Mathematical Methods for Physics and Engineering : A Comprehensive Guide

29.2 CHOOSING AN APPROPRIATE FORMALISM

As an example, the four-element Abelian group that consists of the set{ 1 ,i,− 1 ,−i}

under ordinary multiplication has a two-dimensional representation based on the

column matrix( 1 i)T:

D(1) =

(

10

01

)

, D(i)=

(

0 − 1

10

)

,

D(−1) =

(

− 10

0 − 1

)

, D(−i)=

(

01

− 10

)

.

The reader should check thatD(i)D(−i)=D(1),D(i)D(i)=D(−1) etc., i.e. that

the matrices do have exactly the same multiplication properties as the elements

of the group. Having done so, the reader may also wonder why anybody would

bother with the representative matrices, when the original elements are so much

simpler to handle! As we will see later, once some general properties of matrix

representations have been established, the analysis of large groups, both Abelian

and non-Abelian, can be reduced to routine, almost cookbook, procedures.

Ann-dimensional representation ofGis a homomorphism ofGinto the set of

invertiblen×nmatrices (i.e.n×nmatrices that have inverses or, equivalently,

have non-zero determinants); this set is usually known as the general linear group

and denoted by GL(n). In general the same matrix may represent more than one

element ofG; if, however, all the matrices representing the elements ofGare

differentthen the representation is said to befaithful, and the homomorphism

becomes an isomorphism onto a subgroup of GL(n).

A trivial but important representation isD(X)=Infor all elementsXofG.

Clearly both of the defining relationships are satisfied, and there is no restriction

on the value ofn. However, such a representation is not a faithful one.

To sum up, in the context of a rotation–reflection group, the transposes of

the set ofn×nmatricesD(X)thatmakeuparepresentationDmay be thought

of as describing what happens to ann-component basis vector of coordinates,

(xy···)T, or of functions, (Ψ 1 Ψ 2 ···)T,theΨithemselves being functions

of coordinates, when the group operationXis carried out on each of the

coordinates or functions. For example, to return to the symmetry operations

on an equilateral triangle, the clockwise rotation by 2π/3,R, carries the three-

dimensional basis vector (xyz)Tinto the column matrix








−^12 x+

√

3

2 y

−

√

3

2 x−

1

2 y

z









whilst the two-dimensional basis vector of functions (r^23 z^2 −r^2 )Tis unaltered,

as neitherrnorzis changed by the rotation. The fact thatzis unchanged by

any of the operations of the group shows that the componentsx,y,zactually

divide (i.e. are ‘reducible’, to anticipate a more formal description) into two sets: