# Use of representation theory?

• tgt
It's very short and very cheap. In summary, representation theory is the study of maps between groups and matrices, and it has many applications in mathematics and physics. One example is the use of the determinant representation in the classification of finite simple groups and the proof of Burnside's pq theorem. Another example is the use of the representation of the group C_2 x C_2 x ... x C_2 in the fast Fourier transform. Representation theory is a fundamental concept in modern mathematics and has many practical applications in various fields. It is also a useful tool for studying different types of representations, not just over the complex numbers but over any field.

#### 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

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).

matt grime said:
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.

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,...

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.

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).

matt grime said:
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?

matt grime said:
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?

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.

## 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. It explores the connections between these algebraic objects and the geometric structures they represent. It has applications in physics, chemistry, and computer science, among others.

## 2. What are the main techniques used in representation theory?

The main techniques used in representation theory include the use of linear algebra, group theory, and homological algebra. Linear algebra is used to study the structure of vector spaces and their transformations, while group theory is used to study the symmetries of mathematical objects. Homological algebra is used to study the algebraic topology of spaces and their associated objects.

## 3. What are the applications of representation theory?

Representation theory has a wide range of applications in different fields. In physics, it is used to study the symmetries of physical systems, such as in quantum mechanics and particle physics. In chemistry, it is used to understand the properties of molecules and crystals. In computer science, it is used in the study of algorithms and data structures. It also has applications in coding theory, cryptography, and robotics.

## 4. What are the different types of representations?

There are several types of representations used in representation theory, including finite-dimensional representations, infinite-dimensional representations, and continuous representations. Finite-dimensional representations are used to study finite groups, while infinite-dimensional representations are used to study infinite groups. Continuous representations are used to study topological groups.

## 5. How is representation theory related to other branches of mathematics?

Representation theory has strong connections to other branches of mathematics, such as algebraic geometry, number theory, and combinatorics. It is also closely related to harmonic analysis, functional analysis, and mathematical physics. In addition, representation theory has applications in many other areas of mathematics, including topology, differential geometry, and probability theory.