_{i}. The subspaces 𝔭^{+} and 𝔭^{−} are clearly visible as the left and right half of that diagram. The arrows show how the root vectors of the simple roots β_{1}, …, β_{6} act on the weight spaces. For β_{6} only two arrows are shown. The highest weight vector is X_{−2} and the lowest X_{2}.

Table 9: Some pairs (r,s) with O_{1}^{r} ≯ O_{1}^{s}

\begin{matrix}r & s & d_{i}(j, r) & d_{i}(j, s) \\ 6 & 3 & d_{0}(2, 6) = 15 & d_{0}(2, 3) = 14 \\ 12 & 4 & d_{1}(2, 12) = 10 & d_{1}(2, 4) = 9 \\ 13 & 4 & d_{1}(2, 13) = 10 & d_{1}(2, 4) = 9 \\ 12 & 10 & d_{0}(1, 12) = 8 & d_{0}(1, 10) = 7 \\ 13 & 9 & d_{0}(3, 13) = 15 & d_{0}(3, 9) = 14 \\ 17 & 16 & d_{0}(3, 17) = 13 & d_{0}(3, 16) = 12 \\ 20 & 19 & d_{1}(4, 20) = 12 & d_{1}(4, 19) = 11 \\ 29 & 27 & d_{0}(1, 29) = 5 & d_{0}(1, 27) = 4 \\ 30 & 28 & d_{1}(1, 30) = 2 & d_{1}(1, 28) = 1 \\ 32 & 23 & d_{1}(2, 32) = 4 & d_{1}(2, 23) = 3 \\ 33 & 23 & d_{1}(2, 33) = 4 & d_{1}(2, 23) = 3 \end{matrix}

Let π : 𝔭 → 𝔭^{+} be the projector with kernel 𝔭^{−}. Note that π commutes with the action of SL_{6}. Any Z ∈ 𝔭 can be written uniquely as Z = X + φ(Y) with X, Y ∈ 𝔭^{+}. The orbit SL_{2} · Z consists of all vectors

(aX + bY) + φ(cX + dY)

with ad − bc = 1. Hence

π(SL_{2} · Z) = ⟨X, Y ⟩\{0}

where ⟨X, Y ⟩ denotes the subspace of 𝔭^{+} spanned by X and Y.

Table 10: The isomorphism φ

\begin{matrix}X & φ(X) & X & φ(X) & X & φ(X) & X & φ(X) \\ X_{2} & X_{−35} & X_{19} & −X_{−30} & X_{26} & −X_{−25} & X_{31} & X_{−17} \\ X_{8} & −X_{−34} & X_{20} & −X_{−29} & X_{27} & −X_{−24} & X_{32} & −X_{−14} \\ X_{13} & X_{−33} & X_{22} & X_{−28} & X_{28} & −X_{−22} & X_{33} & −X_{−13} \\ X_{14} & X_{−32} & X_{24} & X_{−27} & X_{29} & X_{−20} & X_{34} & X_{−8} \\ X_{17} & −X_{−31} & X_{25} & X_{−26} & X_{30} & X_{−19} & X_{35} & −X_{−2} \end{matrix}

The pair (SL_{6}×GL_{1}, 𝔭^{+}), where GL_{1} is the maximal torus of the SL_{2} factor of K leaving 𝔭^{+} and 𝔭^{−} invariant, is a regular prehomogeneous vector space (see [15, p. 145]). There are four nonzero orbits in this space:

𝔭^{+} ∩ O_{1}^{6}, 𝔭^{+} ∩ O_{1}^{4}, 𝔭^{+} ∩ O_{1}^{2}, 𝔭^{+} ∩ O_{1}^{1}

with representatives

X_{25} + X_{26}, X_{8} + X_{22} + X_{25}, X_{2} + X_{24}, X_{2}

and dimensions 20, 19, 15, 10, respectively. The singular set of this prehomogeneous vector space is the hypersurface S = 𝔭^{+}\O_{1}^{6}.

[figure: Figure 4: The weight diagram of 𝔭; 2; β_{3}; β_{6}; 8; β_{2}; β_{4}; 13; β_{1}; β_{4}; 14; β_{5}; 17; β_{4}; 19; β_{1}; 20; β_{2}; 22; β_{3}; 24; 25; β_{3}; 26; β_{2}; 27; 28; β_{1}; β_{4}; 29; β_{5}; 30; β_{4}; 31; β_{1}; 32; β_{4}; 33; β_{2}; 34; β_{3}; 35; β_{6}; −35; −34; −33; −32; −31; −30; −29; −28; −27; −26; −25; −24; −22; −20; −19; −17; −14; −13; −8; −2]

We start with the pair (32, 6). Note that O_{1}^{32} ≯ O_{1}^{23} follows immediately from Table 9, but the claim that O_{1}^{32} ≯ O_{1}^{6} that we have to prove is much stronger. Let E ∈ O_{1}^{32} be the representative from Table 5 and write E = X + φ(Y) where

X = X_{29} + X_{30} + X_{31}, Y = −X_{26} − X_{27} − X_{28}.

Since π(SL_{2} · E) ⊂ ⟨X, Y ⟩ ⊂ S, we have π(O_{1}^{32}) ⊂ S and so O_{1}^{32} ⊂ S + 𝔭^{−}. Since S is closed and S ⊂ 𝔭^{+}, it is clear that S + 𝔭^{−} is closed. Hence \overline{O_{1}^{32}} ⊂ S + 𝔭^{−} and consequently O_{1}^{6} ⊄ \overline{O_{1}^{32}}, i.e., O_{1}^{32} ≯ O_{1}^{6}.

