2021-10-18 ↩

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created: 2021-10-18 17:28:48
modified: 2022-11-11 17:39:30

Group theory homework #6

1. write out explicitly the regular representations of the and point groups.

Regular representation of the inversion point group
Regular representation of the cyclic point group of order 3

2. give me explicitly the regular representation of the , , and elements of the point group. (do not forget to write your ordering of the group elements!)

Regular representation of the C₃ᵥ point group

3. show that for any 1d representation, , of the point group: .

1D representation of the C₃ᵥ point group

4. consider the additive group and the mapping.

A) is it a representation?

Is a homomorphism?

Both sides are equal so is a representation.

B) is it faithful?

Faithful is injective
Let's say that , then

, which is only possible if . QED

C) is it unitary?

Is any unitary?
is not unitary!

D) is it reducible?

According to Schur's lemma I, if I find an that commutes with all , then is reducible.
Let , and
From this:

The equation system is solvable with infinite solutions. The one we choose: and

With this , so is reducible!

E) is it fully reducible?

Does an matrix exist such that ?
Let's consider and so that according to Matrix inverse#For 2x2 matrices.

Then

The offdiagonal elements must be so:

But if , all the whole matrix will be beceause and is everywhere.
So

Such matrix that makes the representation diagonal does not exist is not decomposable but reducible is not fully reducible.

F) what are the answers if we replace by ?

Every answer is the same. The only one that could change is c), but in this case it didn't.