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). 7
Previous Page Next Page