Next we consider the pairs (23, 11), (23, 12), (23, 13). Let E be the representative from Table 5 of one of the orbits O_{1}^{11}, O_{1}^{12}, O_{1}^{13}. Write E = X + φ(Y) with X, Y ∈ 𝔭^{+}. It is easy to check that ⟨X, Y ⟩ ⊂ O_{1}^{2} ∪ {0}. Consequently,

π(SL_{2} · E) ⊂ 𝔭^{+} ∩ O_{1}^{2},

and

π(K · E) = SL_{6} · π(SL_{2} · E) ⊂ 𝔭^{+} ∩ O_{1}^{2}.

Since

𝔭_{2}(H^{23}) = 𝔭_{2}^{+}(H^{23}) ⊕ 𝔭_{2}^{−}(H^{23})

and

𝔭_{2}^{+}(H^{23}) = ⟨X_{35}⟩ ⊂ O_{1}^{1} ∪ {0},

we conclude that the orbit K · E does not meet 𝔭_{2}(H^{23}). Now [9, Theorem 3.1] implies that

O_{1}^{23} ≯ O_{1}^{11}, O_{1}^{12}, O_{1}^{13}.

Let us now consider the pair (35, 32). The representative E ∈ O_{1}^{32} from Table 5 can be written as E = X + φ(Y) where

X = X_{29} + X_{30} + X_{31}, Y = −X_{26} − X_{27} − X_{28}.

Hence if E′ ∈ SL_{2} · E then π(E′) is a nonzero linear combination of X and Y. It is easy to verify that all such elements π(E′) belong to O_{1}^{4}. On the other hand

𝔭_{2}^{+}(H^{35}) = ⟨X_{30}, X_{32}, X_{33}, X_{34}, X_{35}⟩ ⊂ \overline{O_{1}^{2}}.

Since dim O_{1}^{4} > dim O_{1}^{2}, we conclude that O_{1}^{4} ∩ \overline{O_{1}^{2}} = Ø and so

SL_{6} · π(E′) ∩ 𝔭_{2}^{+}(H^{35}) = Ø.

This implies that O_{1}^{32} = K · E does not meet 𝔭_{2}(H^{35}). Hence O_{1}^{35} ≯ O_{1}^{32}. Next we consider the pair (11, 4). We have

𝔭_{2}(H^{11}) = V ⊕ φ(V)

where

V = ⟨X_{29}, X_{31}, X_{32}, X_{33}, X_{34}, X_{35}⟩ ⊂ 𝔭^{+}.

It is easy to check that V ∩O_{1}^{2} is an open dense subset of V. As dim O_{1}^{4} > dim O_{1}^{2}, we conclude that

V ∩ O_{1}^{4} = φ(V) ∩ O_{1}^{4} = Ø.

The representative E ∈ O_{1}^{4} from Table 5 belongs to 𝔭^{−}. Hence

SL_{2} · E = ⟨E, φ^{−1}(E)⟩\{0}.

Since

SL_{6} · φ^{−1}(E) ∩ V ⊂ O_{1}^{4} ∩ V = Ø,

and

SL_{6} · E ∩ φ(V) ⊂ O_{1}^{4} ∩ φ(V) = Ø,

we can conclude that O_{1}^{4} = K · E does not meet 𝔭_{2}(H^{11}). Hence O_{1}^{11} ≯ O_{1}^{4}.

We now turn to the pair (25, 20). The centralizer of H^{25} in K is SL_{4} · T_{3} where T_{3} is a 3-dimensional central torus. The space 𝔤_{H^{25}}(1, 2) is a direct sum of three simple modules for this centralizer:

𝔤_{H^{25}}(1, 2) = V_{1} ⊕ V_{2} ⊕ V_{3}.

The basis elements for these modules are:

V_{1} : X_{27}, X_{30}, X_{32}, X_{33}, X_{34}, X_{35};
V_{2} : X_{−31}, X_{−28}, X_{−25}, X_{−20};
V_{3} : X_{−29}, X_{−26}, X_{−22}, X_{−17}.

We write an arbitrary X ∈ 𝔭_{2}(H^{25}) as

X = X^{(1)} + X^{(2)} + X^{(3)} + X′

where X′ ∈ 𝔭_{3}(H^{25}) and X^{(i)} ∈ V_{i} are given by:

X^{(1)} = x_{1}X_{27} + x_{2}X_{30} + x_{3}X_{32} + x_{4}X_{33} + x_{5}X_{34} + x_{6}X_{35},
X^{(2)} = y_{1}X_{−31} + y_{2}X_{−28} + y_{3}X_{−25} + y_{4}X_{−20},
X^{(3)} = z_{1}X_{−29} + z_{2}X_{−26} + z_{3}X_{−22} + z_{4}X_{−17}.

The singular set S of (Q_{H^{25}}, 𝔭_{2}(H^{25})) is the union of two hyperquadrics

S_{1} = {X ∈ 𝔭_{2}(H^{25}) : f(X) = 0},
S_{2} = {X ∈ 𝔭_{2}(H^{25}) : g(X) = 0},

where f and g are the relative invariants defined by

f(X) = x_{1}x_{6} − x_{2}x_{5} + x_{3}x_{4},
g(X) = y_{1}z_{4} − y_{2}z_{3} + y_{3}z_{2} − y_{4}z_{1}.

