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Character table for cyclic group of order 7

  1. Feb 23, 2014 #1

    CAF123

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    Gold Member

    1. The problem statement, all variables and given/known data
    a)Write down all irreducible representations of ##\mathbb{Z}_7##.
    b)How many of the irreducible representations are faithful?

    2. Relevant equations
    Group structure of ##\mathbb{Z}_7 = \left\{e^{2\pi in/7}, \cdot \right\}## for ##n \in \left\{0,....,6\right\}##

    3. The attempt at a solution
    I am trying to construct the character table of ##\mathbb{Z}_7##. Since this is an abelian group, there are 7 conjugacy classes and thus 7 one dimensional irreducible representations. Since the representations are all 1D, the character of the representation is simply the representation itself. My question is are the representations of the form ##e^{2\pi in/7}?## Letting n run from 0 to 6 does indeed give 7 irreducible representations.

    I have in some notes the following statement: Let ##\rho## denote a representation. Then $$\rho(1)^7 = \rho(1^7) = \rho(1) = 1$$ Since ##\rho## is always 1D, it can be written as a complex scalar. Let ##x = \rho(1) \Rightarrow x^7 - 1 = 0## and thus the seven irreducible reps of 1 are of the form ##e^{2\pi in/7}## Do I do a similar analysis for the other elements? I wasn't sure though why ##\rho(1^7) = \rho(1)## though. ##1 \in \mathbb{Z}_7 = \left\{n, + | n \in \left\{0,...,6\right\}\right\}##, so I think ##1^7 = 1 + 1...+1 = 0## mod## 7##, by summing 1 seven times.

    To mentors: This is from a course on symmetries of QM, so that is why I posted it here. Perhaps it is better elsewhere.

    Thanks.
     
  2. jcsd
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