Maths - Representation Theory

Representation theory is related to group theory except that we look at this in terms of linear algebra such as matrices. In other words we are looking at the symmetries of vector spaces. A simple example is Pauli matrices.

morphism

Construction of Representations

From Permutation Groups

Cyclic Group

We can construct a matrix that has the property that [M]n = [1] in several ways including:

The second method produces a very large matrix because the matrix contains only 0's and 1's.

Example of a Cyclic Group Representation using a Program

We can use a computer program to show that this works, here I have used Axiom/FriCAS which is described here.

m:= matrix [[0,0,1],[1,0,0],[0,1,0]]
0 0 1
1 0 0
0 1 0
                                                    Type: Matrix(Integer)
m*m
0 1 0
0 0 1
1 0 0
                                                    Type: Matrix(Integer)
m*m*m
1 0 0
0 1 0
0 0 1
                                                    Type: Matrix(Integer)
(4) ->

Combining Cycles

We can create a representation of any permutation (and therefore of any group) by generating the representation for each cycle separately. That is each one is a seperate generator.

By combining two other Representations

If U and V are representations of a group then

Are also representations.

The identity for vector space is also true for representations:

V⊗(U⊕W) = (V⊗U)⊕(V⊗W)

From Invariant Subspaces

There is a common theme to:

Which is related to the idea of symmetry, somthing which does not change during a transform.

(1) -> t:= matrix [[1,2],[0,4]]

        +1  2+
   (1)  |    |
        +0  4+
                                                    Type: Matrix(Integer)
(2) -> t*t

        +1  10+
   (2)  |     |
        +0  16+
                                                    Type: Matrix(Integer)
(3) -> t*t*t

        +1  42+
   (3)  |     |
        +0  64+
                                                    Type: Matrix(Integer)
(4) ->

From Clifford Algebra

See this page

Generating Representations from other Representations

If we already know a representation of a given group then how can we find other 'equivalent' representations?

group   group representation   other representation
g --> [G] --> f([G])
h --> [H] --> f([H])
|
|
v
 
|
|
v
 
|
|
v
g•h --> [G•H])= [G]*[H] --> f([G•H])= f([G]*f([H])

If we can find some matrix function f() the preseves the multipication so that:

f([G•H])= f([G]*f([H])

Then we can use that to generate a new representation, an example is the determinant,

det([G•H])= det([G]*det([H])

Of course this representation contains a lot less information and so can't reconstruct the original group.

The action of a linear transform has a quadratic form:

quadratic action

More about the quadratic form of rotations on this page.

Two representations are equivalent if they are related by a 'similarity' transformation.

If D(G) is a representation of a group G.
and D'(G) is an equivalent representation of a group G.
then:

D'(g) = T-1•D(g) • T

where:

 


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see also:

 

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