A question on irreducible representation

[sineontheline]

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). :)

 

[meopemuk]

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.

 

[sineontheline]

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'?

 

[meopemuk]

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.