Complex reflection groups by David William Koster

By David William Koster

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96] and rep ro d u ced h e r e : P [4 ]2 , 2 [q ]2 , 3 [3 ]3 , 3[6]2, 3 [4 ]3 , 4 [3 ]4 , 3 [8 ]2 , 4 [6 ]2 , 4 [4 ]3 , 3 [5 ]3 , 5 [ 3 ] 5 , 3 [ 1 0 ] 2 , 5 [ 6 ] 2 , 5 [4 ]3 i n L i s t 1 w i th two v e r t i c e s . Ill] and we g iv e them h e r e a l s o : 6 [ 3 ] 6 , 3 [ 4 ] 6 , 4 [ 4 ] 4 , 2 [ 6 ] 6 , '3 [ 6 ] 3 , 2 [ 8 ] 4 , 2 [1 2 ]3 ( p^ > P2 ^ • Now by th e d e f i ­ n i t i o n o f a su b g rap h i t i s e a sy t o v e r i f y t h a t i f r e C*2 th e n D e t ( r ' ) > D e t ( r ) . anc* *f Thus i f T o c c u rs amongst th o s e graph s j u s t l i s t e d , we h av e D e t ( r ' ) > 0 .

3 and we compute D e t ( r ) < 0 , g i v i n g a c o n t r a d i c t i o n . Suppose Te c + S te p X I. a. Then P i s one o f P r o o f: , Eg, i s co n n ected and c o n t a i n s a bran ch p o in t , Eg. S te p X t o g e t h e r w it h t h e f a c t t h a t has d e te r m in a n t z e ro im ply t h a t Deg(a) = 3 and t h a t t h e su b g rap h o f r formed by a and th e t h r e e v e r t i c e s to which a i s j o i n e d i s D^. F u r t h e r a l l edges o f P a r e u n l a b e l l e d ; o th e r w is e some B*, would o c c u r as a s u b g ra p h .

W| > |a| L et re C . = 2 So we have ou r c la im and th u s ftp . k- 1 k and l e t W = W (r) . I f r ^ and r^ a r e in d i s ­ t i n c t e q u iv a le n c e c l a s s e s u nd er th e e q u iv a le n c e r e l a t i o n no n o n - i d e n t i t y elem en t o f < rj> th e n *-s c o n ju g a te i n W t o any elem en t o f . L et V be a f i n i t e d im e n sio n a l complex v e c t o r s p a c e and l e t f i n i t e subgroup o f GL(V). 6 be a We say ReG i s a r e f l e c t i o n i f R / 1 and i f t h e r e i s a h y p e rp la n e UcrV f ix e d by R.

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