A question on irreducible representation

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Discussion Overview

The discussion revolves around the concept of irreducible representations in the context of group theory, specifically focusing on examples from SU(2) and SO(3). Participants explore the significance of these representations and the conditions under which they are considered irreducible or reducible.

Discussion Character

  • Conceptual clarification
  • Technical explanation
  • Exploratory

Main Points Raised

  • One participant seeks clarification on irreducible representations and their significance, particularly for a physics student with limited interest in mathematical details.
  • Another participant explains that a representation maps group elements to operators in a vector space, and that irreducible representations cannot be separated into independent subspaces.
  • It is noted that reducible representations can be expressed as direct sums of irreducible ones.
  • A question is raised about the nature of the basis used in representations and whether there are systematic methods to find a basis that reveals reducibility.
  • The response indicates that while there are methods to determine reducibility and select bases, no universal approach applies to all cases.

Areas of Agreement / Disagreement

Participants generally agree on the definitions and properties of irreducible and reducible representations, but there is uncertainty regarding the methods for finding appropriate bases and the existence of a unique prescription for all cases.

Contextual Notes

The discussion highlights the complexity of determining the reducibility of representations and the lack of a universal method for identifying bases, which may depend on specific cases or contexts.

sineontheline
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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). :)
 
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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.
 
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:
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.
 

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