For restricted representations, this is easy. Let be a matrix representation of a group , and let be a subgroup. Then for any . We just consider an element of as an element in and construct the matrix as usual. Therefore we can see that
That is, we get the restricted character by restricting the original character.
As for the induced character, we use the matrix of the induced representation that we calculated last time. If is a matrix representation of a group , which is a subgroup , then we pick a transversal of in . Using our formula for the induced matrix, we find
Where we define if . Now, since is a class function on , conjugation by any element leaves it the same. That is,
for all and . So let’s do exactly this for each element of , add all the results together, and then divide by the number of elements of . That is, we write the above function out in different ways, add them all together, and divide by to get exactly what we started with:
But now as varies over the transversal, and as varies over , their product varies exactly once over . That is, every can be written in exactly one way in the form for some transversal element and subgroup element . Thus we find: