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The crystallography of the rotation subgroups of Coxeter groups Broderick, Norma 1975

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THE CRYSTALLOGRAPHY OF THE ROTATION SUBGROUPS OF COXETER GROUPS by NORMA BRODERICK B. S c , University of B r i t i s h Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the department of MATHEMATICS We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1975 I n p r e s e n t i n g t h i s t Jaes is i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an a d v a n c e d degree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f M a t h e m a t i c s The U n i v e r s i t y - o f B r i t i s h C o l u m b i a V a n c o u v e r 8 , Canada ABSTRACT. The theory of space groups has i t s o r i g i n s i n crystallography and s o l i d state physics. In t h i s t h e s i s , we study those space groups i n n-dimensional Euclidean space whose point groups are the r o t a t i o n subgroups of c r y s t a l l o g r a p h i c Coxeter groups. TABLE OF CONTENTS CHAPTER I. 1. Introduction 1 2 . Some tech n i c a l r e s u l t s 5 3 . The crystallography of Coxeter groups 8 CHAPTER I I . 1 . A presentation of K + 12 2 . The case dim V K = 2 14 3 . The normal!ser of K + 15 4 . L a t t i c e s i n v a r i a n t under K. 18 5 . The groups H 1(K*,V/A) 2 2 6 . A p p l i c a t i o n to c l a s s i c a l crystallography 3 4 TABLE I 3 5 BIBLIOGRAPHY 3 7 ( i v ) ACKNOWLEDGMENT I would l i k e t o thank P r o f e s s o r s B . M o y l s and T . A n d e r s o n f o r r e a d i n g t h i s t h e s i s and P r o f e s s o r G . M a x w e l l f o r s u g g e s t i n g t h e t o p i c and h e l p f u l a d v i c e d u r i n g i t s p r e p a r a t i o n . 1 CHAPTER I 1. Introduction. The theory of space groups has i t s o r i g i n s i n c r y s t a l -lography and s o l i d state physics. A f t e r introducing the fundamental concepts, we s h a l l give a b r i e f h i s t o r i c a l o u t l i n e of i t s development. Let E be an a f f l n e space with n-dimensional r e a l vector space of t r a n s l a t i o n s V and a f f i n e group A(E). Once an o r i g i n has been chosen i n E, the elements of A(E) can be expressed i n the form ( t , g ) , where t e V and geGL(V). The composition rule i n A(E) i s then given by (t,g)(r,h) = (t+gr,gh). Let (x,y) be a p o s i t i v e d e f i n i t e symmetric b i l i n e a r form on V, P i t s orthogonal group and P(E) the subgroup of A(E) c o n s i s t i n g of isometries. A subgroup S of the 'Euclidean group' P (E) i s c a l l e d a space group i f S n v i s a l a t t i c e A i n V. The p r o j e c t i o n of S on P i s c a l l e d the point group K of S. Elements of K leave A i n v a r i a n t , f o r i f (a,g) i s a representative i n S of g € K and t € - A , the element ( a , g ) ( t , 1 K a . g ) " 1 = ( g t , 1 ) € S and therefore gt € A. We can thus regard S as an extension of A by K. unique cocycle K-—»V//\ can be associated with S. Let {(s(g),g)) be a system of representatives i n S of a l l g€K. If {(s'(g) ,g)] i s another such system, then (s(g),g)(s ,(g),g)'" 1 = (s(g)-s'(g ) , 1 ) €S, so that the function'B:K—^V / A obtained by reducing the values of s mod A i s uniquely determined by S. Since (s(g),g)(s(h),h) = (s(g)+gs(h),gh), we have 2 1s(gh) = s(g)+gs(h), so that's i s a cocycle. Conversely, given a cocycle " s : K — V / A > the set of elements i n T ( E ) of the form (s(g)+t,g), where g€K, t e A and s:K—=>V i s some l i f t i n g of "s, form the space group with point group K and l a t t i c e A which induces the given oocycle by the previous construction. Two space groups S and S' are considered equivalent i f they are conjugate by an element of A(E). I f S' i s conjugate to S by a t r a n s l a t i o n ( r , D , the.n.every element i n S' i s of the form ( r , 1 ) ( t , g ) ( r , l ) " 1 = (t+r-gr,g), where (t,g)£S. Thus'S» has the same point group and l a t t i c e as S but i t s cocycle "s'rK—^V / A d i f f e r s from s by the coboundary g — ^ r - g j * . Conversely, cohomologous cocycles K —•»V/A correspond to space groups which are conjugate by a t r a n s l a t i o n . Secondly, i f S' i s conjugate to S by an element i n A(E) of the form (0,h), then S' consists of elements (0,h)(t,g)(0,h)" 1 = ( h t , h g h - 1 ) , where ( t , g ) 6 S . I t follows that i n t h i s case A ' = h A , K« = hKh" 1 and s'(g) = h s U ^ g h ) . Consequently, the equivalence classes of space groups can be determined i n the following way. C a l l a subgroup of P c r y s t a l i o g r a p h i c i f i t leaves i n v a r i a n t a l a t t i c e i n V and determine the conjugacy classes of such subgroups i n GL(V). For each c l a s s {K] determine the set L(K) of a l l l a t t i c e s l e f t i n v a r i a n t by K and f i n d the o r b i t s i n t h i s s e t under the natural a c t i o n of the normaliser N(K) of K i n GL(V). Calculate H ^ K j V / A ) f o r each of these o r b i t s £ A ] . Let g.(K,A) be the subgroup of N(K) c o n s i s t i n g of those elements which leave A i n v a r i a n t . Then the o r b i t s of H 1(K,V/ A ) under the 3 action of H(K,_A) given by s (g) = hs(h gh) correspond to the equivalence classes of space groups with point group K and l a t t i c e .A. The determination of the equivalence classes of space groups dates back to the l a s t century f o r n ^ 3. In three dimensions, a space group can be considered as the symmetry group of an i n f i n i t e c r y s t a l and the problem of f i n d i n g a l l inequivalent space groups was attacked by crystallographers t r y i n g to understand possible c r y s t a l structures. In I 8 3 O , Hessel found the 32 classes of point groups; every c r y s t a l could now be placed i n t o one of 32 types according to i t s external symmetry. However, Hessel's work remained unnoticed for over 60 years and these groups were independently derived by Bravais i n 1 8 4 8 , who also found two years l a t e r the 73 inequivalent p a i r s (.A,K). In 1879 Sohncke determined the 5 4 inequivalent space groups whose point groups consisted only of r o t a t i o n s and observed that Jordan had derived them 10 years previously but had not expressed h i s r e s u l t s i n geometric terms. F i n a l l y , the complete l i s t of 219 inequivalent space groups was derived independently by Fedorov i n 1 8 8 5 , Schoenflies i n 1891 and Barlow i n 1 8 9 4 . However, i t was not u n t i l 1912 that Laue was able to confirm these p r e d i c t i o n s of c r y s t a l structure through experiments with X-ray d i f f r a c t i o n . In two) dimensions, a space group can be considered as the symmetry group of a plane pattern repeated p e r i o d i c a l l y i n two d i r e c t i o n s , such as i s common, for example, i n wallpaper and f l o o r t i l i n g designs. However, i t was not u n t i l 1891 4 that Fedorov gave the f i r s t mathematical treatment of these groups and determined the 17 p o s s i b l e inequivalent space groups. Fricke and K l e i n rediscovered these groups i n 1897, as did Po'lya and N i g g l i i n 1924. Examples of a l l 17 symmetry groups can be found among the decorative patterns of the early Egyptians. Indeed, i n Weyl's words, the a r t of ornamentation contains i n i m p l i c i t form the oldest piece of higher mathe-matics known to us. The s o l u t i o n of the problem f o r n ^  3 l e d H i l b e r t to ask i n 1900 (as part of h i s 18th problem) whether there were only f i n i t e l y many inequivalent space groups i n any dimension. Fundamental r e s u l t s of Minkowski and Hermite implied that there were only f i n i t e l y many inequivalent p a i r s ( A , K ) . Joined to the f i n i t e n e s s of H U K .V/A), t h i s enabled Bieberbach and Frobenius to answer H i l b e r t ' s question i n the a f f i r m a t i v e . In 1951, Hurley [ l l ] published a l i s t of 222 point groups for n = 4 ; the l i s t was completed to 227 i h 1967. In 1965, Dade [9] found the maximal subgroups of GL^(Z) by considering automorphisms of p o s i t i v e d e f i n i t e i n t e g r a l quadratic forms. With the help of a computer, Bulow,fteubuser and Wondratschek [o"3 i n v e s t i g a t e d the subgroups of these groups and determined the 710 inequivalent p a i r s ( A ^ K ) . In 1972, Brown, Heubiiser and Zassenhaus found the 4733 inequivalent space groups. Ryskov [ l3] devised a method of constructing a f i n i t e family of p o s i t i v e d e f i n i t e quadratic forms i n n v a r i a b l e s such that a l l maximal f i n i t e subgroups of GL (.2) occurred among the 5 groups of t h e i r i n t e g r a l automorphisms, and a c t u a l l y found the maximal subgroups for n = 5« In the general n-dimensional case, Burkhardt [6] treated i n 1946 the s p e c i a l case of it being a symmetric group of degree n or i t s a l t e r n a t i n g subgroup, with A being a c e r t a i n type of l a t t i c e . Some progress was also made by Hermann fJO] ^ n 194-8* while Zassenhaus [15] developed an algorithm f o r the c a l c u l a t i o n of H^(K,V/ A ) . However, one cannot reasonably expect a general s o l u t i o n since every f i n i t e group w i l l eventually occur as the point group of some space group'. The c l a s s of point groups which are generated by r e f l e c t i o n s , 1. e. are c r y s t a l l o g r a p h i c Coxeter groups, has the advantage of being well understood and yet general enough to account (together with t h e i r subgroups) f o r a l l cases i f n ^  3. i t was studied by Maxwell [jZ] . In t h i s t h e s i s , we s h a l l i n v e s t i g a t e the case when the point group i s the r o t a t i o n subgroup of a cr y s t a l l o g r a p h i c Coxeter group. Together with Maxwell's r e s u l t s , we can then derive p r a c t i c a l l y a l l inequivalent space groups for a ^ 3 from theorems v a l i d i n a l l dimensions. In the remainder of t h i s chapter, we s h a l l summarise some t e c h n i c a l r e s u l t s as well as the s i t u a t i o n when K i s a cr y s t a l l o g r a p h i c Coxeter group. The proofs of these f a c t s can be found i n [l2] . 2. Some t e c h n i c a l r e s u l t s . 2.1 I f K i s a subgroup of P , l e t be the subspace of elements f i x e d by K and V R the orthogonal complement of i n V.. The group K i s c a l l e d e s s e n t i a l i f s 0; i n p a r t i c u l a r , the r e s t r i c t i o n 6 K of K to V K i s e s s e n t i a l . Subgroups K and K' of T are conjugate i n r i f f they are conjugate i n GL(V); i t i s therefore s u f f i c i e n t to determine the conjugacy classes of point groups within T . Furthermore, K and K' are conjugate i n P i f f K and K' are conjugate by an isometry Vg—"^V.^,. As V s and V*K are i n v a r i a n t under H(K), we see- that \ •> N(K)"- =- ~ G L ( V K ) x N ( K ) . One can also prove that N(K) = C(K)N r, (K), where C(K) i s the c e n t r a l i s e r of K" i n GL(V^) and H.^  ( I C ) i s the normaliser of K i n fjj. 2.2 I f A i s i n v a r i a n t under K, we define the weight group J\ to be the subgroup of V con s i s t i n g of a l l elements x such that x-gx€A f o r every g^-K. The exact sequence 0—*A—vV-^V/A.—>0 gives r i s e to the long exact sequence ... —>H°(K, V) -5» H°(K, V/A) —>H 1 (K, A ) —>H 1 (K, V) —>... It follows that i f K i s e s s e n t i a l , we have A * / A = H 1(K,_A), so that _A / A i s f i n i t e and _/\ i s a l a t t i c e . A sub s t a n t i a l part of H*(K,V/_/Sj can be obtained as follows. C a l l a cocycle 1:K—>V/A. .weightlike i f a l l i t s values l i e i n _A*/A. The exact sequence 0—->.A /A—~>V/A — - > V/A _r> 0 gives r i s e to the long exact sequence 0 —>A**/ A -> Hom(K, A*/A )-^>H1 (KiV/ A ) —> The image of «^  w i l l be c a l l e d the group of weightlike classes and denoted by J L ( K , A ) . K 2.3 Suppose A i s a l a t t i c e i n v a r i a n t under K. Let A = A n V*, A K = A H V K and A Q a A K © A K . I f x = y+z € A, where y G V^ and z tV^., then x-gx = z-gz £ AR% s o that z<?A*. The mapping x —>z induces a homomorphism A/A~—~?A*/AV which i s i n j e c t i v e since i f z , y = x-z £_AK and thus x <fA_ 7 I t follows that A / A i s f i n i t e and A i s a l a t t i c e . Let ©(A) be the image of A / A 0 i n -^/•Ar T h e m a P P i r i S x—>y also induces an i n f e c t i v e homomorphism J^/A Q -*> V ^ / y \ K whose image i s isomorphic to ©<A>. Conversely, given l a t t i c e s J\K C and -AR C V K and a subgroup ® of A g / A K which i s isomorphic to a subgroup ®^ of V^/A^» one can construct a l a t t i c e _A ^>A0 = A K @ A £ such that (9(A) = 0. Namely, l e t K : 0—* 0^  be some isomorphism and define where z Q ( r e sP»y^( Q)) a r e representatives of the elements of ©(resp. (H) ) i n V £ (resp.V^). One can show that i f A* i s another l a t t i c e i n v a r i a n t under K, then A* =gA f o r some g£N(K) i f f there e x i s t s h e N(K) which maps -A f i to A£ and induces an isomorphism @(A) —*> Q ( /V ) • Therefore, i n order to obtain a set of representatives of the o r b i t s of L(K) under the acti o n of N(K), i t i s s u f f i c i e n t to f i r s t choose a set of l a t t i c e s A^ representative of the o r b i t s of L(K) under K(K) and an a r b i t r a r y l a t t i c e A K C V^. For each A K > one then chooses a representative set of subgroups © of A* /Aj^ from those which are isomorphic to subgroups of F i n a l l y , one combines j\ , a n d & by (1). 2.4 To c a l c u l a t e H^K.V /A) , we s h a l l e s s e n t i a l l y use the idea of Zassenhaus Cl53» Choose a presentation F/R of K with generators fe^i^I 3 1 1 ( 1 r e l a t i o n s L ^ j l j ^ j * B v f i r s t reducing the elements of F modulo R, one obtains an acti o n of F on V. A cocycle t:F—=>V i s uniquely determined by the vector ( t ( e i ) ) , which may be chosen a r b i t r a r i l y . Since t(uR.u _ 1) » ut(R.j, the values of t on R are determined by the vector ( t ( R . ) ) . 0 8 The cocycle t : F — ^ V induces a cocycle "t:K —>V/A i f f t(R.) £ A J for a l l j € J , and every cocycle K — ? V / A can be obtained i n t h i s way. 2.5 I f the i n v e r s i o n - 1 ^ i s not an element of K, we s h a l l denote by +K the subgroup generated by K and - l y C l e a r l y L(K) = L(+K) and N(K) C N(+K); however, there may be l a t t i c e s inequivalent under N(K) but equivalent under N(+K). I f A €L(K), i t follows from the exact sequence ° ^ H ^ + 1 V , ( V / A ) K ) ^ H ^ K . V / A ) - ^ H H K . V / A ^ V that H ^ ( + K , Y / A ) i s isomorphic to the d i r e c t sum of the group of elements of order ^2 i n H*(K,V/A) and the quotient of A * / A by 2 ( A * / A ) . 3, The crystallography of Coxeter groups. Suppose K i s the Weyl group W(R) of a root system R i n V K. Following Bourbaki L 1 3, we denote by s ^ the r e f l e c t i o n corresponding to the root <*. and by Q(R) and P(R) the root and weight l a t t i c e s of R. Let B = be a b a s i s of R and S = [ s i ] the corresponding set of generators of K, where we extend the function from to V by l e t t i n g i t vanish on V*. Then {(s±s^) ^ = 1] i s a presentation of K, summarised by the Coxeter matrix ( m ^ ) . 3.1 We define X(R) to be the group of those g€GL(V^\ which s a t i s f y goi = a ^ hoi f o r a l l o(f B, where h i s an angle preserving permutation of B and the are p o s i t i v e r e a l numbers s a t i s -f y i n g n( p ,c« ) a Q L = n(h(3 ,hc< f o r a l l c B. Then N(K) i s equal to the semidirect product of *K and'A"'(R). I f R i s i r r e d u c i b l e , one can see that 1(R) = H.^(R), where H i s the group of p o s i t i v e homotheties and AQ(R) the group of graph automorphisms of B except f o r the cases of Bg'^S ^ » when AQ(R) posseses an a d d i t i o n a l element. 