Group Action of C3v on 2x2 Complex Matrices

[filip97]

Homework Statement

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Relevant Equations

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Group action on $2$x$2$ complex matrices of group $C_{3v}$ for all matrices from $C^{22}$, for all $g$ from $C_{3v}$ is given by:

$D(g)A=E(g)AE(g^{-1}), A=\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$,

Where $E$ is $2D$ irrep of $C_{3v}$. I think that representation of rep $D$ is two dimensional, because act on $2$x$2$ matrix $A$.

when I calculate characters

$\chi_D(g)$ by
$tr(D(g)A)=\sum_{i,j=1,1}^{2,2} D(g)^{\dagger}_{ij}(g)A_{ij}=$
$=aD^{*}_{11}(g)+bD^{*}_{21}(g)+cD^{*}_{12}(g)+dD^{*}_{22}(g)$.
I always get for all $g, D^{*}_{11}(g)=D^{*}_{11}(g)=1$ and for
$D^{*}_{21}(g)=D^{*}_{12}(g)=0$, and $\chi_D(g)=2$ for all $g$. And $D=A\oplus A$
Main question is how decompose representation $D$ in irreducible components of group $C_{3v}$

 

[Antarres]

Assuming irreducible representations of the group are known, you find irreducible characters of those representations. Then you take the character of your chosen representation and use the orthogonality relations between the characters to find the number of occurences of the chosen irreducible representation in your reducible one.

For example, one of the irreducible representations of $C_{3v}$ is $A_0$, which is equal to $1$ for all elements of the group. We want to find whether this representation is occurring in our reducible one, and how many times. It's irreducible character for each element is equal to $1$. So we use the relation:

$$\alpha_{A_0} = \frac{1}{\vert G\vert} \sum_g \chi^{A_0}(g)\chi^{D}(g) = \frac{1}{6} \cdot12 = 2$$
Here I assumed that all characters of $D$ are equal to $2$ like you said. Therefore, we see that the IR $A_0$ occurs twice, so $D$ is decomposed into two of those irreducible representations(which in this case means that $D$ is identity transformation on that space).

In general, you can create character tables for irreducible characters for all irreducible representations of a group which are in general known, and then calculate the characters of your given representation, and in the end use the relations above(the orthogonality relations) to find numbers of occurences of each irreducible representation.

If you want to find the basis in which the transformation assumes this block-diagonal form, the so called symmetry-adapted basis, you need to use the group projecton operator technique, which is probably found in books on group representation theory.

 

[filip97]

Problem is biger. How we find representation $D$ ?

I assumed that $D$is $2$x$2$ matrices. $D(C_3)A=E(C_3)AE(C_3^{-1})=\begin{bmatrix} D_{11} & D_{12}\\ D_{21} & D_{12}\end{bmatrix} \begin{bmatrix} a & b\\ c & d\end{bmatrix}=\begin{bmatrix} aD_{11} + cD_{12}&bD_{11} + dD_{12}\\ aD_{21} + cD_{22} & bD_{21} + dD_{22}\end{bmatrix} = \begin{bmatrix} a&b\epsilon^*\\ c\epsilon^* & d\end{bmatrix}.$

Implies that $D_{22}=2$ and $D_{22}=\epsilon$

$\epsilon=e^{\frac{i2\pi}{3}}$

 

[Antarres]

The dimension of your representation will be the dimension of vector space in which it is acting. So here, from what I see, you want to find the way this representation transforms elements of the space of $2\times 2$ complex matrix space. But those matrices can be treated as operators over a two component complex vector space(say, the spin space of 1/2 spin particles). Then you will have the transformation rules:
$$\psi' = D\psi \qquad A' = DAD^{-1}$$
where $D=D(g)$ of course.
But there are many possible representations of this group, you need to define the representation in order to reduce it and inspect it's properties, there isn't just one particular representation for each possible space. Representation is just a function that maps elements of the group into some linear operators. The only rule here is that this mapping preserves group multiplication law. Otherwise, it can be completely arbitrary. So you can't find general $D$ for your space, you must adapt $D$ to the physical problem you're trying to solve, or you must be given a particular $D$ who's properties are sought for.