- #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.
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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