3.2 Every l a t t i c e -A^ £ L(K) i s such that Q(R') P(R«) and A K = P(R') f o r some root system R' s a t i s f y i n g W(R') = K. I f R and R* are two root systems of the same type for which W(R) = W(R«) = K, i t follows from 3.1 that there e x i s t s g€A(R) such that R' = gR. Therefore l a t t i c e s s a t i s f y i n g Q(.fi') ^ A £ C P(R') are equivalent under N(K) to l a t t i c e s A £ such that Q(R) C A £ ci P(R). Consequently, i n order to f i n d a representative set of l a t t i c e s , i t i s s u f f i c i e n t to consider only d i f f e r e n t types of root systems R f o r which W(R) = K and f o r each type R only those l a t t i c e s A s a t i s -K fylng Q(R) C AK C P(R) and -A* = P(R); the l a t t e r condition i s equivalent to r e q u i r i n g that U/2. f A for a l l <* C B. 3.3 By 2.4, a map t:S—->V w i l l induce a cocycle~E:K —*>V/A i f f <2) t U s ^ ) ± V = ( l + s ± s >...+( s ^ ) ^ ) ( t ( s i ) + 8 j L t ( g i ) ) € f o r a l l i , j . Let U s ^ = y ± + ^ k t k i < * k , with y ^ V 1 ^ and define p.^ = < t ( g i ) , . Then t i s a coboundary i f f t ( s . ) = t . . o i . mod A f o r a l l i . One can deduce from (2) X\ X X X. that 2.H 1(K,V/A) = 0. By subtracting from t the map b( s i ) = P-y^  ^j,/2, which induces a coboundary, we can assume that p. . € 2 • Equations (2jl then amount to the following: (3) 2 t ( s i ) € A f o r a l l i (4) p ^ o r f j = p j . ^ i mod A 0 (5) t ( s ) - t ( s . ) = p.. ex. _ p ^ m o d A i f n. . = n.. = 1 0 31 0 i j 1 xj jx 10 (We have abbreviated the Cartan integer n( tf., oi .) by n. ..) X J X j Since the projection of A on Vg. i s contained i n •AK = P(R), we can evaluate **. at both sides of eqns (3)-(5) to obtain u the following necessary conditions for t to induce a cocycle: (6) 2p±i £ H for a l l i , j ( ? ) P i j n j k " * j i n i k ^ ^ k " ^ i j » 0 ( 8 ) Pik-^jk = 5 j i : n 3 k ~ p i j n i k m o d 1 f o r a 1 1 k i f n i j = n j i = Call the last root c<m i n a conjpnent of B of type C exceptional: i t i s characterised by the fact that 0<m/2 i s a weight. The equations (6)-(8) imply that by r e p l a c i n g " with a cohomologous cocycle, we can assume that P^€ 2 except for the following cases (when i / j ) : (i ) cx! and ot. are exceptional and ( . - ot .)/2 4 A , when p i j = p 5 i m G d Z ' ( i i ) o( i s the last and oi. the second last root i n a component X J of type C , m ^ 2, when ^ A A ) = 2L, ( i i i ) ot^ and cx1.. belong to a component of type A^,B^ or B^ and ( ot^- 0^)./2 6 j \ , when the only exceptions are p ^ = p^j = p ^ mod (iv) c*!.. and o( belong to a component of type D. , when the exceptions can be p ^ = p 3 1 = p 2 3 mod ^ , P l Z f = P2fl = P 2 Z f mod £ or p ^ = p ^ = P 2^ + P 2^ fflod- Z • Correspondingly, the elements (cx^- o<5)/2, ( o^- «^ )/2 or (tf-j- ^)/2 must belong to A . 3.4 When p. . 6 Z for a l l i , j , the cocycle *t i s weightlike i n the sense of 2.2. The dimension over W/Z% of the subgroup «TL(K,.A) can be calculated as follows. Let & be the dimension over W/2. Z of the set of elements of order ^ 2 i n ( A P ( E ) ) / A , P the number of connected components i n the Dynkin diagram of B after 11 a l l double and t r i p l e bonds have been removed and T the number of components of type C m i n B. Then dim^-H-CK,A) = <£p - X . 3 . 5 I f TT i s a component i n B of type C^, with m odd and 3 , then <V m_.j£Q(R) so that condition ( i i ) of 3 . 3 i s s a t i s f i e d . Let t n:S-*>V be defined by t ( 6 f f i ) = <•> /% when co^ i s the l a s t root of n and t ( s ^ ) = 0 otherwise,; then t induces a cocycle "t :K—->V//V . Let "C be the subgroup of H 1(K,V /A) generated by the cohomology classes of such cocycles. Then ~t has zero i n t e r s e c t i o n with J 1(K , _ A ) and i t s dimension over 7?/2.H i s equal to the number of components i n B of type C^, with m odd and 3 » Frequently, JL(KtA) <£>C accounts f o r the en t i r e group H 1(K,V /A) . More generally, suppose £R q] i s the set of components i n B of type U-^ , A^,B^,B^ or and R' the remaining part of R. Suppose that A = - A K @ A ^ @ © q - A g ) , w h e r e ( r e s p . A ^ ) i s the i n t e r s e c t i o n of -A^ with the subspace spanned by the elements of R« (resp.R q;. Then n J U j V / A ) = J1(K , A ) @ £ G>B where the dimension of £) over 2 7 2 % i s equal to the number of components fiq of type A ^ » B ^ Q r B^ plus twice the number of components of type for which A^ equals P ( R q ) . For l a t t i c e s not covered by these remarks, the existence of a d d i t i o n a l cocycles can be inve s t i g a t e d i n each case by s t a r t i n g from the equations ( i ) - ( i i i ) of 3 . 3 . 12 CHAPTER I I Let K be a c r y s t a l l o g r a p h i c subgroup of P generated by r e f l e c t i o n s and K + i t s r o t a t i o n subgroup. In t h i s chapter, we s h a l l i n v e s t i g a t e the crystallography of K +. Suppose R i s a root system i n V R for which W(R) = it. Let B = be a basis of R, S = fs^] the set of corresponding generators of K and ML = C^-j) the Coxeter matrix of S. We s h a l l abbreviate the Cartan integer n< o(^, d ) by n ^ and extend the function of^ from V^ . to V by K l e t t i n g i t vanish on V . The case of dim = 1 i s t r i v i a l and w i l l be excluded from discussion. The somewhat a t y p i c a l case of dim = 2 w i l l be discussed i n section 2 ; from then on, we s h a l l assume that dim "7/ 3 » 1 . A presentation of K +. The group K + consists of those elements of K which can be written as a product of an even number of elements of S. Choose any cV ^  £ B and l e t g^ = s i s o f o r ^ °» t l l e n S^€^*» Since g^S^ = 6 i s j * t n e s i 6 g e r i e r a t e K + * They s a t i s f y the following r e l a t i o n s : CD g ^ 1 0 = 1 , iSi&Jy1* = i . 1 .1 Proposition. The r e l a t i o n s ( 1 ) form a presentation of K.+ . Proof. Let G + be the abstract group defined by a set of generators {g^J » subject to the r e l a t i o n s (l). Then fcCg^ = g*[ + 2 defines an automorphism of G such that £ = 1, since a word of the form gj~ i s mapped to i t s inverse by £ , while a word of the form (g.g't 1 ) m i j i s mapped to ^ Ug^g^' i^^g"^ We, can obtain an action of the two-element group £+lJ on G + cby - 1 13 a s s o c i a t i n g t h e automorphism £ t o t h e e lement - 1 . L e t G be t h e s e m i d i r e c t p r o d u c t o f G + and[+l] r e l a t i v e t o t h i s a c t i o n . The f u n c t i o n ^(g^) = ^ s 0 and ^C -1 ) = s Q i n d u c e s a homo-morphism G —>K s i n c e t h e c o n j u g a t e o f by -1 i n G i s mapped t o the same e lement a s fCg^) = g^> namely s Q s ^ . C o n v e r s e l y , t h e f u n c t i o n ^ ( s ^ ) = ( g i , - l ) , f o r 1 / 0 , and ' / ' (SQ ) = ( 1 , - 1 ) i n d u c e s a homomorphiism K — ^ G s i n c e an e l e m e n t o f t h e form 4 s i s 3 ) ' * o r ^ 0 i s m a P P e d t o ( ( g I » - D ( g ; j » - 1 ) ) = ( ( g ^ ] 1 ) ^ 1 ) , w h i l e ( S ^ S Q ) 1 0 , f o r i / 0 , i s mapped t o m i 0 m i 0 ( ( g ^ ' - l K l j - l ) ) = (g^ » 1 ) « A s ^ i s c l e a r l y t h e i n v e r s e o f j& , t h e l a t t e r must be an i s o m o r p h i s m . The r e s t r i c t i o n o f 0 t o G + i s t h e r e f o r e an i s o m o r p h i s m between G + and K + . / / The r e l a t i o n s {1) a r e w r i t t e n down by C o x e t e r and M o s e r [ 8 > P » 1 2 4 ] who do n o t , h o w e v e r , make any comment a s t o whether t h e y c o n s t i t u t e a p r e s e n t a i o n . The above method o f p r o o f i s s u g g e s t e d by an e x e r c i s e o f B o u r b a k i f I ,p .38 J. 1.2 P r o p o s i t i o n , I f K i s n o t o f t y p e B 2, t h e n K + / [ K + , K + ] (Z^Z)?" 