The intersection S_{1} ∩ O_{1}^{22} (resp. S_{2} ∩ O_{1}^{21}) is a dense open subset of S_{1} (resp. S_{2}). Since the orbits O_{1}^{20}, O_{1}^{21}, and O_{1}^{22} have the same dimension, we conclude that O_{1}^{20} does not meet 𝔭_{2}(H^{25}). This implies that O_{1}^{25} ≯ O_{1}^{20}.

The last (and most difficult) pair we have to consider is (36, 23). We shall need some facts about the prehomogeneous vector space (Q_{H^{36}}, 𝔭_{2}(H^{36})). The centralizer of H^{36} in K has the form

Z_{K}(H^{36}) = SL_{2} · T_{5}

where the positive root of this SL_{2} is α_{4} and T_{5} is a 5-dimensional central torus. As a module for this centralizer, the space 𝔤_{H^{36}}(1, 2) is a direct sum of the following five simple modules:

V_{1} = ⟨X_{−22}, X_{−26}⟩, V_{2} = ⟨X_{27}, X_{30}⟩, V_{3} = ⟨X_{−25}, X_{−28}⟩,
V_{4} = ⟨X_{29}⟩, V_{5} = ⟨X_{31}⟩.

The torus T_{5} acts on each of the V_{i} by scalar multiplications.

Write an arbitrary vector X ∈ 𝔭_{2}(H^{36}) as

X = x_{1}X_{−22} + x_{2}X_{−26} + y_{1}X_{27} + y_{2}X_{30} + z_{1}X_{−25} + z_{2}X_{−28} + uX_{29} + vX_{31} + X′

where X′ ∈ 𝔭_{3}(H^{36}). The singular set S of this prehomogeneous vector space is the union of the three hyperquadrics:

S_{1} : y_{1}z_{2} + y_{2}z_{1} = 0; S_{2} : x_{1}z_{2} − x_{2}z_{1} = 0; S_{3} : x_{1}y_{2} + x_{2}y_{1} = 0;

and two hyperplanes:

S_{4} : u = 0; S_{5} : v = 0.

Let

Y_{1} = X_{27} + X_{29} + X_{31} + X_{−22} + X_{−28},
Y_{2} = (X_{31} + X_{−25} + X_{27} + X_{−22} + X_{29}) + (X_{−8}),
Y_{3} = X_{29} + X_{30} + X_{31} + X_{−22} + X_{−28},
Y_{4} = X_{27} + X_{30} + X_{31} + X_{−22} + X_{−28},
Y_{5} = X_{27} + X_{29} + X_{30} + X_{−22} + X_{−28}.

Then Y_{i} ∈ S_{i} and a simple computation shows that the orbit Q_{H^{36}} · Y_{i} is a dense open subset of S_{i}. It is not hard to verify that

Y_{1}, Y_{3} ∈ O_{1}^{34}; Y_{2} ∈ O_{1}^{32}; Y_{4} ∈ O_{1}^{30}; Y_{5} ∈ O_{1}^{29}.

It follows that S ⊂ \overline{O_{1}^{34}}. The upshot of this argument is that the proof of O_{1}^{36} ≯ O_{1}^{23} is reduced to that of O_{1}^{34} ≯ O_{1}^{23}.

We now turn to the pair (34, 23). It is immediate from Table 9 that

O_{1}^{32} ≯ O_{1}^{23}, O_{1}^{33} ≯ O_{1}^{23}.

The centralizer of H^{34} in K is just the maximal torus T_{6} (with Lie algebra 𝔥). The space 𝔭_{2}(H^{34}) has dimension 18 while its subspace 𝔤_{H^{34}}(1, 2) has dimension 5 and a basis:

{X_{30}, X_{−27}, X_{−24}, X_{29}, X_{31}} .

Write an arbitrary vector X ∈ 𝔭_{2}(H^{34}) as

X = x_{1}X_{30} + x_{2}X_{−27} + x_{3}X_{−24} + x_{4}X_{29} + x_{5}X_{31} + X′

where X′ ∈ 𝔭_{3}(H^{34}). The singular set S of the prehomogeneous vector space (Q_{H^{34}}, 𝔭_{2}(H^{34})) is the union of the five hyperplanes: S_{i} : x_{i} = 0}.

Now let

Y_{1} = X_{−27} + X_{−24} + X_{29} + X_{31} + X_{−22} + X_{−25},
Y_{2} = (X_{−22} + X_{29} + X_{−24} + X_{31} + X_{−25}) + (X_{30}),
Y_{4} = X_{30} + X_{−27} + X_{−24} + X_{31} + X_{32} + X_{−25},
Y_{5} = X_{30} + X_{−27} + X_{−24} + X_{29} + X_{32} + X_{33}.

Then Y_{i} ∈ S_{i} and the orbit Q_{H^{34}} · Y_{i} is a dense open subset of S_{i}. It is not hard to verify that Y_{1} ∈ O_{1}^{33} and observe that Y_{2} is the representative E^{32} ∈ O_{1}^{32} listed in Table 5. It follows that

S_{1} ∪ S_{2} ⊂ \overline{O_{1}^{32}} ∪ \overline{O_{1}^{33}}.

We also have

S_{4} ∪ S_{5} ⊂ \overline{O_{1}^{33}}

because a computation shows that each of the orbits G · Y_{4} and G · Y_{5} has dimension 64.

