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By David William Koster

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U is c a lle d th e r e f l e c t i n g 11(G) be t h e c o l l e c t i o n We l e t o f h y p e rp la n e s U in V su ch t h a t U i s th e r e f l e c t i n g h y p e rp la n e f o r some re fle c tio n ReG. Note t h a t i f Uet£* i s th e r e f l e c t i n g ReG and i f TeG, th e n T(U)e u t h e r e f l e c t i o n TRT as i t i s th e r e f l e c t i n g Thus G a c t s on ^ . h y p e r p la n e fo r h y p e rp la n e f o r For Ue t t we l e t C(U) be th e subgroup o f G c o n s i s t i n g o f a l l elem en ts o f G w hich a r e th e i d e n t i t y on U.

Further reproduction prohibited without permission. , s " =s i n n / p " , 32 Case (1) None o f q , q ' , q" i s t h r e e . Then r o , P" i s as u b g ra p h (p e rh a p s n o t p ro p e r) m a t r i x f o r t h e form H(ro) = D et(ro ) Case (2) q = 3. s -a ~a s' -c •b -c s" Now th e Thus, s s ' s " - 2abc - b 2 s ' - c 2s - a 2 s " =» - - i s s ' s " < j is and hence D e t ( r o ) ^ 0 . 0 2 abc - - s ' c c " - —s ' c c " - -“s " c c , , g iv in g a c o n tr a d ic tio n . E x a c tly one o f q ^ ^ q " i s 3 .

A l l t h o s e s c a l a r m a tr ic e s in G. G -» V- by g g L e tZ C G be th e subgroup of Put G = G/g and d en o te th e n a t u r a l map for geG. D efine CG(g) = { X€G gX = g } (t'G(g) = { xeG gX = + g } and . Then (a) I f geG s a t i s f i e s C B(g ) (b ) Gg( g ) (8 ) x 2 m a t r i c e s , and geG s a t i s f i e s t r ( g ) = 0 2 = C'G(g) For b o th (a ) and (b) i t i s c l e a r t h a t th e r i g h t s i d e i s in c l u d e d i n t h e l e f t . \ eZ. / t r (g) = \ tr(g ) , Reproduced with permission of the copyright owner.

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Complex reflection groups by David William Koster


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