1 © (273Z)7" , where p i s t h e number o f c o n n e c t e d components i n t h e D y n k i n  d i a g r a m o f B a f t e r a l l d o u b l e and t r i p l e bonds have been  d e l e t e d and = 0 , u n l e s s K i s o f t y p e A ^ G ^ A ^ o r D ^ , when 1. I f K i s o f t y p e B 2 > the g r o u p K + / f K + , K + J = Z ' A Z . P r o o f . We o b t a i n a p r e s e n t a t i o n f o r K + / ( K + , K + ] by a d d i n g the r e l a t i o n s g . g . = g . g . t o ( 1 ) . Choose o^-. t o be c o n n e c t e d t o a t most one o t h e r r o o t c B . I f °<j, / ^o* °^1 t h e n s i = 1 ' — 1 2 i f t h e r e e x i s t s c^^ d i s c o n n e c t e d f rom oi^t t h e n (g^g^ ) = 2 , so t h a t a l s o g ^ = 1. The r e l a t i o n s ( g 1 g ' ! 1 ) l t t i j = 1 t h e n amount I X J t o s a y i n g t h a t g ± = g_. i f m a 3 and a l s o t h a t a 1 i f « a 3j the conclusion follows. In the remaining cases, i f dim = 2, K i s of type ^ 2 , B 2 or and K + i s correspondingly c y c l i c of order 3,if or 6. I f K i s of type B^, then g^ = 1, g 2 = 1 and ( S ^ g ^ 1 ) ^ = U whence g^ = 1. I f K i s of type A^ or D^, then g^ = i , g | = ,l,and ( g ~ 1 g ± ) 3 = 1 for i > 1 , so that g ± = 1.// We s h a l l often need to use the formula (2) s ^ v x ) = x - <x, - i ^ * ^ - n d i<x, c*\ . I t follows for example that V* = v* since s.s.(x) = x for a l l c/. ^ cx\ i n B implies <x, ©(A = 0 for a l l B, so that X £ V*. 2. The case dim V K = 2. By 1.2.1, H(K +) = GL(V K)N(K +), where K + i s the r e s t r i c t i o n of K + to Y^; i t therefore s u f f i c e s to determine H(K+) i n case Y s: Yg.. 2.4 Proposition. I f K i s e s s e n t i a l . N(K +) i s the group of  s i m i l a r i t i e s of V i f R i s of type A 2,B 2 or G 2 and a l l of GL(Y) i f U i s of type A j X A ^ Proof. I f R i s of type A^XA^, the group. K* consists of|/+1^ so that H(K +) = GL(V). Otherwise, K + i s generated by a r o t a t i o n g such that g 2 / 1. By 1.2.1, we have N(K +) s C ( K + ) N p ( K + ) . The group H p ( K + ) contains the e n t i r e group of r o t a t i o n s P+ since the l a t t e r i s abelian. I f s i s a r e f l e c t i o n of Y> so i s gs and therefore ( g s ) 2 = 1 or sgs = g""1£ K +. Thus B p ( K + ) = P . In general, i f f = pu i s the polar decomposition of an a r b i t r a r y f £N(K) with p hermitian and u €. P, we have p £C(K +> and H ^ ( K + ) . Since the minimum polynomial of g i s of maximum possible degree, C(K +) i s spanned by 1 y and g. Suppose p = a1 y+bg for some a,b€lR ; then the adjoint p* = al^+bg"' = p, which i s 15 only possible i f b = 0 , singe ^ £ 1. Therefore f i s a s i m i l a r i t y of V.// When V, = V K, an elementary argument (e.g. [6,p.55^] ) shows that ( i ) i f R i s of type A^XA^, every l a t t i c e i s invariant under K + ( i i ) i f R i s of type A 2 or Q^t every hexagonal l a t t i c e i s i n v a r i a n t and ( i i i ) i f R i s of type every square l a t t i c e i s i n v a r i a n t . In each case, prop.2.1 shows that every l a t t i c e i s equivalent to the root l a t t i c e Q(R). From eqn . ( 2 ) , one sees # that Q(R) = P(R). When V K £ 0 , the determination of invariant l a t t i c e s proceeds according to the general p r i n c i p l e s of I. 2 . 3 * I f -A. i s a l a t t i c e invariant under K +, the group H 1(K +,V /A ) can be calculated by the general method available for c y c l i c groups. Let g be a generator of K +, of order m. Since 1+g+...+g ~ vanishes on V K, while the r e s t r i c t i o n of 1-g to V K i s i n v e r t i b l e , one sees that H 1(K +,V / A ) = { V e ^ M * I my = o] / ®^A) . 3 . The normaliser of K +. We s h a l l prove i n t h i s section that H(K +) = W(K) i f dim V K > 3 . Since N<K+) = GL(V K)H(K +) and H(K) = GL(V K)H(K), i t i s s u f f i c i e n t to prove the assertion i n case V = Vg. 3 . 1 Proposition. C(K +) = C(K). Proof. I f f € C(K +), then f s s (x) = s s f(x) for a l l i n B and a l l x£V. Using ( 2 ) , t h i s can be written as <x, «p Uoi.) + «x, Z^y- n ^ x , ^ d»f((X i) = <f(x), ^ > o ( . + (4(x),o( i> - ^ . ^ x ) , ^ ) ^ . I f we choose x to s a t i s f y <x, ot ^  = 0 and <x, = 1, we see 16 that f( o(^) belongs to the plane P^ .. spanned by crt^ and c**^ .. I f / ^ » t i l e same argument shows that ^ ^ i ^ C P ^ ; conse-quently f(«* ..) €B±2 H P i k a [RoCj,. Suppose f ( o^) = a ± cy ± for some a^e (R. Letting x = o/^ i n the above equation and comparing the coefficients of ot ., we see that n. .a. = n. .a.. It follows U X J j J X that a^ •= a., whenever and belong to the same component of B. Therefore f€C(K).// Since N(K +) = C(K +)Up(K +) and N(K) = C(K)Np<K) by 1.2.1, i t remains to prove that H p (K ) = Np(K), We need the following well-known result (e.g. [7,p.283): 3 .2 Proposition. Suppose w £ K and l e t V w = (vc V I w(v) = v j . Then w can be expressed as a product of reflections corresponding to  roots i n the orthogonal complement of ¥ w i n V..// For each pair of independent roots cK, R, l e t be the plane spanned by o(. and p and l e t P be the collection of a l l such planes i n V. 3 . 3 Proposition. An element g £ N p ( K + ) permutes the elements of P. Proof. Suppose ol,p € R, with c* / + p , and Consider the element w = gs^ Sp g~ 1£ K+. Since g € P , the subspace V w = g(H^ r> Hp ) i s of codimension 2 i n V and i t s orthogonal complement i n V i s the plane U = gCP^^ ). Prop.3.2 implies that w = 6 ^ Sj- ... for some roots If , J ,... i n U. There are at least two independent roots among the , since otherwise w, being i n K+, would equal 1. Therefore U = P^^ for some roots € U . / / 3 . 4 Proposition. Suppose and F ^ are distinct elements of P such that the subspace L = P^ + P^j- i s three-dimensional. Let R' be the root system R HL. Suppose that either (a) R' i s reducible or (b) there exists a rotation w i n P of order ^ 17 and a r o t a t i o n w2 i n P y j of order 3 or 4 such that ( W j V ^ ) = 1. Then P ^ O P ^ j i s spanned by a root p € R. Proof. I f R' i s reducible, i t i s easy to see that every i n t e r s e c t i o n P^p n Py^ i s spanned by a root. On the other hand, a well-known formula of s p h e r i c a l trigonometry asserts that i f r^ and r 2 are rota t i o n s of L through 0^ and © 2 , and i f t h e i r planes are i n c l i n e d at an angle ft , then the angle © of the r o t a t i o n i s given by cos 0/2 = -cos ©^2 cos © 2 /2 + cos f£ s i n Q^/2 s i n © 2 /2 . Applying t h i s formula to the rot a t i o n s w1 and w2, we conclude that the angle <£> between P^ and P ^ i s given by cos f6 = 1/3 or \/\f5* An e x p l i c i t look at root systems of type A ^ B ^ and now reveals that t h i s i s only possible i f ^  o Py^ i s spanned by a root.// 3.5 Proposition. H p ( K + ) = H p ( K ) . Proof. We have seen i n 3*3 that an element g e H p ( K + ) permutes the elements of P. I f B has no components of length ^3, we choose any three elements o^, <*2, oi^CB and consider the planes P^ p = g P 1 2 and PJ-J- = g P j y Since P 1 2 + P ^ i s three-dimensional and R' ne c e s s a r i l y reducible, prop.3.4 shows that F ^ ^ n i s spanned by a root p . Since ? 1 2 n p i s spanned by ^, we conclude that g maps ^ i n t o a multiple of p and therefore g s ^ " 1 = Sp € K. As g s ^ ^ ' U K * f o r a l l i , we have g s ^ " 1 ^ K, so that g belongs to N p ( K ) . I f B. has a component of length >3, then we can choose the elements c*^ ( i = 1,2,3) i n such a jray that m 1 2 = 3 and m^ = 3 or if. I f we l e t w] = g s ^ g - 1 and W2 = ] S S i s 3 S " ' a n d a P P l y propO'Jf, the argument proceeds as before.// 18 4. L a t t i c e s i n v a r i a n t under JS. . We s h a l l assume that V = V R j the general case can be treated by the general method o i I.2.3* I t turns out that a l a t t i c e i n v a r i a n t under K + i s also i n v a r i a n t under K, apart from the following exceptional cases; (a) Suppose R i s of type Aj X... XA^. Let A 0 be the s u b l a t t i c e of P(R) containing Q(R) and a l l elements of the form ^j ) /2 f o r c<^ y o<^  i n B. When dim V i s even, define Al = A e U < A e + ( %1 A* • A U ( A e + ( ZkcXj.JA • V2) • (b) Suppose R i s the d i r e c t sum of two root systems R^  and R., of types A 1 and A 2 r e s p e c t i v e l y . Let A Q = QC.Rj ) $ P( Rg) and define A l = A e U <^e + .