If v ∈ V , then Gv = {g · v | g ∈ G} is called the orbit of v, and G
v=
{g ∈ G | g · v = v}
is called the stabilizer of v. If the orbit Gv of a point
v ∈ V
is closed, then the stabilizer G
vis reductive and the tangent space of
V
at v decomposes into the direct sum of the tangent space of Gv at v and a
complementary G
v-module N
v. The corresponding representation (G
v, N
v)
is
called slice-representation (cf. [Pop83]).
If S ⊂ G, then N
G(S) = {g ∈ G | gSg
−1= S}
is called the normalizer of S,
and Z
G(S) = {g ∈ G | gsg
−1= sfor all s ∈ S} is called the centralizer of S.
Consider the group T = {
t 00 t−1
| t ∈ C
∗
} ⊂ SL
2
. Then the normalizer of
T
is
N
SL2(T ) = {
t 0 0 t−1,
0 t −t−10| t ∈ C
∗}.
Lemma 6.2.1. ([Pop83, Lemma 1]) Consider v = x
ny
n∈ V
a) SL
2v
is closed in V
2n.
b) SL
2v=
(
T
if n = 1 modulo 2,
N
SL2(T )
if n = 0 modulo 2.
Proposition 6.2.2. ([Pop83, Proposition 4]) Consider G a reductive group of
rank 1 and V a G-module. Consider T a maximal torus of G and suppose that
p, n, r
are the dimensions of the T-weight subspaces of V which have positive,
negative, zero weight respectively.
If pn > 0, then hd O(V )
G≥ (p − 1)(n − 1).
Proposition 6.2.3. ([Pop83, Proposition 4]) Consider G a reductive group
of dimension 1, T a maximal torus of G, with Z
G(T )
diagonalizable and with
Z
G(T ) 6= G. Consider and V a G-module, and V
+, V
−, V
0the sum of those T -
weight subspaces of V which have positive, negative, and zero weight respectively.
Then, there exist bases of Z
G(T )-weight vectors x
1, . . . , x
pand y
1, . . . , y
pin (V
+)
∗and in (V
−)
∗respectively, such that the T -weights of x
i
and y
iare
inverse of each other, and ghx
ii = hy
ii
and ghy
ii = hx
ii, if g /∈ Z
G(T ).
If p > 0, then, hd O(V )
G≥
(p−1)(p−2)2
.
Proposition 6.2.4. ([Pop83, Propositions 6 and 7])
a) If n ≥ 2, then hd O(V
2n)
SL2≥
(
(n − 2)
2if n is odd,
(n−2)(n−3) 2if n is even.
b) If n ≥ 3, then hd O(V
2n−1)
SL2≥ n
2− 2n − 2.
c) If hd O(V
2n)
SL2= dand n ≥ 2, then n ≤
(
√8d+1+52
if n is even,
√
d + 2
if n is odd.
d) If hd O(V
2n−1)
SL2= dand n ≥ 3, then n ≤
√
d + 3 + 1.
Theorem 6.2.5. ([Pop83, Theorem 2]) (A monotony theorem.) Consider G a
reductive group and V a G-module. We have the following:
a) If v ∈ V and Gv is closed in V . Denote by (G
v, N
v)the slice-representation.
Then hd O(N
v)
Gv≤ hd O(V )
G.
b) If V = W
1⊕ . . . ⊕ W
p, then P
pi=1hd O(W
i)
G≤ hd O(V )
G.
c) If W is a submodule of V , then hd O(W )
G≤ hd O(V )
G.
One can use the Poincaré series of the algebra I of invariants of an SL
2-
module V to obtain a lower bound on the number of generators of I, and hence
on the homological dimension of this algebra. The following result was proved
by Brower [BP12]:
Proposition 6.2.6. (Brouwer [BP12]) Table
6.2contains lower bounds on the
homological dimension of the algebra of invariants of several SL
2-modules. In
the table are listed the modules, the Poincaré series of the algebra I of invariants
of these modules, and lower bounds on the number of generators r of I and on
hd I:
module
Poincaré series
r ≥
hd I ≥
V
111 + 2t
4+ 13t
8+ 13t
10+ 73t
12+ 110t
14+ . . .
158
149
V
121 + t
2+ t
3+ 3t
4+ 3t
5+ 8t
6+ 10t
7+ 20t
8+
28t
9+ 52t
10+ 73t
11+ 127t
12+ 181t
13+ . . .
