Representations in group theory

In summary, the speaker is struggling with their mandatory group theory course in their undergraduate physics studies. They find the concepts to be abstract and difficult, with little help from their books or online resources. They have specific questions about representations, such as what makes a representation a "standard representation" and how to find and prove it, as well as techniques for finding irreducible representations and determining the number of irreps. They also mention a lack of clear examples, and request help with more complicated problems.
  • #1
Gulli
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The group theory course I'm taking is driving me crazy. It's a mandatory class in my undergraduate physics studies but it's all very alien and very abstract to me and my books scarcely give any examples when introducing new concepts. It's just so much harder than calculus or physics courses.

I'm studying representations now and have some questions about them that my books nor Google seem to answer in a clear manner:

1) What exactly makes a representation a "standard representation"? How do I find it when I've been given a random representation? How do I prove a representation is a standard representation, how do I prove it's not?

2) What's a general and useful technique to find irreducible representations (dividing the representation matrices into blocks)? How do I prove a representation is reducible, how do I prove it's not (apart from computing the trace of the matrix)? How do I know how many irreducible representations there are?

3) Is there a faster way to construct character tables than deriving all irreducible matrices of the representation?

I'm sorry if this seems like a lot or if the questions seem stupid but I'm just lacking clear examples. All I've got is a nice and easy example that reduces the standard representation of D3 into irreps by finding invariant vectors such as (1,1,1). Although this example taught me the basic concepts it does nothing to help me solve more complicated problems. If I could just see a more useful/less idealized example (such as V4 represented on the complex space in 3-D, or D3xC2) worked out properly that would help me a lot.
 
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  • #2
I'm sorry if this seems like a lot to ask but I'm really stuck and I don't know who else to turn to. Any help would be greatly appreciated.
 

1. What is a representation in group theory?

A representation in group theory is a way of associating elements of a group with linear transformations of a vector space. This allows us to study groups using tools from linear algebra.

2. How do representations help in understanding groups?

Representations help in understanding groups by providing a concrete and visual way to analyze group elements and their interactions. They also allow us to use techniques from linear algebra to study the properties of groups.

3. What are the different types of representations in group theory?

There are two main types of representations in group theory: faithful and non-faithful representations. Faithful representations preserve the group structure and can be used to study the group itself, while non-faithful representations do not preserve the group structure and are useful for studying the group's actions on other objects.

4. How do we construct representations of a group?

Representations of a group can be constructed by assigning group elements to invertible matrices of appropriate size. This is known as the matrix representation method. Alternatively, we can construct representations using the group's action on vector spaces, known as the permutation representation method.

5. What is the importance of representations in group theory?

Representations are important in group theory because they provide a powerful tool for studying groups and their properties. They also have applications in various fields such as physics, chemistry, and computer science, where groups are used to describe symmetries and transformations.

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