If Y ∈ S_{3} then the orbit G · Y has dimension ≤ 58, and so S_{3} ⊂ \overline{O_{1}^{33}}. As O_{1}^{23} is not contained in the union of \overline{O_{1}^{32}} and \overline{O_{1}^{33}}, we infer that it does not meet 𝔭_{2}(H^{34}). We deduce that O_{1}^{34} ≯ O_{1}^{23}.

This completes the proof of the theorem.

An interesting feature of the above prehomogeneous vector space is that the hyperplane S_{3} contains infinitely many Q_{H^{34}} -orbits. Indeed dim S_{3} = 17 while each of the orbits in S_{3} has dimension ≤ 15.

5. Types E III and E IV

Let 𝔤_{0} be of type E III and so 𝔨 is of type D_{5} + C and K = (Spin_{10} × GL_{1})/Z_{4}. We may assume that 𝔥 ⊂ 𝔨. The roots

β_{1} = α_{1}, β_{2} = α_{3}, β_{3} = α_{4}, β_{4} = α_{5}, β_{5} = α_{2}

form a base of the root system of (𝔨, 𝔥). We also set β_{6} = α_{6}.

Table 11: Nonzero nilpotent orbits in 𝔭 (𝔤_{0} = E III)

k i  β_{j}(H^{i})  E^{i}  Type of E^{i}
1 1 00001 0 X_{36}  A_{1}
1 2 00010 −2 X_{−6}  A_{1}
2 3 10000 1 (X_{23}) + (X_{36})  2A_{1}
2 4 10000 −2 (X_{−6}) + (X_{−31})  2A_{1}
2 5 00011 −2 (X_{36}) + (X_{−6})  2A_{1}
4 6 02000 −2 X_{32} + X_{−6}  A_{2}
5 7 11010 −2 (X_{32} + X_{−6}) + (X_{33})  A_{2} + A_{1}
5 8 11001 −3 (X_{36} + X_{−20}) + (X_{−21})  A_{2} + A_{1}
6 9 40000 −2 (X_{23} + X_{−21}) + (X_{36} + X_{−20}) 2A_{2}
8 10 00013 −2 X_{20} + X_{−6} + X_{32}  A_{3}
8 11 00031 −6 X_{−20} + X_{36} + X_{−32}  A_{3}
12 12 02022 −6 X_{−20} + X_{31} + X_{−21} + X_{32}  A_{4}

Table 11 lists the nonzero K -orbits O_{1}^{i} ⊂ N_{1}, 1 ≤ i ≤ 12. Its description is the same as that of Table 5 (except that β_{6}(H^{i}) may be negative).

Theorem 5.1. Let 𝔤_{0} be of type E III or E IV. Then the closure diagram of the orbit space N_{1}/K is as given in Figure 5.

Proof. Assume first that 𝔤_{0} = E IV. Then there are only two nonzero orbits in N_{1} : O_{1}^{1} of dimension 32 and O_{1}^{2} of dimension 48. Since N_{1} is an equidimensional variety, in this case it is irreducible. Hence O_{1}^{2} > O_{1}^{1}.

From now on let 𝔤_{0} = E III.

[figure: Figure 5: Closure diagrams for E III and E IV; E III; 0; 0; 22; 1; .; .; .; .; .; .; 2; 32; 3; .; .; .; 5; .; .; .; 4; 42; 6; 46; 7; 8; .; .; .; .; .; .; 48; 9; 52; 10; .; .; .; .; .; .; 11; 60; 12; E IV; 0; 0; 32; 1; 48; 2]
Table 12: Elements E ∈ 𝔭_{2}(H^{i}) ∩ O_{1}^{j}
i j Type  E  w
12 11 A_{3}  X_{−20} + X_{31} + X_{−21}  s_{15}
12 10 A_{3}  X_{31} + X_{−21} + X_{32}  s_{17}
12 9 2A_{2}  (X_{32} + X_{−21}) + (X_{33} + X_{−20})
9,11 8 A_{2} + A_{1} (X_{36} + X_{−20}) + (X_{−21})
9,10 7 A_{2} + A_{1} (X_{32} + X_{−6}) + (X_{33})
8 6 A_{2}  X_{36} + X_{−20}
8 4 2A_{1}  (X_{−20}) + (X_{−21})
7 6 A_{2}  X_{32} + X_{−6}
7 3 2A_{1}  (X_{32}) + (X_{33})
6 5 2A_{1}  (X_{36}) + (X_{−6})
4,5 2 A_{1}  X_{−6}
3,5 1 A_{1}  X_{36}

Let (i, j) be a pair of vertices in the E III diagram in Figure 5 which are joined by a solid line with i above j. We show that O_{1}^{i} > O_{1}^{j} by exhibiting an element

E ∈ 𝔭_{2}(H^{i}) ∩ O_{1}^{j}.

This is accomplished in Table 12 whose description is the same as that of Table 7.

It remains to show that O_{1}^{i} ≯ O_{1}^{j} when (i, j) is one of the four pairs: (10, 4), (11, 3), (3, 2), and (4, 1). In all four cases this can be verified by applying [9, Theorem 4.1] to the module V (ω_{1}). We omit the details.

6. Appendix

Here is our enumeration of the positive roots of E_{6}. (The simple roots are enumerated as in [2].)

