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Math Help - 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 Ind^{Q}_{H} (\chi), where H is a subgroup of Q of index two and \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 \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 Ind^{Q}_{H} (\chi), where H is a subgroup of Q of index two and \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 \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 \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 \chi is the character of your representation \Phi, then verify that <\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 \rho that sends i to i is a one-dimensional representation such that:

    \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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