10 ELDAR STRAUME
from the weight system Q(O), cf. (1.6) and (1.7) of Chapter II, so all cases with
c(O) = dim^O - dim (G/G
are readily found by some straightforward estimates of dimensions.
Next, assume O is reducible. By direct checking there are just a few combinations
0 = 0 ^ + O2, with c(O^) + c(32) - 3, leading to c(3) 3, namely, when G = SO(n),
G2 or Spin(n), 5 n 9, see Table II and III.
Case (ii) O =
® 42 g g 4 ^ , q 2, O irreducible.
Let K(n) = O(n), U(n) or Sp(n), with standard representation 8
respectively. The isotropy types of the tensor product linear group
( 5 ) (K(m) x K(n),5
), m n ,
are given by the family
( 6 ) A[K(t-|) x • • • x K(t
)] x K(m-t)x K(n-t), 0 Z tj = t m,
where A[ ] denotes the diagonal imbedding A -» (A, A) of the "blocks" K(tj) into
K(t)x K(t) c K(m)x K(n). So the principal isotropy type (G^) corresponds to tj = 1,
t = m in (6), consequently
( 7 ) c(5
8n) = m (m n)
In particular, any linear group (G,0) = (G-j x 02,^ 1 %^2)
w i t n m =
d i m p ^ ^
is a subgroup of the group in (5), so m c(O).
Assume first q = 2, G semisimple and m 2. Since dim^O dim G + 3, it is
straightforward to verify that, in addition to the semisimple component of (5), there
are just two more cases, namely (SO(3)xSpin(7),p3®A7) and (Sp(1)xSp(n),
Assume q 3, G semisimple. Since dim G increases "additively" with q, whereas
dim 3 increases "multiplicatively, the ratio (dim G):(dim O) is generally too small.
By checking "critical" cases such as (8) below, we find that c(O) 3 is only possible
when O = P3 ^ ( v j (% v^) = P3 (% P4, where G = SO(3)xSO(4) is the effective
group. Note for example