Crystallography Reports, Vol. 45, No. 3, 2000, pp. 349–355. Translated from Kristallografiya, Vol. 45, No. 3, 2000, pp. 391–397. Original Russian Text Copyright © 2000 by Burdina. CRYSTALLOGRAPHIC SYMMETRY Complete Normalizer for a Direct Product of Three-Dimensional Rotation Groups V. I. Burdina Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskiœ pr. 59, Moscow, 117333 Russia Received June 19, 1996; in final form, December 1, 1999 Abstract—A normalizer of the symmetry group defined on a three-dimensional sphere S3 of rotation is con- sidered in the four-dimensional Euclidean space E4. The sphere S3 is treated as the first approximation of the 1 & × three-dimensional crystallographic space. The analysis of the normalizer of the direct product = G1 G2 of space crystallographic rotation groups G1 and G2 is reduced to the study of transformations characterized by 1+ 1+ the positive determinants of the subgroups (G1) and (G2). These subgroups correspond to the Euclidean 1 1+ × 1+ normalizers = (G1) (G2) of the components of the direct product. We derived a table including the groups of automorphisms induced by the transformations corresponding to the normalizers under study. Ana- lyzing the general operation of multiplication of three-dimensional rotations in E4, we refined the distribution of the supersymmetry operators of the three-dimensional sphere of rotations, S3, for the symmetry groups con- sidered earlier. © 2000 MAIK “Nauka/Interperiodica”. INTRODUCTION computer-calculated regular divisions. The procedure is illustrated by two examples. A normalizer 1 of the symmetry group & is widely used in crystallography [1, 2] alongside with a more general concept of symmetrizer. In [3, 4], this normal- GENERAL INFORMATION izer was used to interpret the additional symmetry (supersymmetry) arising in the regular divisions of The transformation 3 three-dimensional sphere S of rotation (which involves p = p × p (1) the center of inversion). The normalizer is defined in a 1 2 purely algebraic manner [5] as a group of isometric of the direct product of three-dimensional rotations ϕ ϕ 3 transformations of a sphere of rotation, which induces p1 = P(l1, 1) and p2 = P(l2, 2) on the sphere S about 1&1–1 & & ϕ ϕ automorphisms = of the symmetry group . the unit vectors l1 and l2 through angles 1 and 2 in the By definition, a normalizer can lead to the continuous 4 four-dimensional Euclidean space E (u0, u1, u2, u3) acts spectrum of transformations, and hence to the reduced as a linear orthogonal transformation, which can be dimension of the minimum domain of parameters. The written in the quaternion form [10] as normalizer operators not belonging to the symmetry group of the space under study can be related to the u' = sut, (2) medium “dissymmetry” that causes symmetrization of some lattices formed by the sets of regular, or homolog- where u = u0 + u1i + u2j + u3k and u' = u0' + u1'i + u2'j + ical, points (in full accordance with the generalized u3'k are the arbitrary and the transformed quaternions, ϕ ϕ Curie–Shubnikov principle [6–9]). respectively; s = cos( 1/2) + l1sin( 1/2) and t = ϕ ϕ Developing [4], we consider the general problem of cos( 2/2) + l2sin( 2/2) are the unit quaternions of rota- the determination of the normalizer 1(&) for the direct tions p1 and p2. In the vector–matrix form, we can write product & = G × G of the crystallographic rotation 1 2 u' = Zu, (2') groups G1 and G2. The group G is a symmetry group on the three-dimensional manifold belonging to the sphere where the orthogonal 4 × 4 matrix Z = ST = TS is equal of rotation S3 immersed into the four-dimensional to the product of the commuting matrices Euclidean space E4. We prove the completeness of this normalizer in the sense of the full involvement into it of s0 –s1 –s2 –s3 t0 –t1 –t2 –t3 various automorphisms of the symmetry groups. The s s –s s information contained in the normalizer allows us to S = 1 0 3 2 and T = t1 t0 t3 –t2 construct the minimum parameter domain, and to indi- s2 s3 s0 –s1 t –t t t cate the exact location of the symmetrized lattices of 2 3 0 1 regular points. This allows us to refine the structures of s3 –s2 s1 s0 t3 t2 –t1 t0 1063-7745/00/4503-0349 $20.00 © 2000 MAIK “Nauka/Interperiodica” 350 BURDINA constructed with the use of the elements of the quater- operation, i.e., they also are automorphisms. As a con- Φ × Φ × Φ × nions s = s0 + s1i + s2j + s3k and t = t0 + t1i + t2j + t3k. sequence, we have (E p) = 1(E) 2(p) = E Φ Φ × Φ × Φ Φ × To each rotation P(l, ϕ) there corresponds, in