On a certain class of n-dimensional space groups

Vol. 11 (1977)

REPORTS

ON MATHEMATIC.4t

ON A CERTAIN CLASS OF n-DIMENSIONAL

PIOTR

No. 2

PHYSICS

SPACE GROUPS

WOLOWIEC

Institute of Mathematics, Wroclaw Technical University, Wroclaw, Poland (Received June 22, 1976) For cyclic point group P of order r acting on an n-dimensional lattice T we describe orbits of H*(P, T) (corresponding to non-isomorphic n-dimensional space groups) when this cohomology group is isomorphic to d. The description shows that essentially only one eigenvector of P introduces non-isomorphic extensions. Any n-dimensional space group is (up to an isomorphism) an extension of a lattice T by a point group P acting on T. However, inequivalent extensions (corresponding to different elements of second cohomology group H2(P, T)) may give isomorphic groups. Therefore, in order to obtain the set of all non-isomorphic groups determined by T and P one has to find orbits of HZ(P, T) under the action of normalizer A&,.&P) (where P is embedded in CL@, Z), once a basis of T is fixed), where for c E N~L(~,z)(P) the action on 2-cocycle v E Z2(P, T) is defined by

(MPP; 9Pz) = cqJ(c-‘p, c, c-9,

c),

PIYP2 EP

(I)

(see [3], p. 80, 81). We are concerned here with the case of cyclic point group P, and prove the following theorems. THEOREM1. Let P z 2, act on an n-dimensional lattice T. Suppose that in a certain basis T the generator p of P is of the form

P

4 0 = [ 01,’

1

where q E GL(n-k, 2) has no eigenvalue + I. Then H’(P, T) z 2:. H’(P, T) s Z,“, then p has exactly k reducible eigenvalues + 1.

Conversely,

THEOREM 2. Let P z 2, and H2(P, T) z 2: , k > 1. Then each orbit of H’(P, under the action of N ~L(,,,z) (P) corresponds to an integer j, ivhere jlr. The first part of Theorem theorem :

I is an immediate

consequence

12271

of the following

well-known

if T)

P. WOLOWIEC

228

If P z Z, generated by p acts on Abelian group T, then r-1

H2(P, T) = {t E T: pt = t]/(xp’f:

t E T]

(2)

i-0

(see [2], Theorem 16.10). We also use the’fact that any 2-cocycle q on P is determined its value pl(p-‘,p) and that y(p-‘9 P? = &J-‘5

PI,

(up to coboundary)

0 < I < n.

by (3)

This equality is a direct consequence of formulas (8.7) and (8.9) of the paper [3], p. 64. Note that by (1) the orbit containing [pll necessarily contains [ - ~1. We say that H2(P, T) has a trivial orbit structure if for each v no other elements are in the orbit of [p13. First let us prove the following lemmas: LetPg Z, and H2(P, T) contain a factor of order r. Let tl be an element of T corresponding to it. Then there exist vectors t2 , . . . , 2, E T such that {tl , . . . , tn> forms the basis of T and the generator p of P is expressed in this basis by a matrix of the form LEMMA

1.

Proofz form

If t, is the first vector of a basis, then the matrix corresponding

top has the

la [ OA 3

‘=

and SJ=[b

“sl

and the first row of the latter matrix is divisible by r. Indeed, the first component of each r-1 vector 2 pit is a multiple of r and, by (2), only then one obtains an element generating i.WZO

Z, in H2(P, T). We show that then p is reducible. Note that I-1

1 axA’

p’ =

f=O

i 0 r-1

A’

I ,

r-1

whence a ‘i A’ = aS = 0 and w = a ,gi (r- i)A’-‘, and hence WA - w = a c i-0

-a’?(r-i)A’-l irl

i-l

= as-ra

= -ra.

Let d

(r- i)A -

A CERTAIN

CLASS OF n-DIMENSIONAL

SPACE GROUPS

229

Then d E GL(n, 2) and dpd-’ =

LEMMAS.

1 [0

-w’r+~+wirA]

= [10 z],

Let k>l,OO.

q.e.d.

If r and g.c.d.

(x1,

. . . , xk)

are

i-1

relatively prime, then for some D E GL(k, 2) we have

(modr).

Proof: E g.c.d. (r, x) = 1, then for some integers I,, IL we have II r+12 x = 1, and takiog for D the matrix

we see that

Therefore we may assume, without loss of generality, that g.c.d. (x1, . . . , xk) = 1. Let us proceed by induction on k. If k = 2, then for some integers Ii, I2 we have 1, x1 + +12x2 = 1 and -12

D=xl [ x2

,. 1

1

Assume the lemma to be valid for k < s and let d = g.c.d. (x2, . . . , xJ. Clearly, g.c.d. (x1, d) = 1, so that for some integers kI , k2 we have k, x1 + k2 d = 1. By the inductive assumption, there exists a matrix DI E GL(s- 1,Z) with the first column

Denote by D2 the matrix obtained from D1 by multiplying that column by d, and put