Table 13: Positive roots of E_{6}

\begin{matrix}i & α_{i} & i & α_{i} & i & α_{i} & i & α_{i} \\ 1 & 100000 & 10 & 000110 & 19 & 011110 & 28 & 011211 \\ 2 & 010000 & 11 & 000011 & 20 & 010111 & 29 & 112210 \\ 3 & 001000 & 12 & 101100 & 21 & 001111 & 30 & 111211 \\ 4 & 000100 & 13 & 011100 & 22 & 111110 & 31 & 011221 \\ 5 & 000010 & 14 & 010110 & 23 & 101111 & 32 & 112211 \\ 6 & 000001 & 15 & 001110 & 24 & 011210 & 33 & 111221 \\ 7 & 101000 & 16 & 000111 & 25 & 011111 & 34 & 112221 \\ 8 & 010100 & 17 & 111100 & 26 & 111210 & 35 & 112321 \\ 9 & 001100 & 18 & 101110 & 27 & 111111 & 36 & 122321 \end{matrix}

Consider the structure constants N(i, j) of E_{6} defined by

[X_{i}, X_{j}] = N(i, j)X_{k}

where α_{i} and α_{j} are roots such that α_{i}+α_{j} = α_{k} is also a root. Then N(i, j) = ±1. Table 14 records the above relations for i > 0. The other such relations can be easily written down because N(−i, −j) = N(i, j).

The relation displayed above is recorded in the table by inserting in the i-th box the entry j : N(i, j)X_{k}. For instance we have

[X_{2}, X_{4}] = −X_{8}, [X_{2}, X_{−8}] = X_{−4}, [X_{2}, X_{−13}] = −X_{−9}.

D – oković 409

Table 14: Some defining relations of E_{6}

1 3 : X_{7}, 9 : X_{12}, 13 : X_{17}, 15 : X_{18}, 19 : X_{22}, 21 : X_{23}, 24 : X_{26}, 25 : X_{27},

28 : X_{30}, 31 : X_{33}, −7 : −X_{−3}, −12 : −X_{−9}, −17 : −X_{−13}, −18 : −X_{−15}, −22 : −X_{−19}, −23 : −X_{−21}, −26 : −X_{−24}, −27 : −X_{−25}, −30 : −X_{−28}, −33 : −X_{−31}

2 4 : −X_{8}, 9 : X_{13}, 10 : X_{14}, 12 : X_{17}, 15 : X_{19}, 16 : X_{20}, 18 : X_{22},

21 : X_{25}, 23 : X_{27}, 35 : X_{36}, −8 : X_{−4}, −13 : −X_{−9}, −14 : −X_{−10}, −17 : −X_{−12}, −19 : −X_{−15}, −20 : −X_{−16}, −22 : −X_{−18}, −25 : −X_{−21}, −27 : −X_{−23}, −36 : −X_{−35}

3 1 : −X_{7}, 4 : −X_{9}, 8 : X_{13}, 10 : X_{15}, 14 : X_{19}, 16 : X_{21}, 20 : X_{25},

26 : X_{29}, 30 : X_{32}, 33 : X_{34}, −7 : X_{−1}, −9 : X_{−4}, −13 : −X_{−8}, −15 : −X_{−10}, −19 : −X_{−14}, −21 : −X_{−16}, −25 : −X_{−20}, −29 : −X_{−26}, −32 : −X_{−30}, −34 : −X_{−33}

4 2 : X_{8}, 3 : X_{9}, 5 : X_{10}, 7 : X_{12}, 11 : X_{16}, 19 : X_{24}, 22 : X_{26}, 25 : X_{28},

27 : X_{30}, 34 : X_{35}, −8 : −X_{−2}, −9 : −X_{−3}, −10 : −X_{−5}, −12 : −X_{−7}, −16 : −X_{−11}, −24 : −X_{−19}, −26 : −X_{−22}, −28 : −X_{−25}, −30 : −X_{−27}, −35 : −X_{−34}

5 4 : −X_{10}, 6 : −X_{11}, 8 : X_{14}, 9 : X_{15}, 12 : X_{18}, 13 : X_{19}, 17 : X_{22},

28 : X_{31}, 30 : X_{33}, 32 : X_{34}, −10 : X_{−4}, −11 : X_{−6}, −14 : −X_{−8}, −15 : −X_{−9}, −18 : −X_{−12}, −19 : −X_{−13}, −22 : −X−17, −31 : −X_{−28}, −33 : −X_{−30}, −34 : −X_{−32}

6 5 : X_{11}, 10 : X_{16}, 14 : X_{20}, 15 : X_{21}, 18 : X_{23}, 19 : X_{25}, 22 : X_{27},

24 : X_{28}, 26 : X_{30}, 29 : X_{32}, −11 : −X_{−5}, −16 : −X_{−10}, −20 : −X_{−14}, −21 : −X_{−15}, −23 : −X_{−18}, −25 : −X_{−19}, −27 : −X_{−22}, −28 : −X_{−24}, −30 : −X_{−26}, −32 : −X_{−29}

7 4 : −X_{12}, 8 : X_{17}, 10 : X_{18}, 14 : X_{22}, 16 : X_{23}, 20 : X_{27}, 24 : −X_{29},

28 : −X_{32}, 31 : −X_{34}, −1 : X_{3}, −3 : −X_{1}, −12 : X_{−4}, −17 : −X_{−8}, −18 : −X_{−10}, −22 : −X_{−14}, −23 : −X_{−16}, −27 : −X_{−20}, −29 : X_{−24}, −32 : X_{−28}, −34 : X_{−31}