< V °y/3> U ( A " < °S + °y^> - A e 2 = A e u ( A e + - <x2)/3) tl ( A e - (<x r cx2)/3) , where {c< ^  and ^O1^, °*3$ 3 1 , 6 bases of R1 and Rg. (Crystallograph c a l l these l a t t i c e s •rhombohedric'.) (c) Suppose R i s the d i r e c t sum of two root systems R^  and 82* both of type A_. Let A = P(R) and define 2 e _ A e l s A e u ( A e • ( o y o y / 3 ) U ( A g . ( o t i + ^ 3 > / 3 ) A e 2 = A s ^ ( A Q + ( c < r ^ ) / 3 ) u ( A e - t c y r (X 3 ) /3), where {c^j» o i 2l a J l d ^*3* °Sf^  a r e bases of and R 2« In each of the above cases, the l a t t i c e s A., and A . " el e2 w i l l be c a l l e d o l exceptional type w.r.t, R. One can v e r i f y that + they are i n v a r i a n t under K , but interchanged by an element of H not i n K ( and thus equivalent under fl(K ) ) . 19 Suppose / i i s a l a t t i c e i n V. i n v a r i a n t u n d e r K + . F o r e a c h oi±€ B , l e t -A = A O and d e f i n e = ® ±S^. R e c a l l t h a t A* = ( v C V | v - gv<?A f o r a l l g ^ K + j i s a l s o a l a t t i c e i n V w h i c h c o n t a i n s A . 4.1 P r o p o s j L t i o n . Suppose o ^ , and c < k a r e d i s t i n c t e l e m e n t s o f B s u c h t h a t a... = 0. Then i f x € J\ , we have — — — — — J K (a ) i f n ^ = n ^ = 0, t h e n 2 <x, ^7 <X± ^ -A. (b) i f n 5 J L = -1, t h e n ^ - a ^ n ^ T a ^ n ^ ) < x , e S ^ o ^ A . A * P r o o f . S i n c e x € A , t h e e l e m e n t s x - s ^ C x ) = <x, <&k> * k + « x , oJ ±> - n ^ x , ^ j ^ ) ^ s^SfcCx) - x = -<x, &..> - <x, <*k>0<k a l l b e l o n g t o A . C o n s e q u e n t l y t h e i r sum (2 <x, c ^ > - n J ± < x , ^ > - n ^ <^x, £ k » « ± a l s o b e l o n g s t o A . I f a.. = a. . = 0, we o b t a i n { a ) . O t h e r w i s e , we J x K l a p p l y t h i s c o n c l u s i o n t o t h e e lement y = ( x - s i s j ( x ) ) - ( x - s ; j s : L ( x ) > = - n ^ ^ x , ^ • ^ ) ° L ± + n i j ° ^ i n s t e a d o f x and deduce ( b ) . / / Note t h a t t h e i n t e g e r 4-B.j j n j j , " * n j k n k j i n P a r t c a n o f l l y assume t h e v a l u e s 1,2 o r 3. I f A- i s o f e x c e p t i o n a l t y p e w . r . t . R, one c a n v e r i f y t h a t A # = Q ( R ) . it * 4.2 P r o p o s i t i o n . Suppose A = Q(R) and x <£ A i s s u c h t h a t </x, 4- Z f o r some o(^$B. ( a ) I f 2 <<x, ot^ £ Z , t h e n R i s o f ty,pe A^x ... and e i t h e r _A = -A o r d i m V i s e v e n and A e q u a l s A , o r A ~ . F u r t h e r m o r e . e ~ ~ : e 1 — — e2 * x = ( X ± o < i ) / 4 mod P ( R ) . 20 <b) I£ 3 <x, € Z , then e i t h e r ( i ) R i s of type y Ag, -A = -A e 1 or A e 2 and +x = ( « t + c<2)/3 mod P(R) or ( i l ) R i s of type AgXAg, A = AQ^ or A e 2 and +x = ( 0 ^ + ^) /3 mod P(R). Proof. For any ^ i n B, consider (5) x-s is^(x) s <x, <^j> °^ + (^x, <*±> - n^^x, e^)) ^ 6 A . I f H<x, 2T for some o<,.€B and some integer N, m u l t i p l y i n g (3) by N shows, i n view of A # = Q(R)» that N.<k, o f ^ C Z f o r a l l c*^<B. Thus the hypothesis of the proposition implies that <x, ot. /> 4- 2 for a l l <x\ CB. I t now follows from (3) that V (#) <x, ei ±> - <x, ^ > I Z, since otherwise <x, ©ci.j> would have to be i n Z . v _ Suppose we are i n case (a); then 2<x, c t . ^ Z f o r a l l i-> o ^ C B . Eqn.(J+) implies that n ^ i s always even, which can only happen i f R i s of type A^X ... XA^; furthermore, (3) shows that AeC A. I f y £(A HP(R))\Ae, then y = C XQ / 2 mod A g, contrary to A # = Q(R), so that AnP(R) = A . Since <x, = i mod Z for a l l oi. € B, we have x = ( Z . ) A m o d P(R). I f dim V i s odd, x cannot belong to A- since then 2x = s<g/2 mod A would also belong to A. As R i s of type A. * ... *A.,, prop.if. l(a) shows thatT^2<y, € Z . I f dim V i s odd, i t follows that A. CP(R) so that A = _A. I f dim V i s even and A <f- P(R), then A. must be one of the l a t t i c e s A e 1 and - A e 2 since ( ^  o ^ ) A and ( 5 ^ . 0 ^ ) A + c*Q/2 cannot both be i n A . In case (b), we have 3 <x, C Z for a l l c / ^ B . Eqn.(if) implies that </x, tL^y = <x, t^^"^ mod Z whenever n ^ = - 1 . Further-more, 4.1 shows that ( i ) B does not contain three disconnected roots and ( i i ) B does not contain roots connected by double or t r i p l e bonds or a sequence of three roots connected by s i n g l e 21 bonds, since otherwise <x, o^/ or 2 <x, would belong to "JL for some 6 B. This means that R i s ei t h e r of type A, xA, 1 1 2 or A-*A-. In both of these cases, (3) implies that A--^ -A . c. c. e Suppose R i s of type A^ x, k^. Then x = x^ ^  + x 2 ( < 2^+ ^ ) mod P(R), where x p = +1/3 for p = 1,2 and ^ i s the weight corresponding to © C ^ . Since 6 ^ / 3 = = - ^ / 3 mod P(Rj) and (A^ 2+ ^ ) / 3 = ( <*2+ <*3)/3 = - mod PCR^we have +x = ( cx^i oi 2 ) / 3 mod P(R). Eqn.(3) applied with i = 1 and j = 2 shows that ( 0 ^ + C ^ 2 ) / 3 £ A with the same choice of sign as f o r the value of +x mod P(R). Both of these elements cannot belong to A since t h e i r sum 2 0^/3 ^A. I t follows that e i t h e r -A e |CA or A e 2C_A. I f y £ ( A n P ( R ) ) \ A then y = c^ / 2 mod A contrary to A # = Q(R). Therefore A n P(R) = A . I f ycA\p(R)> choosing i = 2, j" = 3 and k = 1 i n prop.if. 1(b) shows that 3 <y, <* N £ 2T. I t follows that A , = A or A , = -A.// 3 ' e 1 e2 4.3 C o r o l l a r y . I f _/\ i s a l a t t i c e of exceptional type, then j\ can be described as follows: i n case (a), A* = A L^(A + 0L-J2); . ep ep ep 0 • i n case (b)(7) A*p = A e p U (A0p+ 0^/2) and i n case (kHz*/) A*p = A>P> p = 1»2-// 4.4 Proposition. Suppose A i s a l a t t i c e i n v a r i a n t under K +. Then there e x i s t s a root system R1 such that W(R') = K and A i s eith e r of exceptional type w.r.t. R» or else QlRMC A C P ( R ' ) and A = Q( R ,)» In the l a t t e r case. A i s i n v a r i a n t under K and A* = P(R'), unless R« i s of type A. x ... * A. and A= A , when A* = P(R') U (P(R« )+(£" <* ) A ) . Proof. We f i r s t show that A ± t 0 f o r a l l o< € B., so that A , i s a l a t t i c e . I f dim V > 4, there w i l l e x i s t <x\,«r €B 2 2 d i f f e r e n t f rom o< • f o r w h i c h a., = 0. S i n c e A i s a l a t t i c e , we c a n f i n d an element x ^ A f o r w h i c h n e i t h e r o f ^x, and <^ x, <A .y i s z e r o . Prop.if. 1 t h e n shows t h a t j\. ji 0» When dim V = 3, t h i s argument works f o r a t l e a s t one o<_£B. I f ao<^ i s a n o n z e r o element o f A ^ and o*.. CB i s c o n n e c t e d t o , t h e n (1 + s .s. ) ( a o*. ) = n. .a ^ . i s a non z e r o element o f A . - . The o n l y r e m a i n i n g c a s e i s t h a t o f a r o o t o t ^ ^ B w h i c h i s n o t co n n e c t e d t o e i t h e r o f t h e o t h e r two r o o t s , &, € B, b u t f o r x 3 w h i c h we may assume t h a t n ^ = - 1 . L e t > f D e s u c h t h a t <x, <*k/ / 0. S i n c e z = '£if-n. .n.-) <(x, oOc* £A by p r o p . l f . 1 , i t f o l l o w s t h a t ( l + s ^ s^ X z ) = n i j ( ^ " " n j ; - j n - j ^ <*-j> However, s i n c e x - s ^ C x ) = < x , 1 ^ ; j , > ^ + <x, ^ k > ° ^ k ^ A , we deduce t h a t n ^ U - n ^ n ^ ^ x , e ? k > oi fc a l s o b e l o n g s t o A • T h e r e f o r e A ^ . ^ O. Suppose A ^ = Z m ^ o ^ f o r some > 0. S i n c e O-s . - X m.^o^) = (1 + s ^ s i ) ( m i c < i ) = '-o^  jBijL ^  ^ A ^ , A * i s i n v a r i a n t under K and t h e r e f o r e e q u a l t o Q(R') f o r some r o o t s y s t e m R' s u c h t h a t W(R») = K by 1 . 3 . 2 . A p p l y i n g prop.if.2, we see t h a t e i t h e r A CJ\ = P(R') o r A i s o f e x c e p t i o n a l t y p e w . r . t . R 1 o r R' i s o f t y p e A. X ... X A , and A = A „ . I n t h e l a s t c a s e , t h e v a l u e o f j\* l i e ' e a l s o f o l l o w s from p r o p . 4 . 