## Inducing the Trivial Representation

We really should see an example of inducing a representation. One example we’ll find extremely useful is when we start with the trivial representation.

So, let be a group and be a subgroup. Since this will be coming up a bunch, let’s just start writing for the trivial representation that sends each element of to the matrix . We want to consider the induced representation .

Well, we have a matrix representation, so we look at the induced matrix representation. We have to pick a transversal for the subgroup in . Then we have the induced matrix in block form:

In this case, each “block” is just a number, and it’s either or , depending on whether is in or not. But if , then latex g(t_jH)=(t_iH)$. That is, this is exactly the coset representation of corresponding to . And so all of these coset representations arise as induced representations.

[...] now we can define the -module by inducing the trivial representation from the subgroup to all of . Now, the are not all irreducible, but we will see how to identify a [...]

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