8 3 : −X_{13}, 5 : −X_{14}, 7 : −X_{17}, 11 : −X_{20}, 15 : X_{24}, 18 : X_{26}, 21 : X_{28},

23 : X_{30}, 34 : −X_{36}, −2 : −X_{4}, −4 : X_{2}, −13 : X_{−3}, −14 : X_{−5}, −17 : X_{−7}, −20 : X_{−11}, −24 : −X_{−15}, −26 : −X_{−18}, −28 : −X_{−21}, −30 : −X_{−23}, −36 : X_{−34}

9 1 : −X_{12}, 2 : −X_{13}, 5 : −X_{15}, 11 : −X_{21}, 14 : X_{24}, 20 : X_{28}, 22 : −X_{29},

27 : −X_{32}, 33 : X_{35}, −3 : −X_{4}, −4 : X_{3}, −12 : X_{−1}, −13 : X_{−2}, −15 : X_{−5}, −21 : X_{−11}, −24 : −X_{−14}, −28 : −X_{−20}, −29 : X_{−22}, −32 : X_{−27}, −35 : −X_{−33}

10 2 : −X_{14}, 3 : −X_{15}, 6 : −X_{16}, 7 : −X_{18}, 13 : X_{24}, 17 : X_{26}, 25 : −X_{31},

27 : −X_{33}, 32 : X_{35}, −4 : X_{5} − 5 : −X_{4}, −14 : X_{−2}, −15 : X_{−3}, −16 : X_{−6}, −18 : X_{−7}, −24 : −X_{−13}, −26 : −X_{−17}, −31 : X_{−25}, −33 : X_{−27}, −35 : −X_{−32}

11 4 : −X_{16}, 8 : X_{20}, 9 : X_{21}, 12 : X_{23}, 13 : X_{25}, 17 : X_{27}, 24 : −X_{31},

26 : −X_{33}, 29 : −X_{34}, −5 : −X_{6}, −6 : X_{5}, −16 : X_{−4}, −20 : −X_{−8}, −21 : −X_{−9}, −23 : −X_{−12}, −25 : −X_{−13}, −27 : −X_{−17}, −31 : X_{−24}, −33 : X_{−26}, −34 : X_{−29} 410 D – oković

Table 14: (continued)

12 2 : −X_{17}, 5 : −X_{18}, 11 : −X_{23}, 14 : X_{26}, 19 : X_{29}, 20 : X_{30}, 25 : X_{32},

31 : −X_{35}, −1 : X_{9}, −4 : X_{7}, −7 : −X_{4}, −9 : −X_{1}, −17 : X_{−2}, −18 : X_{−5}, −23 : X_{−11}, −26 : −X_{−14}, −29 : −X_{−19}, −30 : −X_{−20}, −32 : −X_{−25}, −35 : X_{−31}

13 1 : −X_{17}, 5 : −X_{19}, 10 : −X_{24}, 11 : −X_{25}, 16 : −X_{28}, 18 : X_{29}, 23 : X_{32},

33 : X_{36}, −2 : X_{9}, −3 : X_{8}, −8 : −X_{3}, −9 : −X_{2}, −17 : X_{−1}, −19 : X_{−5}, −24 : X_{−10}, −25 : X_{−11}, −28 : X_{−16}, −29 : −X_{−18}, −32 : −X_{−23}, −36 : −X_{−33}

14 3 : −X_{19}, 6 : −X_{20}, 7 : −X_{22}, 9 : −X_{24}, 12 : −X_{26}, 21 : X_{31}, 23 : X_{33},

32 : X_{36}, −2 : X_{10}, −5 : X_{8}, −8 : −X_{5}, −10 : −X_{2}, −19 : X_{−3}, −20 : X_{−6}, −22 : X_{−7}, −24 : X_{−9}, −26 : X_{−12}, −31 : −X_{−21}, −33 : −X_{−23}, −36 : −X_{−32}

15 1 : −X_{18}, 2 : −X_{19}, 6 : −X_{21}, 8 : −X_{24}, 17 : X_{29}, 20 : X_{31}, 27 : −X_{34},

30 : −X_{35}, −3 : X_{10}, −5 : X_{9}, −9 : −X_{5}, −10 : −X_{3}, −18 : X_{−1}, −19 : X_{−2}, −21 : X_{−6}, −24 : X_{−8}, −29 : −X_{−17}, −31 : −X_{−20}, −34 : X_{−27}, −35 : X_{−30}

16 2 : −X_{20}, 3 : −X_{21}, 7 : −X_{23}, 13 : X_{28}, 17 : X_{30}, 19 : X_{31}, 22 : X_{33},

29 : −X_{35}, −4 : X_{11}, −6 : X_{10}, −10 : −X_{6}, −11 : −X_{4}, −20 : X_{−2}, −21 : X_{−3}, −23 : X_{−7}, −28 : −X_{−13}, −30 : −X_{−17}, −31 : −X_{−19}, −33 : −X_{−22}, −35 : X_{−29}

17 5 : −X_{22}, 10 : −X_{26}, 11 : −X_{27}, 15 : −X_{29}, 16 : −X_{30}, 21 : −X_{32},

31 : −X_{36}, −1 : X_{13}, −2 : X_{12}, −7 : X_{8}, −8 : −X_{7}, −12 : −X_{2}, −13 : −X_{1}, −22 : X_{−5}, −26 : X_{−10}, −27 : X_{−11}, −29 : X_{−15}, −30 : X_{−16}, −32 : X_{−21}, −36 : X_{−31}

