Representation theory

In summary, the speaker is seeking help with understanding the Gell Mann matrices and representation theory. They mention that they have online versions of Hall's and Humphrey's books, but are struggling to understand the concepts. They mention knowing some information about the root system of su(3) and its Weyl group, but are unsure of how to determine the Weyl group from the root diagrams. They also mention a connection between the root systems of A_n and the alternating group, but are unsure of the specifics. The speaker is seeking clarification on the algorithm for finding the Weyl group, Cartan subalgebras, and determining representations of Lie algebras.
  • #1
Jim Kata
197
6
I'm trying to derive the Gell Mann matrices for fun, but I really don't know representation theory. Can somebody help me? I have Hall's and Humphrey's books, but only online versions and staring at them makes me go crosseyed, and I'm to broke to buy a real book on representation theory. Let me tell you what I kind of know, and maybe you can help me from there.

So, I know the root system of su(3) is [tex]A_2[/tex], ok? Now the Weyl group for [tex]A_2[/tex] is given by dihedral three, the symmetries of an equilateral triangle, fine. I don't really understand how you go from looking at [tex]A_2[/tex] to determining that its Weyl group is dihedral three. I understand it has rotational symmetry like cyclic six, but whatever. From the root diagrams, I should be able to determine the weights of the representation and hence the matrices that make up the Cartan subalgebra of su(3), I guess the vertices of the triangle being the weights I think. The elements of the Weyl group act like ladder operators on the Cartan subalgebra?
Can someone simply spell out the algorithm from looking at root diagrams to finding the Weyl group, Cartan subalgebras and determining the representations of the lie algebras. I understand su(2) pretty well. I also heard (this weeks finds John Baez) that the root systems of [tex]A_n[/tex] were related to the alternating group, but I don't see it. Dihedral three is not alternating two.
 
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  • #2
I may have quoted Baez wrong, I think he was talking about the root systems of [tex]A_n[/tex] being the same as the symmetries of an n simplex
 
  • #3



Representation theory is a complex and important topic in mathematics that has applications in various fields such as physics, chemistry, and computer science. It deals with the study of how groups, such as symmetries, act on vector spaces. In your case, you are specifically interested in the representation theory of Lie algebras, which are mathematical structures that have applications in physics, particularly in quantum mechanics.

To start, let's define what a root system is. A root system is a set of vectors in a vector space that satisfy certain properties. In the case of su(3), the root system is called A_2 and it consists of two vectors that form an equilateral triangle. The Weyl group, as you mentioned, is a group that is associated with a root system and it describes the symmetries of that root system. In the case of A_2, the Weyl group is dihedral three, which is the group of symmetries of an equilateral triangle.

Now, to answer your question about how to go from looking at A_2 to determining its Weyl group, Cartan subalgebras, and representations of the Lie algebra, there are several steps involved. First, you need to understand the structure of the root system and how it relates to the Lie algebra. This involves studying the properties of the root system, such as the length and angle between roots, and how they correspond to the structure of the Lie algebra.

Next, you need to determine the Cartan subalgebra, which is a subalgebra of the Lie algebra that contains the maximal set of commuting elements. In other words, it is a set of elements that can be simultaneously diagonalized. In the case of su(3), the Cartan subalgebra is spanned by two elements, which correspond to the two roots in the root system A_2.

Once you have determined the Cartan subalgebra, you can use it to construct the weight space of the representation. The weight space is a vector space that contains all the possible weights of the representation. In the case of su(3), the weights are given by the vertices of the equilateral triangle in the root system.

Finally, you need to determine the action of the Weyl group on the weight space. As you mentioned, the elements of the Weyl group act as ladder operators on the Cartan subalgebra, which in turn affects the weights of
 

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 concretely as matrices or linear transformations on vector spaces. It provides a powerful tool for understanding the structure and behavior of these abstract objects.

2. How is Representation Theory used in other fields?

Representation theory has applications in various fields, including physics, chemistry, and computer science. In physics, it is used to study symmetries and conservation laws in quantum mechanics and relativity. In chemistry, it is used to understand the electronic structure of molecules. In computer science, it is used to design efficient algorithms for solving problems related to symmetry and group actions.

3. What are the main concepts in Representation Theory?

The main concepts in Representation Theory include group representations, character theory, irreducible representations, and tensor products. Group representations are mappings that preserve the group structure, while character theory studies the properties of these mappings. Irreducible representations are those that cannot be decomposed into smaller representations, and tensor products allow us to construct new representations from existing ones.

4. What are some real-world examples of Representation Theory?

One example of Representation Theory in the real world is the study of crystal structures in materials science. The symmetry of a crystal can be described by a group, and the different ways this symmetry can be represented using matrices or linear transformations can provide insights into the physical properties of the material. Another example is the use of representation theory in coding theory, where error-correcting codes are constructed based on group representations.

5. What are the current research topics in Representation Theory?

Current research topics in Representation Theory include the study of modular representations, which arise when working with finite groups over fields with positive characteristic. Other areas of active research include the representation theory of Lie algebras and algebraic groups, and the connections between representation theory and other areas of mathematics, such as combinatorics and topology.

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