# What is the derivation of the character formula for SU(2) representation?

• matematikawan
In summary, the dimension of the representation is the character evaluated on the unit matrix. This is easy to verify using the character formula for SU(2).
matematikawan
I'm trying to understand this paper on the representation of SU(2).

I know these definitions:
A representation of a group G is a homomorphism from G to a group of operator on a vector space V. The dimension of the representation is the dimension of the vector V.
If D(g) is a matrix realization of a representation, the character $$\chi (g)$$ is the trace of D(g).

The paper I'm reading state that the dimension of the representation is the character evaluated on the unit matrix. (***)

I try to confirm this with the character formula for SU(2) which is given as
$$\chi^j (\theta)=\frac{\sin(j+\frac{1}{2} )\theta}{\sin \frac{\theta}{2} }$$
where j labelled the irreducible representation.

So at unit matrix $$\chi (0) = 2j + 1$$ which is the correct dimension for the irreducible representation.

My question is how do we go about proving (***). I can't find the literature that proved this statement. Any clues ?

Think about the trace of the identity matrix. Count how many 1's it has down its diagonal.

The trace for the identity matrix (2j+1)X(2j+1) is 2j+1. That's easy!

Thank you so much ThirstyDog.

Really sorry I have to ask again. I'm already clear of my initial problem. My problem now is to understand the derivation for the character formula of SU(2) which is given by.
$$\chi^j (\theta)=\frac{\sin(j+\frac{1}{2} )\theta}{\sin \frac{\theta}{2} }$$

One book I'm reading now derived the above formula in the context of SO(3) as follows ( I think it should be ok because SU(2) and SO(3) share the same Lie algebra )

$$\chi^j (\theta)= \sum_m D^j[R_3(\theta)]_m^m = \sum_{m=-j}^{m=j} e^{-im\theta} =\frac{\sin(j+\frac{1}{2} )\theta}{\sin \frac{\theta}{2} }$$

I don't understand where does the exponential $$e^{-im\theta}$$ comes from?

Again any clues for this?

I'm in a different time zone. It is about 2am now. I have to :zzz: and hope someone could help.

## 1. What is the purpose of studying the character of a representation?

The character of a representation is studied in order to better understand the properties and behavior of a system or phenomenon being represented. It can provide insights into underlying patterns and relationships, and help make predictions about future outcomes.

## 2. How is the character of a representation determined?

The character of a representation is determined by analyzing the various components or elements that make up the representation, as well as their interactions and relationships. This can involve methods such as data analysis, modeling, and experimentation.

## 3. Can the character of a representation change over time?

Yes, the character of a representation can change over time as new information is gathered or as the system being represented evolves. Additionally, different perspectives or interpretations can also influence the character of a representation.

## 4. What factors can influence the character of a representation?

The character of a representation can be influenced by various factors such as the type of data being used, the methods and tools used to analyze and represent the data, the context in which the representation is being used, and the biases and assumptions of the researchers or scientists involved.

## 5. How can understanding the character of a representation be beneficial in the scientific community?

Understanding the character of a representation can be beneficial in the scientific community by providing a deeper understanding of complex systems and phenomena, facilitating more accurate and reliable predictions and interpretations, and promoting collaboration and communication among researchers and scientists.

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