Classifying, Up To Equivalence, the Irreps of a Cyclic Group
Point of post: In this post we classify, up to equivalence, the irreps of a finite cyclic group. Of course as a consequence we shall have attained the character table for such a group.
Hidden secretly in our last post was the information to obtain the character table for any cyclic group. But, since it’s such a big statement (not in terms of difficulty, but for what comes later) that it’s probably prudent to put it in a post of its own. The basic idea is simple, namely since all cyclic groups are isomorphic to for some it suffices to find the irreps up to equivalence of . For that we can just exhibit them and since we know we’ll know we’re done when we’ve exhibited non-equivalent ones. The reason that this relates to the last post is that we implicitly already defined them there!
Irreps Up to Equivalence of a Finite Cyclic Group
Let be a finite group with . We shall now exhibit, for each , an element from where we will have a set of representatives from each equivalence class in . We shall then have the set of all irreducible characters by considering . But, since it will suffice to do this for . Indeed:
Theorem: Let , then for each define (where is given the usual structure) by . Then, is an irrep of and whenever . Moreover, for each there exists some such that .
Proof: The fact that each is irreducible is evident since they’re linear. To see that for we merely note that they induce different characters. The last conclusion follows since and since we may conclude that .
From this we are able to construct the character table for any . Namely the character table is the matrix where . Graphically,