Why E's representation marix is unit matrix

Click For Summary

Discussion Overview

The discussion centers on the representation of the identity element in group representation theory, specifically why the representation matrix of the identity element is the unit matrix. The scope includes theoretical aspects of group representation and the properties of matrices involved.

Discussion Character

  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant asks for an exact explanation of why the identity element's representation matrix is the unit matrix.
  • Another participant explains that group representation theory involves using matrices to represent group elements, suggesting that the identity element corresponds to the identity matrix.
  • A third participant expresses a desire for a proof of this assertion.
  • A later reply discusses the properties of the group identity, indicating that for any group member represented by a matrix, the identity matrix must satisfy certain equations, leading to a question about the representation matrix E.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the proof of why the identity element's representation matrix is the unit matrix, as some seek clarification while others provide explanations without resolving the inquiry.

Contextual Notes

The discussion lacks a formal proof and does not clarify the assumptions or definitions underlying the properties of the identity element and its representation.

xiaoxiaoyu
Messages
5
Reaction score
0
In group representation theory, identity element's representation matrix is unit matrix. But why? Could you give me an exact explanation? Thany you.
 
Physics news on Phys.org
Welcome to Physics Forums.

Group representation theory essentially boils down to using matrices to represent group elements. In other words, a group is described as a set of linear transformations on vector spaces. In this respect it is obvious that the identity element will be the identity matrix.
 
I know this, but I don't know how to prove that exactly
 
The group identity has the property that ae= a and ea= a for any group member a. Suppose e is "represented" by the matrix E and a by the matrix A. Then you must have AE= A and EA= A, for any matrix, A, in the group representation. Therefore E= ?
 

Similar threads

Undergrad How is unit norm axis rotation represented and derived in R3?
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Similarity transformation changing the determinant to 1
  • · Replies 20 ·
Replies
20
Views
3K
Undergrad Need help with tensors and group theory
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Adjoint Representation Confusion
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Representations of finite groups -- Equivalent representations
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Decomposing arbitrary representations
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Adjoint representation and the generators
  • · Replies 10 ·
Replies
10
Views
4K
Graduate Question about ##FG## modules
  • · Replies 14 ·
Replies
14
Views
4K
Graduate What is a Group Representation and How Does it Act on a Vector Space?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Jacobi matrix diagonalization problems
  • · Replies 5 ·
Replies
5
Views
3K