Homework question regarding irreducible representations

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Discussion Overview

The discussion revolves around a homework question concerning irreducible representations of the C2 group, as well as related tasks for the C3 and C3v groups. Participants are exploring methods to find irreducible representations and diagonalizing matrices.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant expresses difficulty in finding irreducible representations for the C2 group after determining the regular representations.
  • Another participant suggests that finding the matrix M first, which diagonalizes D(reg), will lead to identifying the eigenspaces that correspond to the irreducible representations.
  • Some participants propose specific mappings, such as sending element a to 1 or -1, and discuss the applicability of these mappings over different fields, including R, C, and Q.
  • There is mention of the possibility of working with fields of characteristic 2, with one participant noting that a simple representation exists in this case, resulting in a double representation.

Areas of Agreement / Disagreement

Participants express varying views on the approach to finding irreducible representations and the role of the diagonalizing matrix. There is no consensus on the best method or the implications of working over different fields.

Contextual Notes

Participants have not resolved the dependencies on field characteristics and the implications for the representations. The discussion includes assumptions about the nature of the representations and the diagonalization process.

PhysKid24
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Hi,

I keep having problems with a homework question regarding irreducible representations. For the C2 group,which has only two elements,say, e and a, Iwas able to find the regular representations for them, yet i don't know how to find an irreducible representation for them. I'm also supposed to find a matrix M which diagonalizes D(reg)? Do I find M first and then the irreducible representation?Can anyone help?? I'm also to do the same procedure for group C3 and C3v. Thanks.
 
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PhysKid24 said:
Hi,

I keep having problems with a homework question regarding irreducible representations. For the C2 group,which has only two elements,say, e and a, Iwas able to find the regular representations for them, yet i don't know how to find an irreducible representation for them. I'm also supposed to find a matrix M which diagonalizes D(reg)? Do I find M first and then the irreducible representation?
Find M first. The eigenspaces will be the irreducible reps
 
Presuming you mean over R ro C or even Q.
what about the map sending a to 1, and the map sending a to -1 in any of these spaces?
 
matt grime said:
Presuming you mean over R ro C or even Q.
I think any field of characteristic other than 2 will work.
 
it even works with char 2. there is one simple rep in char 2, this gives it twice.
 

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