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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; Apr 28th 2009 at 06:58 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 $\displaystyle B=\{\alpha_1, \alpha_2 \}$ and $\displaystyle B'=\{1,i \},$ the standard basis of $\displaystyle \mathbb{C}$ over $\displaystyle \mathbb{R}.$ then $\displaystyle [T_{\alpha}]_B=P[T_{\alpha}]_{B'}P^{-1},$ for some $\displaystyle 2 \times 2$ matrix $\displaystyle P.$ thus $\displaystyle N(\alpha)=\det [T_{\alpha}]_{B'}$ and $\displaystyle Tr(\alpha)=Tr [T_{\alpha}]_{B'}.$

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

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

    this proves part (a) of your problem. also it's clear that $\displaystyle N(2 - 3i)=4+9=13$ and $\displaystyle Tr(2-3i)=4.$
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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 $\displaystyle G$ is a finite group, the order of $\displaystyle a$ is finite, i.e. $\displaystyle a^m=1_G,$ for some integer $\displaystyle m.$ now if $\displaystyle \lambda$ is an eigenvalue of $\displaystyle \rho(a),$ then $\displaystyle \rho(a)x=\lambda x,$ for some non-zero vector $\displaystyle x.$

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

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

    eigenvalues of $\displaystyle \rho(a^{-1})$ are: $\displaystyle \overline{\lambda_1}=\frac{1}{\lambda_1}, \cdots , \overline{\lambda_n}=\frac{1}{\lambda_n}.$ therefore: $\displaystyle \chi_{\rho}(a^{-1})=\sum \overline{\lambda_j}=\overline{\sum \lambda_j}=\overline{\chi_{\rho}(a)}.$
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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 $\displaystyle \pi$ to prove two things:

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

    2) for any $\displaystyle g_1,g_2 \in G: \ \ \pi(g_1g_2)=\pi(g_1) \pi(g_2),$ which will prove that $\displaystyle \pi$ is a homomorphism and thus a representation of $\displaystyle G.$
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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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