Representations of finite groups: Irreducible and reducible

In summary, a representation of a finite group is a way of describing group elements through linear transformations on a finite-dimensional vector space. An irreducible representation cannot be broken down into smaller representations, while a reducible representation can. The character of a representation can be used to determine if it is irreducible or reducible. Representations of finite groups have various applications in mathematics, physics, and other fields.
  • #1
LagrangeEuler
717
20
Matrix representation of a finite group G is irreducible representation if
[tex]\sum^n_{i=1}|\chi_i|^2=|G|[/tex].
Representation is reducible if
[tex]\sum^n_{i=1}|\chi_i|^2>|G|[/tex].
What if
[tex]\sum^n_{i=1}|\chi_i|^2<|G|[/tex].
Are then multiplication of matrices form a group? If yes what we can say from ##\sum^n_{i=1}|\chi_i|^2<|G|##. ##\chi_i## are characters (traces of matrices).
 
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  • #2
You cannot have that situation.
 

1. What is a representation of a finite group?

A representation of a finite group is a way of describing the group's elements using matrices. It allows us to study the group's structure and properties through linear algebra.

2. What does it mean for a representation to be irreducible?

An irreducible representation is one that cannot be broken down into smaller representations. In other words, there are no non-trivial subspaces that are invariant under the group's action.

3. How do you determine if a representation is reducible?

A representation is reducible if it can be broken down into smaller representations. This can be determined by finding non-trivial subspaces that are invariant under the group's action.

4. What is the significance of irreducible representations in the study of finite groups?

Irreducible representations are important because they provide a way to decompose a group into simpler parts, making it easier to understand and analyze. They also have applications in physics, chemistry, and other areas of mathematics.

5. Can a finite group have more than one irreducible representation?

Yes, a finite group can have multiple irreducible representations. In fact, the number of irreducible representations of a group is equal to the number of conjugacy classes in the group.

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