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Thread: Induced Representations

  1. #1
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    Induced Representations

    I have to show that the two-dimensional irreducible representation of the quaternion group is isomorphic to an induced representation $\displaystyle Ind^{Q}_{H} (\chi)$, where $\displaystyle H$ is a subgroup of $\displaystyle Q$ of index two and $\displaystyle \chi: H \rightarrow C^{\times}$ is a one-dimensional character.

    For starters, I don't even know the two-dimensional irreducible representation of the quaternion group! And I'm not really sure how to progress past that, even, since I'm not comfortable with the induced representation over $\displaystyle \chi$ and not a vector space...any help is much appreciated!

    About the only thing I realize here is that the subgroup H is normal in Q...but I'm not even sure if or how that helps me.
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  2. #2
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    Quote Originally Posted by jpps1 View Post
    I have to show that the two-dimensional irreducible representation of the quaternion group is isomorphic to an induced representation $\displaystyle Ind^{Q}_{H} (\chi)$, where $\displaystyle H$ is a subgroup of $\displaystyle Q$ of index two and $\displaystyle \chi: H \rightarrow C^{\times}$ is a one-dimensional character.

    For starters, I don't even know the two-dimensional irreducible representation of the quaternion group! And I'm not really sure how to progress past that, even, since I'm not comfortable with the induced representation over $\displaystyle \chi$ and not a vector space...any help is much appreciated!

    About the only thing I realize here is that the subgroup H is normal in Q...but I'm not even sure if or how that helps me.
    Choose H = {1, -1, i, -i} and its one-dimensional representation as a representation <i> that sends i to i.

    Now you need to induce this one dimensional representation to two dimensional representation.



    Use this induced representation formula, where $\displaystyle \rho$ is your one dimensional representation. Here, x_1, ..., x_t denotes the representatives for distinct left cosets of H in G. Thus choose x_1 = j and x_2 = 1.

    Now you need to find the representation of each element in Q_8 using the above formula. Verify that it is an indeed representation.

    You also need to show that the representation that you'll obtain is irreducible. If the $\displaystyle \chi$ is the character of your representation $\displaystyle \Phi$, then verify that $\displaystyle <\chi, \chi> =1 $ in order to show it is irreducible.
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  3. #3
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    So let me see if I follow.

    I only need to choose H={1,-1,i,-i}? I don't need to choose any other subgroups of Q?

    By <i>, you just mean the identity representation? I'm kind of confused what you mean here...might just be the notation.

    Anyways, once that is sorted out, I form 2x2 matrices for each g? Are the elements g coming from H or from Q? Why are the distinct left cosets x_1=j and x_2=1?

    Sorry, I know I'm really retarded...as soon as I'm asked to do something explicitly rather than link theorems together, I don't know what to do.
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  4. #4
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    Quote Originally Posted by jpps1 View Post
    So let me see if I follow.

    I only need to choose H={1,-1,i,-i}? I don't need to choose any other subgroups of Q?

    By <i>, you just mean the identity representation? I'm kind of confused what you mean here...might just be the notation.

    Anyways, once that is sorted out, I form 2x2 matrices for each g? Are the elements g coming from H or from Q? Why are the distinct left cosets x_1=j and x_2=1?

    Sorry, I know I'm really retarded...as soon as I'm asked to do something explicitly rather than link theorems together, I don't know what to do.
    <i> = H and H is the cyclic group generated by i. For instance, the representation $\displaystyle \rho$ that sends i to i is a one-dimensional representation such that:

    $\displaystyle \rho(1)=1, \rho(-1)=-1, \rho(i)=i, \rho(-i)=-i$.

    The g you mentioned comes from Q. You can find the figure I used in Dummit & Foote p 893. To find the induced representation from the subgroup H to G, you need to find the representatives for the left cosets of H in G. There are examples and descriptions in Dummit's.
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