2 . / / 5. The g r o u p s H 1 (K + ,V /_A) . Suppose A i s a l a t t i c e i n V = © Vg i n v a r i a n t under K + . Then A . c o n t a i n s t h e s u b l a t t i c e A ^ =. A K ® A ^ and t h e p r o j e c t i o n o f A on V £ is c o n t a i n e d i n A * . We may assume t h a t A K i s a s d e s c r i b e d i n p r o p . i f . b u t w i t h R 1 = R. I t w i l l be c o n v e n i e n t t o i n t r o d u c e t h e number (5) eij = mij/^-aijV 23 for tt., o(.£B. I t equals §,1,2 or 6 according to whether i s disconnected from C< or connected to i t by a s i n g l e , double or t r i p l e bond. Choose CX Q to be connected to at most one other root Oi^ and l e t G = {g^ = S ^ S Q j i / oJ be the corresponding set of generators of K + . Suppose t i s a function G — ^ V . I f t(g^) = y i + ^ k i ^ k ' w i t h y i € v K » d e f i n e u i j = < t < g i ) , ^ j > = 5.1 P r o p o s i t i o n . The function t induces a coboundary t:K +—>V/LA. i f f there e x i s t s a constant c e IR such that t C g ^ = c otQ + t ^ C X ^ mod_A for a l l i /0. Proof. I f a = y + ^ k ^ k ^ k ^ ' w i t n V t n e co° o uhdary corresponding to a mod A maps g^ to i - g ^ t = < Z " k V k 0 ) * Q + ( 1 ^ - B Q iZ^^o) <x± mod -A. Conversely, suppose t(S^) = c (x^ + t j ^ 0 ^ mod A. The system of equations ^ k W o = c Z k W i " n o i z k \ \ o " * ! ! can be written i n the form aH. = t, where a = (a^) , K i s the Cartan matrix of R and t i s the vector with 0-th coordinate equal to c and i - t h coordinate, f o r i / 0, equal to t - y / ^ o i 0 . Since H i s i n v e r t i b l e , the system has a unique s o l u t i o n a and "t coincides with the coboundary defined by a = ^ " ^ a ^ ^ ^ * / / I f e x i s t s , l e t c and d be the solutions of the system of equations n Q 1 c + 2d = u n 2c • n 1 0 d = u 1 Q ; 24 o t h e r w i s e , l e t c = 0. By s u b t r a c t i n g f rom t t h e f u n c t i o n gf —> + g±__•> c c ^ 0 + ^ i ^ ^ C M D we may assume t h a t u ^ = 0 f o r a l l i and a l s o t h a t u ^ Q = 0. 5.2 P r o p o s i t i o n . The f u n c t i o n t i n d u c e s a c o c y c l e ~t:K*—5> V / A i f f the f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d f o r a l l i , j " 4 0: (6) 2 t ( g ± ) = u i Q < * 0 mod A i f i / 1 (7) * l 0 t ( 6 i > 6 -A-(8) ^ ( t C g ^ - t C g ^ = • i J f - < 2 » a i * a 3 1 « 1 j ^ i + ( 2 t tir n ij uji^j] m o d  P r o o f . I f h£ K + i s o f o r d e r m, we s h a l l d e n o t e 1+h+.. ,+hm" 1 by 31(h). Our f i r s t a im i s t o e s t a b l i s h t h e f o r m u l a (9) N ( s i s . ) t ( g i ) = m ^ t C ^ ) • e ^ U ^ o ^ - 2 u . . c V j ) f o r i 4 0. I n d e e d , s i n c e s ^ ^ S ^ ) = t ;^Si^~uij ^j+njiuij ° * i t y (2), t h e l e f t s i d e o f (9) e q u a l s m ^ i C t C g ^ - v ) , f o r some v i n Ej^t S i n c e t h e l e f t s i d e o f (9) i s a l s o i n v a r i a n t u n d e r s ^ s ^ » t h e v e c t o r v must i n f a c t be t h e o r t h o g o n a l p r o j e c t i o n o f t ( g i ) on t h i s p l a n e . C o n s e q u e n t l y , i f v = v . + v . <X . , we have the e q u a t i o n s 0 = u ± i = < t ( g i ) , \ > = < v , ^ ± > = 2v ± + n ^ uij = < t C g i ) , ^ i > = < v , ^ > = n ^ + 2v. , whose s o l u t i o n s a r e v i = - ^ / ^ J V V J = 2vu-*ijnji} ; t h i s e s t a b l i s h e s 19). A c c o r d i n g t o the p r e s e n t a t i o n CO o f K + , t h e f u n c t i o n t w i l l i n d u c e a c o c y c l e " t : K * — * V /A i f f t i S j L ) = N ( g i ) t ' ( g i > = N ( s i s 0 ) t ( g i ) ^A. and tCCgj.g]1) i 3 ) = H C g ^ K t C ^ ) - s±s1}Ug.))e A . Since g.g" 1 = s. s. and H(s.s.) = B{s.s.), the l a t t e r r e l a t i o n x. 3 x J X J J X can also be expressed as M (s j.s^)t(g i) = HCs^s^tCg^) mod A . Using 19), one sees that a r e l a t i o n of the f i r s t type amounts to (6) or (7), while that of the second type amounts to (8).// We s h a l l postpone u n t i l the end of t h i s s e c t i o n the discussion of the cases when A„ i s of exceptional type or when R i s of XV type A * . . . * A, and Au = A . This enables us to assume i l xv e that A * = P(R), so that oL±iA) C. TL for a l l oC± < B. 5.3 Proposi t i o n . For a l l i , j / 1 , we have (aj 2 u i 5 £ P. (b) 2u. 1 u a 0 1 u i 0 mod / ; m 1 0 u 1 1 6 1'. (c) 2 i * i 0 = 2u.-0 mod Z i f m±^ = 2 or if; 5u±Q = 3^0 mod 21 i f = 3 or 6. Proof. Evaluating *L . at (6), we obtain i a ) ; to obtain ( b ) , J evaluate and <x.^  at (6) and (7) r e s p e c t i v e l y . v _ I f = 2, evaluating <X-Q a t (8) shows that 2 ( u i o " u j O ^ * S i m i l a r l y , i f = 3, evaluating UQ at 18) shows that 3(U;J.0-U.J0)£ ^  . I f m.^ = if or 6, then i s even and therefore the r i g h t side of (8) i s i n A by (a) . Using (6), we conclude that. -Jm^^  ( H^Q-U j Q) oCQ £ A, which can only happen i f i m ^ t u ^ - u . ^ ) ^  J T . / / I f R i s of type B, XA, x ... * A , , we define A to be the 3 1 1 e su b l a t t i c e of P(R) which contains Q(R) and a l l elements ( o * Q + o<2+ ^^)/Zy where oiQ and o<2 are the f i r s t and t h i r d roots i n the component of type B^ and i s i n a component of type A j . 26> ( I f there are no factors of type A., l e t Aa = Q(R).) I e 5.§ Proposition. We have ^ - ^ i ^ i o ^ O € % for a l l i ^  1, unless (a) n l Q = -2 and n ^ = 0, when * ± 0 g Z . (b) OLQ belongs to a component of type A^.D^.B, or B^(in the l a s t  two cases., c* 0 i s the f i r s t and the t h i r d root) and (<*0+ *L%)/2 6 A , when 6u i Q£ Z. (c) R i e of type X-Aj * ... * A 1 and - A £ x A e , when 6u 2 Q6 Z . Proof. Eqn.(8) with j = 1 gives OO) m i l ( t ( g i ) - t ( g 1 ) ) = e . 1 [ - ( 2 u l i + n l i u i l ) ^ i + ( 2 u 1 1 + n : L 1 u l i W 1 ) mod A . . Evaluating ^ Q at t h i s equation, we deduce that (11) m i 1 u i 0 = e i l n 1 0 ( 2 u 1 1 + n i 1 u 1 i ) mod Z . Suppose = 0. I f » 1 Q = -2, (11) shows that 2u i Q = -211^  mod ff; since n Q 1 = -1, 5.3(b) implies that u i Q £ Z. Otherwise, multiplying <1l) by 2 and using 5.3(b), we conclude that ( ^ " n o i a i 0 ^ u i O £ Z . I f n ^ 0 and n 1 Q = -2, we have = 3, so that (11) becomes the equation 3u^0 = ~ * * u j j + 2 u i i m o d & • M u l t i p l y i n g t h i s by 2 and using 5.3(b), we again conclude that (if-nQ 1n^ 0)u^ 0£ Z» Suppose n ^ / 0 and n 1 Q = -1. M u l t i p l y i n g (11) by m 1 Q and using 5.3(b), the conclusion can be expressed as 2(4-n 0^n^ 0)u i Q € Z . I f n Q 1 = -2, we see from 5.3(b) that both kVL^^ and Z f U ^ are i n Z . M u l t i p l y i n g (10) by Zf and using (7), we deduce that 12t(g i ) 6 J A ; i t therefore follows from (6) that 6uiQ€ Z and hence 2u j L 0 = ( H 0 1 n 1 0 h i 0 ^ . I f n Q 1 = -1 and m^ s 3, 5.3(b) implies that 12uil and 3^^ are i n Z • M u l t i p l y i n g <10) by 6 and using (7) and 5.3(b), we deduce that l8t ( g i ) = -3^0 ^ m o d i n view of (6) and the fact that 6u i Q€ 2?, t h i s means that 3 t t i 0 ( . V" o t i ) T h u s e i t h e r 3 u i 0 = ( Z f ~ n O l n i O ) u i O € ^ o r 27 ( c<0+ ot~)/2. € A . However, one can easily see that ( ^)/2 can be a weight only i n the cases specified i n (b). Finally, suppose n(^ 1 = -1 and = 4. I f n^ = -1, multiplying (10) by 3 and using 5.3(e)* (6) iUnd (7), we see that 3u i 0«' i^A, so that 3 u i Q = ^ ~ a o i n i 0 ^ u i O € ^ * T n e r e m a i n i n S case i s when c<0 and o<^  are the f i r s t and third roots i n a component of type B^. If R i t s e l f i s of type B^ and (ot Q+ ^±)/Z A. R must be the l a t t i c e Q(R) = A . Otherwise, suppose . i s not i n this e j component; eqns.(6) and (8) show that (u. ~-u ._) „ = - u c ^ - . + u. , . mod -A- . x 10 oO 0 J i i w i j Q * Multiplying this equation by 3 and using 5.3(a) and the fact that 3UJQ£ Z„ we deduce that 3 u i o ( > ' o = u j i °*i + u i j m o d Evaluating oL^ at this equation shows that u ^ = 3uJ_Q» I i 3 u i 0 ^ 1 and (o* Q+ o*±)/2 ^  A, then we must have (c* Q+ ^ i+<KJ)/2^ 'A However, such an element can only be a weight for every j only i f (joi .] i s of type A.; this produces the exception noted i n (c), since A i s a maximal sublattice of P(R) w.r.t. the property (A e) # = Q(R).