113
103
V
131 + 2t
4+ 22t
8+ 33t
10+ 181t
12+ 375t
14+ . . .
502
491
V
141 + t
2+ 3t
4+ 10t
6+ 4t
7+ 31t
8+ 27t
9+
97t
10+ 110t
11+ . . .
182
170
V
151 + 3t
4+ t
6+ 36t
8+ 80t
10+ 418t
12+ . . .
425
412
V
161 + t
2+ t
3+ 3t
4+ 4t
5+ 13t
6+ 18t
7+ 47t
8+
84t
9+ 177t
10+ . . .
198
184
V
181 + t
2+ 4t
4+ t
5+ 16t
6+ 13t
7+ 71t
8+ 99t
9+
161
145
V
201 + t
2+ t
3+ 4t
4+ 5t
5+ 20t
6+ 35t
7+ 102t
8+
123
105
V
221 + t
2+ 4t
4+ t
5+ 24t
6+ 26t
7+ 144t
8+ . . .
164
144
V
241 + t
2+ t
3+ 5t
4+ 7t
5+ 29t
6+ 62t
7+ 201t
8+
242
220
V
281 + t
2+ t
3+ 5t
4+ 8t
5+ 40t
6+ 97t
7+ 365t
8+
440
414
V
321 + t
2+ t
3+ 6t
4+ 10t
5+ 54t
6+ 153t
7+ . . .
201
171
V
2⊕ V
81 + 2t
2+ t
3+ 5t
4+ 5t
5+ 15t
6+ 17t
7+
41t
8+ 54t
9+ 108t
10+ . . .
35
26
V
3⊕ V
81 + t
2+ t
3+ 3t
4+ 4t
5+ 9t
6+ 16t
7+ 30t
8+ . . .
37
27
V
4⊕ V
81 + 2t
2+ 4t
3+ 8t
4+ 16t
5+ 35t
6+ 60t
7+ . . .
42
31
V
5⊕ V
81 + t
2+ t
3+ 3t
4+ 6t
5+ 15t
6+ 31t
7+ . . .
43
31
V
6⊕ V
81 + 2t
2+ 2t
3+ 10t
4+ 14t
5+ 46t
6+ 82t
7+ . . .
88
75
V
1⊕ 2V
31 + t
2+ 13t
4+ 26t
6+ . . .
26
19
V
2⊕ 2V
31 + 2t
2+ 3t
3+ 9t
4+ 12t
5+ 26t
6+ 44t
7+ . . .
26
18
V
1⊕ 2V
2⊕ V
31 + 3t
2+ 6t
3+ 15t
4+ 30t
5+ 65t
6+ . . .
34
25
V
2⊕ V
3⊕ V
41 + 2t
2+ 3t
3+ 7t
4+ 14t
5+ 29t
6+ 52t
7+ . . .
43
34
V
1⊕ 2V
2⊕ V
41 + 4t
2+ 6t
3+ 18t
4+ 33t
5+ . . .
27
17
3V
2⊕ V
41 + 7t
2+ 8t
3+ 42t
4+ 64t
5+ . . .
37
26
2V
3⊕ V
41 + 2t
2+ 2t
3+ 9t
4+ 16t
5+ 37t
6+ 71t
7+ . . .
69
59
V
3⊕ 2V
41 + 3t
2+ 4t
3+ 10t
4+ 22t
5+ 49t
6+ 96t
7+ . . .
45
34
V
3⊕ V
51 + 6t
4+ 7t
6+ 36t
8+ . . .
28
21
V
4⊕ V
51 + t
2+ t
3+ 2t
4+ 4t
5+ 8t
6+ 12t
7+ 22t
8+
37t
9+ 56t
10+ . . .
59
51
2V
51 + t
2+ 7t
4+ 14t
6+ 72t
8+ 168t
10+ . . .
105
96
V
3⊕ V
61 + t
2+ t
3+ 3t
4+ 4t
5+ 8t
6+ 12t
7+ 21t
8+ . . .
24
16
V
4⊕ V
61 + 2t
2+ 2t
3+ 7t
4+ 8t
5+ 24t
6+ 31t
7+ 68t
8+
33
24
V
5⊕ V
61 + t
2+ t
3+ 3t
4+ 5t
5+ 12t
6+ 22t
7+ . . .
31
21
2V
61 + 3t
2+ 12t
4+ 6t
5+ 44t
6+ 40t
7+ 150t
8+ . . .
29
18
Table 6.2: Bounds from the Poincaré series
In document
Invariants of binary forms
(Page 121-123)