18 2 : −X_{22}, 6 : −X_{23}, 8 : −X_{26}, 13 : −X_{29}, 20 : X_{33}, 25 : X_{34}, 28 : X_{35},

−1 : X_{15}, −5 : X_{12}, −7 : X_{10}, −10 : −X_{7}, −12 : −X_{5}, −15 : −X_{1}, −22 : X_{−2}, −23 : X_{−6}, −26 : X_{−8}, −29 : X_{−13}, −33 : −X_{−20}, −34 : −X_{−25}, −35 : −X_{−28}

19 1 : −X_{22}, 4 : −X_{24}, 6 : −X_{25}, 12 : −X_{29}, 16 : −X_{31}, 23 : X_{34}, 30 : −X_{36},

−2 : X_{15}, −3 : X_{14}, −5 : X_{13}, −13 : −X_{5}, −14 : −X_{3}, −15 : −X_{2}, −22 : X_{−1}, −24 : X_{−4}, −25 : X_{−6}, −29 : X_{−12}, −31 : X_{−16}, −34 : −X_{−23}, −36 : X_{−30}

20 3 : −X_{25}, 7 : −X_{27}, 9 : −X_{28}, 12 : −X_{30}, 15 : −X_{31}, 18 : −X_{33},

29 : −X_{36}, −2 : X_{16}, −6 : X_{14}, −8 : −X_{11}, −11 : X_{8}, −14 : −X_{6}, −16 : −X_{2}, −25 : X_{−3}, −27 : X_{−7}, −28 : X_{−9}, −30 : X_{−12}, −31 : X_{−15}, −33 : X_{−18}, −36 : X_{−29}

21 1 : −X_{23}, 2 : −X_{25}, 8 : −X_{28}, 14 : −X_{31}, 17 : X_{32}, 22 : X_{34}, 26 : X_{35},

−3 : X_{16}, −6 : X_{15}, −9 : −X_{11}, −11 : X_{9}, −15 : −X_{6}, −16 : −X_{3}, −23 : X_{−1}, −25 : X_{−2}, −28 : X_{−8}, −31 : X_{−14}, −32 : −X_{−17}, −34 : −X_{−22}, −35 : −X_{−26}

22 4 : −X_{26}, 6 : −X_{27}, 9 : X_{29}, 16 : −X_{33}, 21 : −X_{34}, 28 : X_{36}, −1 : X_{19},

−2 : X_{18}, −5 : X_{17}, −7 : X_{14}, −14 : −X_{7}, −17 : −X_{5}, −18 : −X_{2}, −19 : −X_{1}, −26 : X_{−4}, −27 : X_{−6}, −29 : −X_{−9}, −33 : X_{−16}, −34 : X_{−21}, −36 : −X_{−28}

D – oković 411 Table 14: (continued)

23 2 : −X_{27}, 8 : −X_{30}, 13 : −X_{32}, 14 : −X_{33}, 19 : −X_{34}, 24 : −X_{35},

−1 : X_{21}, −6 : X_{18}, −7 : X_{16}, −11 : X_{12}, −12 : −X_{11}, −16 : −X_{7}, −18 : −X_{6}, −21 : −X_{1}, −27 : X_{−2}, −30 : X_{−8}, −32 : X_{−13}, −33 : X_{−14}, −34 : X_{−19}, −35 : X_{−24}

24 1 : −X_{26}, 6 : −X_{28}, 7 : X_{29}, 11 : X_{31}, 23 : X_{35}, 27 : X_{36}, −4 : X_{19},

−8 : X_{15}, −9 : X_{14}, −10 : X_{13}, −13 : −X_{10}, −14 : −X_{9}, −15 : −X_{8}, −19 : −X_{4}, −26 : X_{−1}, −28 : X_{−6}, −29 : −X_{−7}, −31 : −X_{−11}, −35 : −X_{−23}, −36 : −X_{−27}

25 1 : −X_{27}, 4 : −X_{28}, 10 : X_{31}, 12 : −X_{32}, 18 : −X_{34}, 26 : X_{36}, −2 : X_{21},

−3 : X_{20}, −6 : X_{19}, −11 : X_{13}, −13 : −X_{11}, −19 : −X_{6}, −20 : −X_{3}, −21 : −X_{2}, −27 : X_{−1}, −28 : X_{−4}, −31 : −X_{−10}, −32 : X_{−12}, −34 : X_{−18}, −36 : −X_{−26}

26 3 : −X_{29}, 6 : −X_{30}, 11 : X_{33}, 21 : −X_{35}, 25 : −X_{36}, −1 : X_{24}, −4 : X_{22},

−8 : X_{18}, −10 : X_{17}, −12 : X_{14}, −14 : −X_{12}, −17 : −X_{10}, −18 : −X_{8}, −22 : −X_{4}, −24 : −X_{1}, −29 : X_{−3}, −30 : X_{−6}, −33 : −X_{−11}, −35 : X_{−21}, −36 : X_{−25}

