Abelian Group Theory by D. Arnold, R. Hunter, E. Walker

By D. Arnold, R. Hunter, E. Walker

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12), l are sets of the following types: type a. An orbit of p6 i, where 6 i is a f u n d a m e n t a l weight, 6 i ~ N. type b. The union of (0) and the orbit of p~i, where ~i is a fund a m e n t a l weight, ~i 6 N. Proof of (1). First we note that for every ~ C N, y ~ &, the weights y and way = y - ~ are d i r e c t l y N-connected. So the N - c o n n e e t e d components are invariant under the action of W. If ~i is a root, then 0, p~i are d i r e c t l y E-connected. that sets of type a or b are E-connected.

The r e p r e s e n t a t i o n (Pi)M1. 0 Fr i n LM . z d e n o t e s l the vector space LM. , viewed as r e p r e s e n t a t i o n space of oi). 9). denote (see Lemma 5O So every non-zero weight of Pi is in the orbit of ~i, and has multiplicity 1 (see [ 7 [, Expos@ 16, Proposition 20, Proposition 1 and Expos@ 4). Suppose zero is a weight of Pi" Then 6 i is Z-connected W6 i U (0), so 6i is a multiple a root, because of a root. it is a minimal formula, or from [25]. In fact ~i has to be dominant weight.

Z*~,6 = such that ~+8 is on ~+8. It is c l e a r : 0). Hence while ~+8 we that consider = Y+~ is n. 6, suppose (iii)). = < ~ + 8 , 8 > = 2. So = i, a contradiction. (~,~) : ! 1. X*Y-8' image G2, > (8,8), Ny_8, ~ : ! 1. For the = -N _~,6 for X6, an 7) Y into only ~ 1, < y + 6 , B > = -3, = 0 shows that to E Z, to get (~,8) N6,~_~ (ker n. distinct = (6,~). ~ 1, <~,8> Now s(~,B) (~,8) (8,8) y-6 E ~. Then 7 is c o n t a i n e d that independent depends are respect and we h a v e suppose Z* ~,8,y,6 with = (Y,Y) We h a v e (~ e ~), to some (We r e q u i r e that degenerate (~,~) come inverse [X * , X ~] , if ~ ,B are respect In this sums ker .

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