// 5.5 Proposition. 6.H1(K+,Y/A) = 0, unless R i s of type A, X ... * A,, B_ *A, X ... X A, or B, XA..X ... XA, and, i n the 1 1 d 1 1 "~~ 3 • I . last case = A g . In the last two cases. 6H (K ,V/A) i s of  order ^ 2 . Proof. Suppose B contains a component of type 6^. I f we choose vlQ to be the f i r s t root of such a component, 5.4 says that U i 0 6 ^ f o r 3 1 1 1 ^ U so that 2t(g ±)£A by (6). In view of (7), we have 6t » 0. Secondly, suppose B contains a component of type d O other than A and Bg and choose o£ Q to be the f i r s t root of such a component; then mQ^  = 3» We s © e from 5 . 4 that 3u^ Q £ Z for i / 1,, and thus by ( 6 ) , 6 t ( g ^ £ A, apart from c e r t a i n exceptional cases,, when 6t(g±) = c* 0 /2 mod A . I f ( oiQ+ *±)/2. «A, then S U g ^ ) = <*.j/2 mod A; since 3 t(g t)*A by ( 6 ) , prop . 5 . 1 shows that 6t i s a coboundary. The only remaining exception occurs when R i s of type B^* A^ x ... X A^ and -A^  = A e ? F i n a l l y , suppose B contains a component of type Bg. Choosing Q to be the second root of such a component, so that n^ Q = - 2 , we see from 5.1+ that; Q 6 7Z for a l l i / 1, and thus 2t(g i)<£ A. Eqn . ( 8 ) with j = 1 then says that (12) 2 t ( g i ) = u l t^ t - u i lcyL1 mod A. Since J+t^g^eA by (7) and 2 u i l«Z by 5.3(b), we must also have au.. 6 I f there e x i s t i , j ^ 1 such that n. . = - 1 , apply-in g d&tj to (12) shows that \x^€"E then 2 t(gj) = - u i l c X 1 and 2t i s a coboundary. This f a i l s only i f R i s of type B ^ X A ^ ... ^ A ^ when we may assume that 2t-'(g^) = oC^/2 mod A f o r any i / 1. The only Remaining case i s when R i s of type A^y. ...X-A^// In each of the excluded cases, one can f i n d examples of l a t t i c e s A for which 6H 1 (K+,V/_A-) / 0 . For instance, suppose R i s of type Aj X-A^  with 5 f a c t o r s and that .A contains (0*2+ * y / 2 , ( ^ + ^ 5 ) / 2 and ( c*0+ <Xg+ ^ ) / 2 . Define t(gg) = t(g 3> = (<*0+ * + ^ 5 ) A and t(g^) = t ( g 5 ) = (oCg+ * ) A ; then t i s a cocycle but 6t i s not a coboundary. 5 . 6 Proposition, i+.tt' (K+,Y/A ) = 0 , unless R i s of type A, x A , A,XG 2, A 3 , A , x A 3 , A 2 X A 2 , A 2 x G 2 , A^, D^, A ^ x ^ , A,XD^ , A_. D^ or A. x D,.. 29 Proof. Suppose R contains a component of type Bffl or C m f o r some m ^ 2 . Let ©< Q be the l a s t root i n such a component. By 5«4* we have Zn±Q6 ^ f o r a l l i / 1, so that 2ft ( g ^ ) 4 A . From (7), we also have A , and thus 4t = 0. Secondly, suppose R contains a component TT of type I f c^0 i s the f i r s t root i n TI , 5.4 shows that u i Q € 2? f o r a l l i / 1, so that 2t(g^)€-.A . Eqns.(6) and (8) together imply that for a l l k / 1, (13) 2t( g ]) . -u^ + u l k * k mod A, Suppose there e x i s t two disconnected roots ., ^ . ^BNTT. L e t t i n g v k = j i n 03) and applying ck^, we conclude that 2M^€ "Z • Since 2u i l £ "E by 5.3(b), we see from (13) with k = i that 4~t = 0. I f B \ H" = i s of type G,, t h i s method of argument shows that 2u^ = - 3 u ^ mod H and 2u^ = - u ^ mod 2? \ together, these equations again imply that 2u^ £ %. The only other p o s s i b i l i t i e s are A ^ G 2 and k^yC Otherwise, i f cx^  e x i s t s , we have = 3. Exclude f o r the moment the cases F^ and Dg i n addit i o n to the l i s t e d exceptions. Let B« =(^±6-B 1 i i 0,l} ,and choose some o/^£B' which i s connected to at most one other o^^B'. I f b ^ ^ B ' i s d i f f e r e n t from «*• 5.3(c) says that 2u i Q = 2u k Q mod . On the other hand, our assumption implies that there e x i s t s ^ t B 1 disconnec-ted' from <y-.., so that 2u^ = Za^ = Zn±Q mod Z% Thus the numbers 2u i Q have, mod 2? , a common value c for i / 1. I f bl^ i s disconnected from f e q n s > ( 6 ) and (8) show that (14) 2t( g l) = u w 0 « 0 - u ^ o ^ + u ^ c ^ mod «A. 30 Again, our assumption implies the existence of two elements o^, i n B 1 disconnected from ^ and ffom each other. Applying ©C i to (14) with m = j shows that 2.u^€ u . Then, on m u l t i p l y i n g (14) )>y 2, with m = i , we deduce that 4 t(g^) = e c > t o ~ 2 u i 1 °S mo<*A ? Since 4t(g ; i L) = c ^ Q mod A f o r i / 1, 5i\ sKows^that 4*t i s a coboundary. The cases and Dg w i l l be handled during the next argument.// Since there i s no overlap i n the exceptional cases of the preceding two propositions, we conclude that either 4 or 6 always a n n i h i l a t e s H 1 ( K + , V / A ) . The group 4 H 1 ( K + , V / A ) i s therefore an n i h i l a t e d by 3 i n exceptional cases of prop.5.6.The exact sequence 0 - ^ V A - ^ V / A - ^ V / A - > 0 , where A * = V K © F ( R ) , gives r i s e to the exact sequence Hom(K+, A * / A ) ^ > H 1 (K +,V / A >->H1 (K +,V K/P(R)) . 1.2 shows that unless R i s of type A^ or D^, the f i r s t group i n t h i s sequence has no 3-torsion. Thus the 3-torsion component of H 1(K*,V/ A ) maps i n j e c t i v e l y i n t o that of H 1(K +,V K/P(R)). When R i s of type *^ or Dg, - 1 y <£ K + and 1 + ^ therefore H (K ,V K/P(R)) i s a n n i h i l a t e d by 2. Otherwise, one fi n d s by e x p l i c i t computation that the 3-torsion component of E 1(K +,V K/P(R)) i s isomorphic to Z / 3 Z i n a l l cases except that of AgXAg, when i t i s isomorphic to (2V3jP)^» When R i s of type &^  or D^, the 3-torsion component of H 1(K +,V K/P(R)) vanishes, but the f i r s t group i n the sequence i s isomorphic to the group of elements i n (V s Q P ( R ) ) / A an n i h i l a t e d by 3. We thus have i n a l l cases a bound for the dimension of 4H 1(K +,V/A) over 2?/3 Z. 31 As K + i s normal i n K and A i s thy assumption) i n v a r i a n t under K., we have the exact sequence 0 - > H 1 C K A + , ( V M ) K + ) i ^ H L ( K , V / A ) £ 2 ^ H 1 ( K + , Y / A > Since K A + acts t r i v i a l l y on ( V / A ) K = ( V ? © P ( R ) ) / A * the f i r s t group* can be i d e n t i f i e d with the group of elements i n ( V ^ + P ( R ) ) / A a n n i h i l a t e d by 2. On the other hand, since 2 H 1 ( K , V / A ) = 0 by 1.3.3, the image of res i s contained i n the subgroup H 1 ( K + , V / - A ) 2 of elements a n n i h i l a t e d by 2. 5.7 Proposition. The image of res i s of index 1 or 2 i n H 1 ( K + , V / _ A ) 2 . In the l a t t e r case, the components of R can only  be of type A 1 or Ay Furthermore, i f R Q i s a component of  type Ay then Q ( R Q ) C A n P C R Q ) C P d y . Proof. Suppose ~t i s a cocycle J t + — > V / A such that 2t i s a coboundary, i . e . -(15) 2tig ±) = coC Q + mod . A , for some c,d^ € /R. We s h a l l f i r s t show that one can, assume c = 0. I f R i s of type Aj * ... XA^, we can subtract from t the coboundary inducing function g^—^i-coC^, which does not a f f e c t the assumption that % . Otherwise, we can choose o i Q such that Cx^  e x i s t s and n Q 1 = -1. Evaluating <*Q and oi 1 at (15) with i = 1, we conclude that Zc+n^d^ and -c+2d,j both belong to Z . I f n l Q = -2 or -3, i t follows that c c S.» I f n l 0 = -1, then 3c € £ and d^ = 2c mod ~E. Therefore, adding to t the coboundary inducing function g^—=> c <xQ ( i / 1), g 1—=>cc^ Q + 200^ does not a f f e c t the assumption that VL±± and U10 a r e i n ~^ * t u t e n a b l e s u s to assume that c = 0. The argument also permits us to assume that d, =0, unless n,„ = 32 when d 1 = 0 o r i . E v a l u a t i n g c<^ a t (15)> we c o n c l u d e t h a t d^^  can be assumed t o be e i t h e r 0 o r i. Eqn.