Abstract Algebra Theory and Applications - Computer Science ...

10.2 SYMMETRY 183Given two bases for the same lattice, say {x 1 , x 2 } and {y 1 , y 2 }, we canwritey 1 = α 1 x 1 + α 2 x 2y 2 = β 1 x 1 + β 2 x 2 ,where α 1 , α 2 , β 1 , and β 2 are integers. The matrix corresponding to thistransformation is( )α1 αU =2.β 1 β 2If we wish to give x 1 and x 2 in terms of y 1 and y 2 , we need only calculateU −1 ; that is,( ) ( )U −1 y1 x1= .y 2 x 2Since U has integer entries, U −1 must also have integer entries; hence thedeterminants of both U and U −1 must be integers. Because UU −1 = I,det(UU −1 ) = det(U) det(U −1 ) = 1;consequently, det(U) = ±1. A matrix with determinant ±1 and integerentries is called unimodular. For example, the matrix( ) 3 15 2is unimodular. It should be clear that there is a minimum length for vectorsin a lattice.We can classify lattices by studying their symmetry groups. The symmetrygroup of a lattice is the subgroup of E(2) that maps the lattice toitself. We consider two lattices in R 2 to be equivalent if they have the samesymmetry group. Similarly, classification of crystals in R 3 is accomplishedby associating a symmetry group, called a space group, with each type ofcrystal. Two lattices are considered different if their space groups are notthe same. The natural question that now arises is how many space groupsexist.A space group is composed of two parts: a translation subgroup anda point group. The translation subgroup is an infinite abelian subgroupof the space group made up of the translational symmetries of the crystal;the point group is a finite group consisting of rotations and reflections ofthe crystal about a point. More specifically, a space group is a subgroup ofG ⊂ E(2) whose translations are a set of the form {(I, t) : t ∈ L}, where L is