Representation theory question

In summary: A function belongs to a particular row of an irreducible representation if and only if the index of that row and the function's index within that representation are both good quantum numbers.
  • #1
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I'm self-studying representation theory for finite groups using "Group Theory and Quantum Mechanics" by Michael Tinkham. Most of it makes sense to me, but I'm having difficulty understanding what is meant by saying a function "belongs to a particular irreducible representation", or "has the symmetry of a particular irreducible representation". I assume this has something to do with a particular set of functions acting as a basis for an irreducible representation, but I still don't understand what is meant by "the symmetry of an irreducible representation". To make my question a bit more concrete, what do the last columns two columns in this character table (labeled "linear, rotations" and "quadratic") mean?

I have a couple other questions, but they're more specific to this book. I understand it's a fairly popular text, so if someone has a copy of it handy and wouldn't mind entertaining another question or two, I'll post them in a follow up comment.

Thanks!
 
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  • #2
Alright, I'm just going to go ahead and post my follow up question and hope it catches someone's eye. I'm actually feeling a bit more comfortable with what I originally asked about, with the exception of one thing. Here's the theorem I'm trying to understand:

"Matrix elements of an operator H which is invariant under all operations of a group vanish between functions belonging to different irreducible representations or to different rows of the same representation."

So, that would mean the (possibly) non-vanishing matrix elements are between functions belonging to the same row of the same irreducible rep. But... aren't just those just the same function? Once you choose a basis, I thought that the index of a particular irreducible representation and a row index within that rep uniquely identify a single function. Isn't that the point of the rep index and row index being "good quantum numbers"? The only way I can think of it kind of making sense of it is if the two functions correspond to a different choice of basis. But that doesn't really make sense either: suppose I have a three dimensional irreducible representation and I choose basis functions {ψ123}. According to the theorem, <ψ1|H|ψ2> = 0. However, I could choose a "new" basis {[itex]\phi[/itex]1,[itex]\phi[/itex]2,[itex]\phi[/itex]3} = {ψ213}. Now, going purely indices, ψ1 and [itex]\phi[/itex]1 would be considered belonging to the same row of the same irreducible rep. This would seem to imply, again by the theorem, that <ψ1|H|[itex]\phi[/itex]1> = <ψ1|H|ψ2> does not necessarily vanish. So, clearly my understanding of the notion "belong to a particular row of a particular irreducible representation" is not correct.

I hope I've explained the issue in a way clear to someone without access to the same text as me. However, for anyone who does, the theorem in question is in section 4.9 of Tinkham and the relevant material is developed in 3.8.

Can anyone help?
 
  • #3
LastOneStanding said:
So, that would mean the (possibly) non-vanishing matrix elements are between functions belonging to the same row of the same irreducible rep. But... aren't just those just the same function?

Yes.

I think your theorem is Schur's lemma, one statement of which is:

"Any operator that commutes with all operations of a group must be proportional to the identity within any given irreducible representation."

That is, if H commutes with all group operators, and |ψ> belongs to some irreducible representation R of the group, then H|ψ> = C(R)|ψ>, where C(R) is a number (not an operator) that depends on the particular irreducible representation R (but does not depend on the row within the representation).

I suspect you're trying to make this more complicated than it is. The point is that any operator that commutes with the whole group must act essentially trivially within any irreducible representation.

A generalization of this is Wigner-Eckhart theorem, which in part says that if H is an operator that belongs to a particular irreducible representation R1, and |ψ> is a state that belongs to a particular irreducible representation R2, then H|ψ> belongs to the direct product representation R1 x R2. In your case R1 is the trivial representation, so R1 x R2 = R2, that is H|ψ> belongs to the same representation as |ψ>.
 
  • #4
Thank you for the response, but I'm afraid I didn't understand it very well. I mean... I mostly follow what you're saying (though I'm not sure I understand what it means for an operator to belong to a representation), I'm just not making the connection between it and my question :-S Could you elaborate? I'm sorry, I wish I just I could ask a more specific question, but I'm just not seeing it. Could you start by explaining what it means for a function to belong to a particular row of an irr. representation? I thought I understood that, but from your response it seems that maybe I don't...
 
  • #5


I would first like to commend you for taking the initiative to self-study representation theory for finite groups. It is a complex and important subject in mathematics and physics, and your efforts will surely pay off.

To answer your question, let me start by explaining what an irreducible representation is. In representation theory, we study how groups act on vector spaces. An irreducible representation is a representation that cannot be broken down into smaller, simpler representations. In other words, it is a representation that has no proper invariant subspaces. This means that any vector in the representation, when acted upon by the group, will not be transformed into a vector outside of the representation.

Now, when we say that a function belongs to a particular irreducible representation, we mean that the function can be expressed as a linear combination of basis functions that transform under the same irreducible representation. In other words, the function shares the same symmetry as the irreducible representation. This is why the last two columns in the character table are labeled "linear, rotations" and "quadratic" - they represent the symmetries of the functions that belong to the corresponding irreducible representations.

As for your other questions, it would be helpful if you could provide more context or specific examples from the book. I would be happy to assist you further in understanding representation theory. Keep up the good work!
 

1. What is representation theory?

Representation theory is a branch of mathematics that studies how abstract algebraic structures, such as groups, rings, and algebras, can be represented by matrices and linear transformations. It provides a powerful tool for understanding and studying these structures in a more concrete way.

2. What are some practical applications of representation theory?

Representation theory has many applications in physics, chemistry, and engineering, where it is used to model and solve problems involving symmetry and group actions. It is also used in coding theory, cryptography, and data compression algorithms.

3. What is the difference between a finite and infinite-dimensional representation?

A finite-dimensional representation is one where the matrices representing the algebraic structure have a finite size, while infinite-dimensional representations have an infinite number of rows and columns. Finite-dimensional representations are easier to work with and have more applications, but infinite-dimensional representations are important in fields like quantum mechanics.

4. How is representation theory related to other areas of mathematics?

Representation theory has connections to many other branches of mathematics, including linear algebra, group theory, and algebraic geometry. It is also closely related to other areas of mathematical physics, such as quantum mechanics and statistical mechanics.

5. What is the importance of representation theory in modern mathematics?

Representation theory has become an essential tool in many areas of mathematics, providing powerful techniques for studying and understanding complex algebraic structures. It has also led to important breakthroughs in fields like number theory and algebraic geometry, and continues to be an active area of research with many open problems and applications.

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