(6) a l s o shows t h a t f o r i / 1, u i 0 = d i m o d ^ a n d <16) d i = i i m p l i e s < °<Q+ o*±)/Z € A . I f n 1 Q = -3, t h e n u i Q e J by 5.4 and t h u s d ± = 0 . I f n^Q = -1 and we a r e n o t i n c a s e (bj> o f 5.4, i t a g a i n f o l l o w s t h a t d i = 0 . I f O ( Q i s t h e l a s t r o o t i n a component o f t y p e B 2, t h e n d i = 0 f o r i / 1 s i n c e (tf Q+ ^ . ^ / S c a n n o t be a w e i g h t . ITg s u c h t h a t < x , ~ Q , 5.8 Lemma. Suppose t h e r e e x i s t s an x £ V K s u c h t h a t <x, <^  ^ ^ * 2 <^x, 6 J , < x , = d± mod Z l o r i ^ 1 and 2x £ A . Then t h e f u n c t i o n T ( s Q ) = x , T ( s j L ) = tig/) - x - £ ^ 6 ^ ' + a c * 0 , where a = d^ - <x , ^ ^ , i n d u c e s a c o c y c l e T : K — » > V / A s u c h  t h a t the r e s t r i c t i o n o f T t o K + i s cohomologous t o * t . P r o o f . Note t h a t p Q 0 s ^ x , o^ Q> € Z and p ^ = \x^± - ^ x , * ? ^ + d^ + a ^ Q i 6 % f o r a l I i » t h e r e f o r e , we have o n l y t o v e r i f y t h e c o n d i t i o n s <3)-(5) o f I.3.3. I t i s c l e a r t h a t 2 T ( s 0 > s 2x €J\ and 2 T ( s ± ) eA f o r a l l i . I f i 4 1, P Q i = d± mod Z and p i Q = u i Q mod Z , so t h a t p Q i c*^  = P i 0 m o d " ^ " * I f ^ and a r e d i s c o n n e c t e d r o o t s i n B d i f f e r e n t f r om oC Q , we have p... = u ± f d y P ^ = and - d . ^  = u ^ . . - u ^ o ^ mod A by <8), so t h a t p ^ ot = P j 1 ° t i mod A I f m 0 1 = 3, t h e n 3t (g j )«A by (7) and 2 t ( g t ) c A by (15), so t h a t t (g 1 )<sA. T h e r e f o r e p Q l = a mod Z , P 1 0 £ J and T ( S l ) . T ( S o ) = p 0 1 o t o . p 1 0 (* t mod A . I f o t ± and ^ a r e d i f f e r e n t f rom and a r e s u c h t h a t m. . = 3, t h e n p. . = \ $ " d j + * d i m o d ^ » p j i = u j i " d i + m o d & • A p p l y i n g 33 . to (15)» v/e have 2u. . = -d. and s u b s t i t u t i n g t h i s i n (8), we obtain t(g±) - t(g..) = ( d ; f u i j * ( * i - " ' * d i ~ u ; J i ^ J m o d - ^ - » so that TCs^^} - T(Sj) = - P ^ O ^ mod A . Therefore T induces a cocycles • 7c :.&—-> V/A ; since T(g A) = TCs.^) + S ^ C S Q ) = "tCg.^ + ao< 0 - <-Jdj. + ^ x , ^ . ^ )<* i mod A , the r e s t r i c t i o n of T to K + i s cohomologous t o ~ t . / / Returning to the proof of the p r o p o s i t i o n , suppose d^ = 0 f o r i 4 1; we can apply 5»8 with x = 0 to reach the conclusion. I f oLQ and are the f i r s t and t h i r d roots i n a component of type or B^ and i f d^ = -Jr, we can l e t x s -fe^g, since by (16). The same kind of argument works for a component iQ of type A ^ or D^ i f A H P C R Q ) = P ( R Q ) . In case RQ i s type and A O P(RQ) =- q(RQ) U (Q(RQ) + for some j , we can choose o(Q = ot^ . and again have d^ = 0. There remains the case when a l l components of R are of type A 1 or A^. Define a homomorphism ^r: H 1{K +,V/A) 2->i7/2 by C ( t ) = X . . / , d ' mod Z . I f c r(t) = 0, then d , = 0 i n case there i s a component of type A^» so that the previous argument applies, or the number of d i = i i s even i n case R i s of type A j * . . . x* Aj • However, v/e can then l e t x = 2l <^ ^ i n 5»8 since 2x = Z d i^^^-A i n view of ( l 6 ) . / / One can give examples i n which the image of res i s a c t u a l l y of index 2 i n H 1 (K+,V/A)->» Suppose f i r s t that R i s of type A, and A i s the l a t t i c e A g J then the function t(gj') = 0, t(gg) = 0*.|/2 + c 2^/4 provides a counterexample. Secondly, suppose R i s of type k^ ... XA^, with k f a c t o r s , and A contains (^Q-*- °^ 34 and (<* 0+ &^)/2; t h e n we c a n d e f i n e t ( £ g 2 ) = ( &Q+ + c>c2f)A and t ( g ^ = t ( g ^ ) = 0. I f we e x c l u d e a l l the e x c e p t i o n a l c a s e s o f the p r e c e d i n g 3 p r o p o s i t i o n s , we see t h a t H 1 ( K + , V / A ) i s a n n i h i l a t e d by 2 1 1 + and t h a t t h e r e s t r i c t i o n honiomorphism H ( K , V / A ) — > H (K , V / A ) i s s u r j e c t i v e . G i v e n a c o c y c l e t : K+—"> V / A , one can t h e r e f o r e f i n d a c o c y c l e T:K—=>V//\ s u c h t h a t T J g + = t . A space group w i t h p o i n t group K + c o r r e s p o n d i n g to t c a n t h e n be v i e w e d a s t h e s u b g r o u p o f d i r e c t i s o m e t r i e s o f a space group w i t h p o i n t group K c o r r e s p o n d i n g t o T . F o r l a t t i c e s o f e x c e p t i o n a l t y p e , e x p l i c i t c o m p u t a t i o n shows t h a t i n case ( b ) , H 1 ( K + , V / A ) = I ~Z/Z ^\tw©(A). = 0 a n d ( Z / 2 Z ) d i m / ' w K " 1 i f (9(A) = w h i l e i n c a s e ( c ) , H 1 ( K + . , V / A ) = iJ/Z I ) 6 1 * 1 We s h a l l n o t d e a l w i t h t h e c a s e when R i s o f t y p e A j X . . . * A j s i n c e t h e s i t u a t i o n i s n o t c l e a r even i n t h e o r d i n a r y c a s e . 6. A p p l i c a t i o n t o c l a s s i c a l c r y s t a l l o g r a p h y . I n t h e two and t h r e e d i m e n s i o n a l c a s e s , t h e above r e s u l t s e n a b l e one t o g i v e a ' t h e o r e t i c a l ' d e r i v a t i o n o f some space g r o u p s w e l l - k n o w n i n c l a s s i c a l c r y s t a l l o g r a p h y . I n two d i m e n s i o n s , t h e r e i s a l w a y s o n l y one e q u i v a l e n c e c l a s s o f l a t t i c e s and the group H (K , V / A ) i s a l w a y s z e r o . The t h r e e d i m e n s i o n a l s i t u a t i o n i s p r e s e n t e d i n T a b l e I , u s i n g t h e S c h o e n f l i e s n o t a t i o n f o r ease o f c o m p a r i s o n w i t h j . The e n t r i e s f o r g r o u p s o f t h e form +K*" were deduced f rom 1.2.5. 35 Table I K + A © < A ) C l a s s i c a l H ^ K ^ Y / A ) ^ o r b i t s Q ( R ) 0 1 2 V 2 2 2 P ( R ) ** ^/2 Z 2 Q ( R ) * * ' 3 P ( R ) o r Z/zZ 2 'Q(R). ° P 1/zZ 2 Q ( R ) D3c7 Z / 3 Z 2 Q ( A 1 ) + P ( A 2 ) ha 2 7 3 Z 2 A , n 0 1 el i ( A 1 X B 2 ) + \ ^ Z/K7ZG1/ZZ 6 { (^ 1 +^ 2)./ 2] 2?tZZ 2 ( A 1 X G 2 ) + < Q ( R ) D6 Z/Zl® 273Z ( A 1 X A 1 >< A 1 ) + Q ( R ) Z/Z1@T/Z1® T/Z2Z. 4 7Z/ZZ 2 Z/Z7Z 2 0 1 Q ( R ) 0 2 7 3 ^ 2 Q ( R ) P ( R ) / Q ( R ) 0 1 <Q( R ) 0 3 + q(R) P ( R ) / Q ( R ) ^ / 2 Z 2 Q ( R ) 0 ° 6 Z/Zl®!/?,? 4 H Q ( R ) 0 C20( Z/z-z 2 Q ( R ) { o , ( c / 1 + 6i2)/23 0 1 A1 any °1 0 1 36 T a b l e I ( c e n t ) as f o r At T -r>, 3 3 Thp 272 Z 2 t A 2 a s f o r A * ^ . j Q ^ C3 i * 0 1 +B+ a s f o r B+ l/ZlQI/zV f k +G+ a s f o r G+ ? / 2 Z 2 +A. a s f o r A t C . 0 1 — I i i I n t h e above t a b l e , we have abandoned o u r c o n v e n t i o n r e g a r d i n g oiQ and 1 and s i m p l y numbered the e l e m e n t s o f B c o n s e c u t i v e l y s t a r t i n g w i t h oi . n o t a t i o n s u c h as d e n o t e s t h e l a t t i c e Q(R) U ( Q ( R ) + t O ) . 37 BIBLIOGRAPHY 1 . N.Bourbaki, Groupes et algebres de L i e , Chap.IV-VI, Hermann, P a r i s , 1 9 6 8 . 2 . H.Brown,J.Neubuser and H.Zassenhaus, On i n t e g r a l groups,I, Numer.Math. 19.(1972), pp.386-399. 3. i b i d . . On i n t e g r a l groups I I , Numer.Math. 20(1972), pp.22-31 4. i b i d . . On i n t e g r a l groups I I I , Math. Of Computation, 22(1973), pp. 167-182. 5. R.Bulow,J.Neubuser, On some a p p l i c a t i o n s of group t h e o r e t i c a l programmes to the de r i v a t i o n of the c r y s t a l classes of R^, Computational problems i n abstract algebra (Proc.Conf.,Oxford, 1967), Pergarmon Press, Oxford, 1970, pp.131-135. 6. J.Burkhardt, Die Bewegungsgruppen der K r i s t a l l o g r a p h i e , Birkhauser, Basel, 1 9 6 6 . 7. R.Carter, Simple groups of L i e type, Wiley, 1972. 8 . H.S.M.Coxeter and W.Moser, Generators and r e l a t i o n s for d i s c r e t e groups, Springer, 1 9 7 2 . 9. E.Dade, The maximal f i n i t e groups of 4*4 matrices, I l l i n o i s ff.Matk. 2(1965), pp .99-122. 1 0 . C.Hermann, K r i s t a l l o g r a p h i e i n Raumen b e l i e b i g e r Dimensionszahl, Acta Cryst. 2(1949), PP.139-145. 1 1 . A.Hurley, F i n i t e r o t a t i o n groups and crystal'.classes i n 4 dimensions, Proc.Camb.Phil.Soc. 4JZ.095D, pp . 6 5 0 - 6 6 l . 1 2 . G.Maxwell, The crystallography of Coxeter groups, J.Algebra, to appear 1975. 13. S.Ryskov, On maximal f i n i t e groups of integer n * n matrices, Dokl.Akad.Nauk SSSR, 204C1972). no.3,; t r a n s l a t e d i n Soviet Math.Dokl. !2(1972j>, pp.720-724. H.Weyl, Symmetry,Princeton, 1952. H.Zassenhaus, Uber einen Mgorithmus zur Bestimmung der Raumgruppen, Comment. Math. Hei v . 2J.( 1948), pp. 117-1J+1. 

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