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  1. #1
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    Question Representation Theory problems -->

    i've got a few adv. linear algebra problems that i need to do, but i'm not real knowledgeable on representation theory.
    here is the first and easiest problem of the bunch: problem #1

    also, with that i need to find the trace and norm of 2 - 3i.

    any help would be greatly appreciated.
    Last edited by TreyD0g; April 28th 2009 at 07:58 PM.
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  2. #2
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    here's another problem i'm working on: http://i475.photobucket.com/albums/r...de0flife/2.jpg
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  3. #3
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    Quote Originally Posted by TreyD0g View Post
    i've got a few adv. linear algebra problems that i need to do, but i'm not real knowledgeable on representation theory.
    here is the first and easiest problem of the bunch: problem #1

    also, with that i need to find the trace and norm of 2 - 3i.

    any help would be greatly appreciated.
    let B=\{\alpha_1, \alpha_2 \} and B'=\{1,i \}, the standard basis of \mathbb{C} over \mathbb{R}. then [T_{\alpha}]_B=P[T_{\alpha}]_{B'}P^{-1}, for some 2 \times 2 matrix P. thus N(\alpha)=\det [T_{\alpha}]_{B'} and Tr(\alpha)=Tr [T_{\alpha}]_{B'}.

    now let \alpha=x+yi, \ \beta=z+ti. then T_{\alpha}(1)=\alpha=x+yi and T_{\alpha}(i)=\alpha i = -y + xi, which gives us: [T_{\alpha}]_{B'}=\begin{bmatrix}x & -y \\ y & x \end{bmatrix}. thus N(\alpha)=x^2+y^2 and Tr(\alpha)=2x.

    similarly N(\beta)=z^2+t^2 and Tr(\beta)=2z. now we have: \alpha \beta = xz - yt + (xt + yz)i and hence: N(\alpha \beta)=(xz - yt)^2 + (xt + yz)^2=(x^2 + y^2)(z^2 + t^2)=N(\alpha) N(\beta).

    this proves part (a) of your problem. also it's clear that N(2 - 3i)=4+9=13 and Tr(2-3i)=4.
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  4. #4
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    ok, this problem has given me the most trouble so far. i'm not really sure where to start:
    http://i475.photobucket.com/albums/r...ber3ans010.jpg

    and again, i still haven't been able to solve this problem: http://i475.photobucket.com/albums/r...de0flife/2.jpg
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  5. #5
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    Quote Originally Posted by TreyD0g View Post

    ok, this problem has given me the most trouble so far. i'm not really sure where to start:
    http://i475.photobucket.com/albums/r...ber3ans010.jpg
    since G is a finite group, the order of a is finite, i.e. a^m=1_G, for some integer m. now if \lambda is an eigenvalue of \rho(a), then \rho(a)x=\lambda x, for some non-zero vector x.

    thus x=\rho(1_G)x=\rho(a^m)x=\lambda^m x. hence \lambda^m=1. so every eigenvalue \lambda of \rho(a) is a root of unity and thus \frac{1}{\lambda}=\overline{\lambda}. now \lambda is an eigenvalue of \rho(a^{-1}) if and only if

    \rho(a^{-1})x=\lambda x, for some vector x \neq 0, if and only if \rho(a)x=\frac{1}{\lambda}x, i.e. if and only if \frac{1}{\lambda} is an eigenvalue of \rho(a). so if \lambda_1, \cdots, \lambda_n are the eigenvalues of \rho(a), then the

    eigenvalues of \rho(a^{-1}) are: \overline{\lambda_1}=\frac{1}{\lambda_1}, \cdots , \overline{\lambda_n}=\frac{1}{\lambda_n}. therefore: \chi_{\rho}(a^{-1})=\sum \overline{\lambda_j}=\overline{\sum \lambda_j}=\overline{\chi_{\rho}(a)}.
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  6. #6
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    how do you prove something is a representation?
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  7. #7
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    Quote Originally Posted by TreyD0g View Post

    how do you prove something is a representation?
    you need to prove that the map is a group homomorphism.
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  8. #8
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    do you know how to do that for the last problem i posted?
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  9. #9
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    Quote Originally Posted by TreyD0g View Post
    do you know how to do that for the last problem i posted?
    the reason that i haven't answered that question is that it's quite long and, well, you should do some of the questions yourself!

    anyway, for part (a), you need to use the definition of \pi to prove two things:

    1) \pi(g'), or as the notation in your problem \pi_{g'}, is an automorphism of V for any g' \in G. this proves that \pi is well-defined.

    2) for any g_1,g_2 \in G: \ \ \pi(g_1g_2)=\pi(g_1) \pi(g_2), which will prove that \pi is a homomorphism and thus a representation of G.
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  10. #10
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    You're right. I should be doing these questions on my own, and i've spent hours focusing on this problem alone and haven't really gotten anywhere with it. I've even done a little research on Representation theory and i can't seem to find what i need to be able to solve this question.
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