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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

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- Thread starter tgt
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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.

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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

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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).

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matt grime said:

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.

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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.

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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).

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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?

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dvs said: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.

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matt grime said:

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

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

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.

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.

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.

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.

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.

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