A question on irreducible representation

In summary, irreducible representations are simpler representations of groups, and it is possible to find a basis for them if one exists.
  • #1
sineontheline
18
0
I'm having trouble grasping what an irreducible representation is. Can someone explain what this is through an example using SU(2) and/or SO(3) w/o invoking the use of characteristic? Like I'm reading a bunch of stuff but I'm not catching the significance of any of it. ...Imagine you were explaining to a physics student (not so interested in the math, so much what information it gives you). :)
 
Physics news on Phys.org
  • #2
sineontheline said:
I'm having trouble grasping what an irreducible representation is. Can someone explain what this is through an example using SU(2) and/or SO(3) w/o invoking the use of characteristic? Like I'm reading a bunch of stuff but I'm not catching the significance of any of it. ...Imagine you were explaining to a physics student (not so interested in the math, so much what information it gives you). :)

Hi sineontheline,

welcome to the Forum and Happy new Year!

Basically, a representation of a group is a mapping which assigns an operator (or a matrix) in a vector space to each element of the group, so that all group properties are preserved. In the vector space you can choose different bases, so the actual form of matrices changes, but the vital properties of the representation are not affected. Sometimes you can stumble upon a basis in which all matrices of the representation take the block-diagonal form simultaneously. Then, the representation is called "reducible". In this case the vector space gets separated into two subspaces, such that each subspace has its own independent representation of the group. The reducible representation is said to be a direct sum of two (or more) representations.

It might happen that it is impossible to find a basis in which the above separation occurs. Then the representation is called "irreducible". So, irreducible representation are in some sense "simplest" ones. Reducible representations can be built as direct sums of any number of irreducible ones.

Eugene.
 
  • #3
hey u noticed it was my first post!
ok, that makes sense -- thx

is there anything special about the basis? is there a way to find it if it exists, or do you just 'stumble on it'?
 
Last edited:
  • #4
sineontheline said:
hey u noticed it was my first post!
ok, that makes sense -- thx

is there anything special about the basis? is there a way to find it if it exists, or do you just 'stumble on it'?

There are various methods to decide whether a representation is reducible or not. There are also ways to select the basis for the block-diagonal form. However, I don't think there exists a unique prescription working for all cases.

Eugene.
 

1. What is an irreducible representation?

An irreducible representation is a mathematical concept used in group theory that describes the different ways in which a group of symmetries can act on a set of objects. It is the simplest and most basic representation of a group, meaning it cannot be broken down into smaller, independent parts.

2. How is an irreducible representation different from a reducible representation?

A reducible representation can be broken down into smaller, independent parts, while an irreducible representation cannot. In other words, a reducible representation can be decomposed into a direct sum of irreducible representations, whereas an irreducible representation cannot be further decomposed.

3. Why are irreducible representations important in science?

Irreducible representations are important because they allow us to understand the symmetries present in a system. They provide a mathematical framework for studying the properties and behaviors of physical systems, such as molecules, crystals, and particles. They also have applications in many areas of science, including physics, chemistry, and materials science.

4. How are irreducible representations used in quantum mechanics?

In quantum mechanics, irreducible representations are used to describe the symmetries of particles and their interactions. They are an essential tool for understanding the properties of particles and predicting their behavior. In addition, the concept of irreducible representations is used in the study of quantum states and operators, which are fundamental to quantum mechanics.

5. Can irreducible representations be visualized?

Yes, it is possible to visualize irreducible representations using diagrams or matrices. For example, in the study of molecular symmetry, irreducible representations are often represented by character tables, which are a visual representation of the symmetry operations and corresponding irreducible representations. However, since irreducible representations are abstract mathematical concepts, they cannot always be directly visualized.

Similar threads

  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
3
Views
2K
  • Special and General Relativity
Replies
22
Views
2K
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
881
  • Linear and Abstract Algebra
Replies
1
Views
2K
  • Topology and Analysis
Replies
5
Views
2K
Back
Top