Can Representation Theory Be Applied to Homomorphisms and Finite Abelian Groups?

[tgt]

What's the use of it? Anyone show a simple but illustrative example of the usefulness of representation theory?

I can see how faithful representations might be useful but not fully. What I can't imagine is how unfaithful representations can be of any use.

Thanks

 

[matt grime]

I find these questions hard to answer, not being a physicist for example, which is the most obvious 'application' of rep theory.

But something sprung to mind. Surely you agree that the determinant of a matrix is a useful thing? Well, that's an unfaithful representation for you.

In general one wants to study all representations, and not just over the complex numbers but any field. This was an integral part of the classification of finite simple groups (and note that a simple group is precisely a group with one simple non-faithful representation).

 

[tgt]

I find these questions hard to answer, not being a physicist for example, which is the most obvious 'application' of rep theory.

But something sprung to mind. Surely you agree that the determinant of a matrix is a useful thing? Well, that's an unfaithful representation for you.

In general one wants to study all representations, and not just over the complex numbers but any field. This was an integral part of the classification of finite simple groups (and note that a simple group is precisely a group with one simple non-faithful representation).

How about just answer this question. Applying to groups. "Anyone show a simple but illustrative example of the usefulness of representation theory?"

There are many examples in textbooks but it would be good if someone can show one representation and state why it's important.

 

[matt grime]

Just look at the representation theory of the very simple group C_2 x C_2 x ... x C_2 . That gives you the fast Fourier transform. Will that do? Audrey Terras has written an entire book about some elementary applications of reps of finite abelian groups, such as the FFT, spectra of graphs,...

 

[dvs]

A famous application of rep theory to group theory is the proof of Burnside's pq theorem, see: http://en.wikipedia.org/wiki/Burnside_theorem. And as matt mentioned, another famous example is the classification of finite simple groups, which wouldn't have been completed without the use of rep theory.

In any case, representations are ubiquitous in all of math, from knot theory to differential equations to algebraic geometry to combinatorics to... It's a basic idea in modern math to look at maps between things.

The rep theory of Lie groups has many applications to physics and even to chemistry.

 

[matt grime]

I'm still not sure why the OP didn't accept the det representation as being an important one.

The back of James and Liebeck shows how to work out something to do with the energy levels in some molecule via the representations of S_3 (I think - it is some years since I read it and I no longer own a copy).

 

[tgt]

I'm still not sure why the OP didn't accept the det representation as being an important one.
QUOTE]

That's because I don't understand it. Group representations are maps between groups and matrices. How does the determinant come in? Would you be able to explain that example in more detail?

 

[tgt]

A famous application of rep theory to group theory is the proof of Burnside's pq theorem, see: http://en.wikipedia.org/wiki/Burnside_theorem.

That example is a bit too complicated.

 

[tgt]

Just look at the representation theory of the very simple group C_2 x C_2 x ... x C_2 . That gives you the fast Fourier transform. Will that do? Audrey Terras has written an entire book about some elementary applications of reps of finite abelian groups, such as the FFT, spectra of graphs,...

How does C_2 x C_2 x ... x C_2 give the fast Fourier transform?

 

[matt grime]

The map det: GL_n(k) --> k is a homomorphism, i.e. a representation of GL_n (and hence any subgroup of GL_n.

For more the FFT get hold of a copy of Terras's book on Fourier Analysis of Finite Abelian groups.