Mathematical Methods for Physics and Engineering : A Comprehensive Guide

REPRESENTATION THEORY

of a particular group of symmetry operations has been brought to irreducible

form, the implications are as follows.

(i) Those components of the basis vector that correspond to rows in the

representation matrices with a single-entry block, i.e. a 1×1block,are

unchanged by the operations of the group. Such a coordinate or function

is said to transform according to a one-dimensional irrep ofG.Inthe

example given in (29.10), that the entry on the third row forms a 1× 1

block implies that the third entry in the basis vector (xyz···)T,

namelyz, is invariant under the two-dimensional symmetry operations on

an equilateral triangle in thexy-plane.

(ii) If, in any of thegmatrices of the representation, the largest-sized block

located on the row or column corresponding to a particular coordinate

(or function) in the basis vector isn×n, then that coordinate (or function)

is mixed by the symmetry operations withn−1 others and is said to

transform according to ann-dimensional irrep ofG. Thus in the matrix

(29.10),xis the first entry in the complete basis vector; the first row of

the matrix contains two non-zero entries, as does the first column, and so

xis part of a two-component basis vector whose components are mixed

by the symmetry operations ofG. The other component isy.

The result (29.11) may also be formulated in terms of the more abstract notion

of vector spaces (chapter 8). The set ofgmatrices that forms ann-dimensional

representationDof the groupGcan be thought of as acting on column matrices

corresponding to vectors in ann-dimensional vector spaceVspanned by the basis

functions of the representation. If there exists aproper subspaceWofV,such

that if a vector whose column matrix iswbelongs toWthen the vector whose

column matrix isD(X)walso belongs toW, for allXbelonging toG,thenit

follows thatDis reducible. We say that the subspaceWis invariant under the

actions of the elements ofG. WithDunitary, the orthogonal complementW⊥of

W,i.e.thevectorspaceVremaining when the subspaceWhas been removed, is

also invariant, and all the matricesD(X) split into two blocks acting separately

onWandW⊥.BothW andW⊥may contain further invariant subspaces, in

which case the matrices will be split still further.

As a concrete example of this approach, consider in plane polar coordinates

ρ, φthe effect of rotations about the polar axis on the infinite-dimensional vector

spaceVof all functions ofφthat satisfy the Dirichlet conditions for expansion

as a Fourier series (see section 12.1). We take as our basis functions the set

{sinmφ,cosmφ}for integer valuesm=0, 1 , 2 ,...; this is an infinite-dimensional

representation (n=∞) and, since a rotation about the polar axis can be through

any angleα(0≤α< 2 π), the groupGis a subgroup of the continuous rotation

group and has its ordergformally equal to infinity.