Chapter I . Linear groups of cohomogeneity 4
Closed subgroups of O(n) are called linear groups of degree n, and they will be
presented as pairs (G,I), where I: G O(n) is an almost faithful representation of a
compact Lie group. We shall identify representations whose corresponding subgroups in
O(n) are conjugate. The purpose of this chapter is to determine all connected linear
groups of cohomogeneity 4, that is
c(O) = dim [Rn/G 4
In the case c(O) = 1 all groups (G,a) are well known; they are listed in Table I for
convenience. In general, the nonspiitting linear groups are the "building blocks", see
(A) below. For c(O) = 2 and 3 these are tabulated in Table II and III, respectively.
Since c(O) is an additive function with respect to outer direct sum ©, we obtain all
pairs (G.O) satisfying c(O) 3 by suitably choosing at most three summands from the
tables. These are our linear models.
(A ) Generalities Two linear groups (Gj,Oj), i =1,2, are called c-equivalent if
they are both subgroups of the same linear group (G,0), i.e., Oj = o|Gj , and moreover
c(O) = c(Oj), i = 1, 2. For connected groups it is easy to see that c-equivalence means
that both groups (up to conjugacy in O(n)) have precisely the same orbits in
[Rn.
Furthermore, in each c-equivalence class there is a unique maximal group. Observe
that "maximal" has the same meaning as in [HL].
A linear group is called splitting if it can be represented as an outer direct sum
(G,0) = (G-j x G
2
,Oi © 0
2
), dim Oj 1.
Here Gj may be trivial. If such a decomposition is impossible, (G,0) is called
nonspiitting . On the other hand, the inner direct sum of the two G-representations Oj is
the restriction
(G,®1 + 0
2
) = ( 0 x 0 , 0 ! © 0
2
) I AG
where G is regarded as the diagonal subgroup AG of G x G. We have in general
( 1 ) c(0- | © I2 ) = c(O-j) + c ( 0 2 ) ,
c(0-j + J2 ) c(O-j) + c(X2).
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