Mathematical Methods for Physics and Engineering - Matematica.NET

29.2 CHOOSING AN APPROPRIATE FORMALISMgenerates a new column matrix u ′ =(u ′ 1 u′ 2 ··· u′ n) T . Having established u and u ′we can determine the n × n matrix, M(X) say, that connects them byu ′ = M(X)u. (29.2)It may seem natural to use the matrix M(X) so generated as the representativematrix of the element X; in fact, because we have already chosen the conventionwhereby Z = XY implies that the effect of applying element Z is the same as thatof first applying Y and then applying X to the result, one further step has to betaken. So that the representative matrices D(X) may follow the same convention,i.e.D(Z) =D(X)D(Y ),and at the same time respect the normal rules of matrix multiplication, it isnecessary to take the transpose of M(X) as the representative matrix D(X).Explicitly,and (29.2) becomesD(X) =M T (X) (29.3)u ′ = D T (X)u. (29.4)Thus the procedure for determining the matrix D(X) that represents the groupelement X in a representation based on basis vector u is summarised by equations(29.1)–(29.4). §This procedure is then repeated for each element X of the group, and theresulting set of n × n matrices D = {D(X)} is said to be the n-dimensionalrepresentation of G having u as its basis. The need to take the transpose of eachmatrix M(X) is not of any fundamental significance, since the only thing thatreally matters is whether the matrices D(X) have the appropriate multiplicationproperties – and, as defined, they do.In cases in which the basis functions are labels, the actions of the groupelements are such as to cause rearrangements of the labels. Correspondingly thematrices D(X) contain only ‘1’s and ‘0’s as entries; each row and each columncontains a single ‘1’.§ An alternative procedure in which a row vector is used as the basis vector is possible. Definingequations of the form u T X = u T D(X) are used, and no additional transpositions are needed todefine the representative matrices. However, row-matrix equations are cumbersome to write outand in all other parts of this book we have adopted the convention of writing operators (here thegroup element) to the left of the object on which they operate (here the basis vector).1079