Definition:  If there exists a set T of linear operators T( G_i) in a vector space L which correspond to the elements G_i of a group G such that

T( Ga) T( Gb) = T( GaGb)

then this set of operators is said to form a representation of the group G in the space L.

Note:  These representations can be of arbitrary dimension.  We will always use matrices to denote representations.

Ex.  Let G = {G_q} be the group of counter-clockwise rotations by an angle q about the z-axis (i.e G = SU( 2) ).  Then if we work in the vector space R² with unit vectors e_i = {[^(x)],[^(y)]} we can construct the representation which is the set of two-by-two unitary matrices with positive determinant T(G_q) = á e_i,G_qe_j ñ = (

cosq	sinq
-sinq	cosq

) .  Note that T( G_a) T( G_b) = T(G_aG_b) .  If we work in the vector space R³ with unit vectors e_i = {[^(x)],[^(y)],[^(z)]} then we can create the representation á e_i,Ge_j ñ = (

cosq	sinq	0
-sinq	cosq	0
0	0	1

) .

Generalization to function spaces:  

Thm.  Let L be the space of functions y( [(r)\vec]) and G = {G_i} be the group of coordinate transformations such that if y( [(r)\vec]) Î L then y( G_a^-1[(r)\vec]) Î L.  We can then define a representation T in the function space L such that

T( Ga) y æ è ® r   ö ø = y æ è Ga-1 ® r   ö ø .

Pf.  First define y^¢( [(r)\vec]) = y( G_b[(r)\vec]) .  Note that

T( Ga) T( Gb) y æ è ® r   ö ø = T( Ga) y æ è Gb-1 ® r   ö ø = T(Ga) y¢ æ è ® r   ö ø = y¢ æ è Ga-1 ® r   ö ø = y æ è Gb-1Ga-1 ® r   ö ø = y æ è (GaGb) -1 ® r   ö ø = T( GaGb) y æ è ® r   ö ø .

Since T( G_a) T( G_b) = T( G_aG_b) , this is a valid representation.

The matrix representation T( G_a) can be obtained by simply calculating T( G_a) = á y_i( [(r)\vec]) ,G_ay_j( [(r)\vec]) ñ where y_i( [(r)\vec]) is some basis of the function space L.  After we find this matrix, we can calculate y^¢( [(r)\vec]) = T( G_a) y( [(r)\vec]) = y( G_a^-1[(r)\vec]) .  

So we see that we can make group representations for spatial variables and, as a more general case, functions.  For the remaining of this presentation, when I write e_i as a basis, it represents both a basis for a ``normal'' space such as R³ and a basis for a function space.

Definition:  If L₁ is a subspace of L such that for all e_i Î L₁ it is the case that T( G_a) e_i Î L₁ then we say that L₁ is invariant with respect to the transformations induced by G, or an invariant subspace.

Ex.  Once again let G = {G_q} be the group of counter-clockwise rotations by an angle q about the z-axis.  We found that if our basis e_i =  {[^(x)],[^(y)],[^(z)]}, then a representation of G is the set of matrices (

cosq	sinq	0
-sinq	cosq	0
0	0	1

) .  If we define L₁ = span{[^(x)],[^(y)]} and L₂ = span{[^(z)]} then L₁ and L₂ are invariant subspaces.

Reducibility:  

Definition:  Let L be a space which is invariant with respect to the transformations T( G_a) induced by some group G = {G_a}.  If L₁,L₂ Ì L such that L₁,L₂ are invariant subspaces and L₂ is the orthogonal complement of L₁, then T is reducible with respect to L.  

Generally, we can determine whether a matrix representation is reducible by looking at the off diagonal elements in the matrix.  If they are zero, then the representation is reducible.  If they are not zero, then the representation is not reducible.  

Theorem:  If L₁,L₂ Ì L such that L₂ is the orthogonal complement of L₁, then L₁ is an invariant subspace wrt a unitary representation T if and only if L₂ is an invariant subspace wrt to T.

Pf:  Let the basis of L₁ and L₂ be {e_i} and {e_j} respectively where the basis vectors can be functions or coordinates.  From the invariance of L₁, we have that áT( G_a) e_i,e_j ñ = 0 for all G_a.  Since T is unitary, the invariance condition on L₁ can be transformed to á e_i,T( G_a^-1) e_j ñ = 0, which means that L₂ is invariant wrt T.

Note:  We will prove later that if T(G_a) is a transformation in the set of representations of a group G, then T( G_a) can be written as a unitary matrix (i.e. T( G_a) ^-1 = T( G_a) ^t).

As a consequence of this theorem, if we have orthogonal subspaces such that L = L₁+L₂+L₃+... and each of the L_i's is irreducible with respect to a transformation T( G_a) , then we can write the reduction of the representation as

T( Ga) = T(1)( Ga) +T(2)( Ga)+T(3)( Ga) +...

where T⁽ⁱ⁾( G_a) represents the irreducible representation induced in the space L_i (the + sign does not represent normal addition).  In this case, the reduced form of T( G_a) can be written in matrix form as

æ ç ç ç ç è T(1) T(2) T(3) ··· ö ÷ ÷ ÷ ÷ ø

where each T⁽ⁱ⁾ is an irreducible matrix representation of the group G.

Ex.  To create the irreducible matrix representation of G = {G_q} the group of counter-clockwise rotations by an angle q about the z-axis in R³, we simply note that L₁ = span{[^(x)],[^(y)]} and L₂ = span{[^(z)]} are orthogonal complements.  If we look at the matrix (

cosq	sinq	0
-sinq	cosq	0
0	0	1

) , we see that the upper left two-by-two matrix has off-diagonal entries.  Therefore the representation is irreducible in the {[^(x)],[^(y)]} and {[^(z)]} basis.  Therefore we write L = L₁+L₂ and T = T⁽¹⁾+T⁽²⁾ = (

cosq	sinq
-sinq	cosq

) +1 so that (

cosq	sinq	0
-sinq	cosq	0
0	0	1

) = (

T(1)
	T(2)

) .  This is a bad example since the matrix is already in irreducible form, but it gives the general idea.