addi- 2(p) and (p E) = 1(p) 2(E) = 1(p) E. In the tion to quaternion cos(ϕ/2) + lsin(ϕ/2), also the quater- Euclidean space E4, the matrix Z = S of the linear trans- nion of the opposite sign, which arises upon the change formation (2') is transformed into the matrix s by auto- of the rotation angle ϕ by ϕ + 360° not changing the morphism Φ, whereas the matrix Z = T is transformed rotation. Therefore, to the direct product (1) of rotations into the matrix t. Coming back to direct product (1) of there correspond ±s, ±t, ±Z, ±S, and ±T and two trans- rotations, we consider some geometrical characteristics formations, (2) and (2'), related by inversion, which of the resulting operator in E4 [11]. The existence of the transform the point u ∈ E4 either into u' or –u', and a linear orthogonal transformation of the four-dimen- pair of diametric points u and –u' either into u' and –u' sional Euclidean space corresponding to direct product ϕ × ϕ or –u' and u', i.e., into the same pair. Therefore, a P(l1, 1) P(l2, 2) of rotations corresponds to the exist- unique transformation of the sphere of rotation S3 and ence of two invariant orthogonal two-dimensional sub- 3 π π one transformation of tangential space T (1, x1, x2, x3) spaces, or planes and ' of rotation through angles (ϕ – ϕ )/2 and (ϕ + ϕ )/2, which are equal to the half- correspond to product (1). Here, xi = ui/u0, i = 1–3. The 1 2 1 2 diametric points at this sphere are assumed to be sym- difference and the half-sum of initial rotation angles, metrically identical, because in the transition from S3 to respectively, determined by the directions the tangential space, the diametric points u and –u are π 2 [ , ] projected to a single point. : L1 = l1 – l2, L2 = 1 + ------------------2 l1 l2 (3) × × l1 – l2 The multiplication of pq = (p1 p2)(q1 q2) of oper- × × ators p = p1 p2 and q = q1 q2 is performed from right and to left; the corresponding double quaternion transfor- 2 u s s ut t s s u t t π ' ' [ , ] mation has the form: ' = p( q q) p = ( p q) ( q p). ': L1 = l1 + l2, L2 = 1 – ------------------2 l1 l2 , (4) Since the multiplication of two quaternions corre- l1 + l2 sponds to the multiplication of rotations described by whose locations are independent of the initial rotation these quaternions, then the direct product is written as angles ϕ and ϕ . If one of the latter angles changes by × 1 2 p1q1 q2p2. Thus, the multiplication of direct products 360°, the rotations themselves remain unchanged, but of rotations the rotation angles in the invariant planes π and π' × × × 4 (p1 p2)(q1 q2) = p1q1 q2p2 change by 180°, and the transformation in E is com- plemented with inversion. reduces to the multiplication of left-hand factors in the π same direction, whereas the right-hand factors are mul- In the general case, the first invariant plane (see tiplied in the opposite direction. The product of trans- (3)) in the projective tangential space T 3 is represented by the line of intersection passing through the point formations (2') and matrices Z = Z1 and Z = Z2 reduces | |2 to the linear transformation corresponding to the matrix A(2[l1, l2]/ l1 – l2 ) along the l1 – l2 direction, whereas π Z = Z2Z1. the second plane ' (see (4)) is represented by the point | |2 In addition to individual multiplications of transfor- B(–2[l1, l2]/ l1 + l2 ) and by the (l1 + l2) direction. ϕ × mations (1), we also consider the group In the particular case of direct product P(l1, ) ϕ ϕ ϕ ϕ 3 = P × P P(l2, ) of the rotations of the same order, 1 = 2 = , 1 2 the first plane π remains stationary, since the rotation of direct multiplication of three-dimensional rotation angle in this plane is (ϕ − ϕ )/2 = 0, whereas the rota- Φ 1 2 groups P1 and P2. Let be the automorphism, that is, tion angle in the plane π' is equal to (ϕ + ϕ )/2 = ϕ, i.e., 3 1 2 the one-to-one mapping of group onto itself retain- to the common angle of the initial rotations. In T 3, we ing the multiplication operation have the fixed “rotation axis” of the order of 360°/ϕ. If, Φ(pq) = Φ(p)Φ(q), where p, q ∈ 3. in addition, the initial axes differ only in signs, l2 = –l1, then the vector product becomes zero [l , l ] = –[l , l ] = Here, Φ(e) = e, e is the unity element of the group (e = 1 2 2 1 × 0, the point A is displaced to the origin of coordinates E E) which is transformed into itself. For the element 3 p × p Φ p × p in the tangential hyperplane T , i.e., the point A coin- p = 1 2 in the direct product, we have ( 1 2) = 3 × ∈ cides with the pole (1, 0, 0, 0) of the sphere S .