27 4 : −X_{30}, 9 : X_{32}, 10 : X_{33}, 15 : X_{34}, 24 : −X_{36}, −1 : X_{25}, −2 : X_{23},

−6 : X_{22}, −7 : X_{20}, −11 : X_{17}, −17 : −X_{11}, −20 : −X_{7}, −22 : −X_{6}, −23 : −X_{2}, −25 : −X_{1}, −30 : X_{−4}, −32 : −X_{−9}, −33 : −X_{−10}, −34 : −X_{−15}, −36 : X_{−24}

28 1 : −X_{30}, 5 : −X_{31}, 7 : X_{32}, 18 : −X_{35}, 22 : −X_{36}, −4 : X_{25}, −6 : X_{24},

−8 : X_{21}, −9 : X_{20}, −13 : −X_{16}, −16 : X_{13}, −20 : −X_{9}, −21 : −X_{8}, −24 : −X_{6}, −25 : −X_{4}, −30 : X_{−1}, −31 : X_{−5}, −32 : −X_{−7}, −35 : X_{−18}, −36 : X_{−22}

29 6 : −X_{32}, 11 : X_{34}, 16 : X_{35}, 20 : X_{36}, −3 : X_{26}, −7 : −X_{24}, −9 : −X_{22},

−12 : X_{19}, −13 : X_{18}, −15 : X_{17}, −17 : −X_{15}, −18 : −X_{13}, −19 : −X_{12}, −22 : X_{9}, −24 : X_{7}, −26 : −X_{3}, −32 : X_{−6}, −34 : −X_{−11}, −35 : −X_{−16}, −36 : −X_{−20}

30 3 : −X_{32}, 5 : −X_{33}, 15 : X_{35}, 19 : X_{36}, −1 : X_{28}, −4 : X_{27}, −6 : X_{26},

−8 : X_{23}, −12 : X_{20}, −16 : X_{17}, −17 : −X_{16}, −20 : −X_{12}, −23 : −X_{8}, −26 : −X_{6}, −27 : −X_{4}, −28 : −X_{1}, −32 : X_{−3}, −33 : X_{−5}, −35 : −X_{−15}, −36 : −X_{−19}

31 1 : −X_{33}, 7 : X_{34}, 12 : X_{35}, 17 : X_{36}, −5 : X_{28}, −10 : −X_{25}, −11 : −X_{24},

−14 : X_{21}, −15 : X_{20}, −16 : X_{19}, −19 : −X_{16}, −20 : −X_{15}, −21 : −X_{14}, −24 : X_{11}, −25 : X_{10}, −28 : −X_{5}, −33 : X_{−1}, −34 : −X_{−7}, −35 : −X_{−12}, −36 : −X_{−17}

32 5 : −X_{34}, 10 : −X_{35}, 14 : −X_{36}, −3 : X_{30}, −6 : X_{29}, −7 : −X_{28},

−9 : −X_{27}, −12 : X_{25}, −13 : X_{23}, −17 : −X_{21}, −21 : X_{17}, −23 : −X_{13}, −25 : −X_{12}, −27 : X_{9}, −28 : X_{7}, −29 : −X_{6}, −30 : −X_{3}, −34 : X_{−5}, −35 : X_{−10}, −36 : X_{−14}

33 3 : −X_{34}, 9 : −X_{35}, 13 : −X_{36}, −1 : X_{31}, −5 : X_{30}, −10 : −X_{27},

−11 : −X_{26}, −14 : X_{23}, −16 : X_{22}, −18 : X_{20}, −20 : −X_{18}, −22 : −X_{16}, −23 : −X_{14}, −26 : X_{11}, −27 : X_{10}, −30 : −X_{5}, −31 : −X_{1}, −34 : X_{−3}, −35 : X_{−9}, −36 : X_{−13}

Table 14: (continued)

34 4 : −X_{35}, 8 : X_{36}, −3 : X_{33}, −5 : X_{32}, −7 : −X_{31}, −11 : −X_{29},

−15 : −X_{27}, −18 : X_{25}, −19 : X_{23}, −21 : X_{22}, −22 : −X_{21}, −23 : −X_{19}, −25 : −X_{18}, −27 : X_{15}, −29 : X_{11}, −31 : X_{7}, −32 : −X_{5}, −33 : −X_{3}, −35 : X_{−4}, −36 : −X_{−8}

35 2 : −X_{36}, −4 : X_{34}, −9 : X_{33}, −10 : X_{32}, −12 : −X_{31}, −15 : −X_{30},

−16 : −X_{29}, −18 : X_{28}, −21 : X_{26}, −23 : −X_{24}, −24 : X_{23}, −26 : −X_{21}, −28 : −X_{18}, −29 : X_{16}, −30 : X_{15}, −31 : X_{12}, −32 : −X_{10}, −33 : −X_{9}, −34 : −X_{4}, −36 : X_{−2}

36 −2 : X_{35}, −8 : −X_{34}, −13 : X_{33}, −14 : X_{32}, −17 : −X_{31}, −19 : −X_{30},

−20 : −X_{29}, −22 : X_{28}, −24 : X_{27}, −25 : X_{26}, −26 : −X_{25}, −27 : −X_{24}, −28 : −X_{22}, −29 : X_{20}, −30 : X_{19}, −31 : X_{17}, −32 : −X_{14}, −33 : −X_{13}, −34 : X_{8}, −35 : −X_{2}

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D. Ž. D – oković Department of Pure Mathematics University of Waterloo Waterloo, Ontario, Canada N2L 3G1 djokovic@uwaterloo.ca

Received April 24, 2000 and in final form March 4, 2001