Applied optics and optical engineering,Vol.II by Rudolf Kingslake

By Rudolf Kingslake

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Also, E is nonhomogeneous, because dim E rF N. (2) Now, let X be a simple regular representation; then by (ii) or (iii) of Definition 17X = 0 unless X ~ E or X ~ E I . In fact 17EI = - 17E because LO 0, the regular representation V contains a representation Y isomorphic to E. In view of 17X = 0 and 17E > 0, X and Yare nonisomorphic.

Now, we are going to consider the individual extended Dynkin diagrams. In all cases, condition (i) is satisfied trivially and, in most cases, it is also very easy to see that the condition (iii)' is satisfied: one uses the properties of a simple regular representation X with T/(X) = 0 which are listed in the last column of the tables. These properties are satisfied for every simple homogeneous object, and therefore for every homogeneous representation at all. As a r~ult, we are mainly concerned with conditions (0) and (ii)'.

Similarly, T(R') contains a copy of E~, since (0, 0, 1, 0, 1, 1) = (0, 0, 1, 0, 0, 0) + (0,0,0,0, 1, 1) belongs both to C and to (A 3 n B 2 ) + B t . Finally, T(R") contains a copy of E~ whose central vector space is 0 U 0 U 0 U, and whose vector space C is generated by the two elements (0, 0, 0, 1, 0, -1) and (0, 1, 0, 1, 0, 0). In the remaining four cases, we shall use the concept of an interior and that of a closure of a G-subspace in a vector F-space (where G is a subfie1d of F) defined in the introduction to the tables in Chapter 6 (for details, see [4]).

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