P. woLowIEc

230

Clearly, deleting the first row and column of D we get the matrix obtained from D, by multiplying the first column by k1 and putting it as the last one; therefore its determinant is (- l)S-‘k, det Dl and as det D, = d* detl),, we get detD = x1(-l)“-‘k,detD1+(-l)“+lk,d*detD,

= (-I)“-‘detD,,

hence D E GL(s, Z), q.e.d. Proof of Theorem

1:

If q has no eigenvalue + 1, then, by (2), the vectors &p-‘, p)

have the form

t= r-1

Next, note that c q’ = 0 because for every

t' E

T'

E

Zn-k,

i-0

r-1 q(Cqi)t’ i=O

and hence for arbitrary

r-1

r-l =

(&‘)f, i=o

so

(-&‘)t’=

0,

i=o

t” E T we have

-0

-

rt;

J

Thus H2(P, T) z Z,“, what proves the first part of the theorem. Conversely, we must show that an irreducible eigenvalue + 1 of p does not yield a factor 2, in W(P, T). This easily follows from Lemma 1, because the appearance of the factor 2, in the cohomology group implies reducibility of one eigenvalue + 1 in p. Proof of Theorem 2:

Note that if

generates P, then the normalizer NGL~,&P)

[I t1

Let q(p-',p)

=

i

tn

.

contains all matrices of the form

A CERTAIN

CLASS OF n-DIMENSIONAL

231

SPACE GROUPS

The normalizer of the cyclic group P in H2(P, T) a&s according to the formula h-k+1 Df

,

I t,

I

follows from Theorem 1, formulas (l), (2), (3), and the equality c-‘PC = p’. By Lemma 2, matrices of the form C act transitively on a set of representatives of generators of H’(P, T). From this property and from the facts that the elements of NoLC,,n(P) act as automorphisms of W(P, T) and any element of order r in H2(P, T) is contained in a set of generators, we see that the orbits contain all elements of the same order, and only these elements; thus, orbits with respect to action of matrices of the form C coincide with the orbits with respect to the action of AutH2(P, T), what completes the proof.

as

THEOREM 3. IfP = ((p)), P z 2, andp ha.s exactly one eigenvalue + 1, then H*(P, z Z, has a trivial orbit structure.

T)

Proof Any vector ~(p-‘, p) is a multiple of the shortest vector v,cP-‘, p) generating the line in R” invariant under p. The class [v] generates then H’(P, T), and coboundaries are of the form y = r’@, r’ being the order of H*(P, T) and I E Z. Due to this, we have by (1) and (3)

c&+,P)

= (Sfr’l)qti-‘,

P),

c

ENGL(n,Z)(Ph

0 <

bi

<

r’.

(4)

Note that the possible eigenvalues of c are f 1. In fact, if m were the eigenvalue of c, then in a certain basis of T, c would have the first column of the form

il m 0

and det c would be divisible by m. Hence, by (4), it follows that sfr’l COROLLARY.

+ 1, and r,,

I’,

= +l,

’

what implies s = +I,

q.e.d.

If HZ (P , T) “= z, “z P, the generator p of P has exactly one eigenvalue are extensions of T by P, corresponding to the orbits of elements

[[Jl,

[[J+P,T)

and

g.c.d. (t,t’)

= 1

then Remark

1. The orbit structure of HZ(P, T) may be sometimes determined directly the assumption that P is cyclic. Namely, if AutH2(P, T) = {id, -id}, then H’(P, T) has necessarily a trivial orbit structure, This occurs in the case of cyclic H’(P, T) of orders 1,2,3,4,6; therefore in R2 and R3 all cyclic groups H2(P, T) have trivial orbit structure (see [3]), the same applies to all cyclic groups H*(P, T) in R4 for P z Z, without

(see [In.

232

P. WOLOWIEC

THEOREM4. If Z, E (

> = P c GL(n, Z), H2(P, T) z Z,!, k > 1, p’ is the image of P under the embedding GL(n, Z) + GL(n +m, Z), I’ is the extension of T by P determined by the jth orbit in H2(P, T) = Z,! (where j/r) and F’ the extension of T’ by P’ determined by the jth orbit in H2(P’, T’) z Z:+‘“, then r’ E I’QZ”. The proof of Theorem

4 follows directly from Theorem

2.

Remark 2. Theorem 2 may be applied even in low dimensions. Namely, in the [I] there are described six 4-dimensional point groups (numbers 5, 6, 9, 40, 53, 61) satisfying the assumptions of Theorem 2. The number of elements in the corresponding groups H2(P, T) is 77, and by Theorem 2 the number of non-isomorphic space groups is 15. REFERENCES [1] Fast, G., Janssen, T.: Technical Report 6-68, Katholieke Universiteit, (21 Huppert, B.: Endliche Gruppen, Springer Verlag, Berlin 1967. [3] Mozrzymas, J.: Reports Math. Phys. 6 (1974), 29.

Nijmegen.