A subrepresentation of a representation
Then
This has a block matrix interpretation: if we choose a basis for
Here
A representation
Let
the subrepresentation where the numbers on all faces are equal
the subrepresentation where the numbers on opposite faces sum to zero
the subrepresentation where the numbers on opposite faces are equal and all the labels sum to zero.
Note that in lectures we did the related example of edges of the tetrahedron.
Suppose that
This also has a block matrix interpretation: if we choose bases for
We can generalize this definition to a finite number of
subrepresentations
Consider the permutation representation of
Let
Let
I claim that
If
The next definition is optional: we will avoid using quotient representations.
Suppose that
Remember that
Suppose that
for all
In other words,
There is another word that is sometimes used for
A
If there is a
Suppose that
If
Suppose
for all
Exercise. ∎
Given
We know that they are subspaces, so we just have to show that they are preserved by the action of
For the kernel: suppose that
so
For the image: suppose
is also in the image
of
Let
Then let
Let
The matrix of
and now we just have to check that
which is true, and a similar equation coming from
In fact we could see this without calculation; by the way we defined the isomorphism
for all
The kernel of the homomorphism
to
If
for all
is an isomorphism.
We haven’t yet covered this section; we will come back to it.
If
for
This representation is also written
If
Let
of the permutation representation. Let
The action of
Exercise: what is the restriction of
If
Then
Take
defined as follows: label the three non-identity elements of
and
Then the homomorphism is surjective with kernel
If
This is just the kernel of
the homomorphism
If
Exercise! ∎
If
We prove the equivalent statement that