1. ## 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.

2. Originally Posted by jpps1
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.

3. 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.

4. Originally Posted by jpps1
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.