Irreducible R-characters of a Finite Group in Terms of its Irreducible C-characters
Point of post: In this post we combine our results over the last few posts to give a complete description of the irreducible -characters of a finite group in terms of its irreducible -characters.
We’ve spent a lot of time in our last few posts finding a bijection between a subset (which we denoted by ) equivalency classes of of -representations and irreducible -representations, finding a way to create a bunch of elements of , and showing a necessary condition for membership in . Thus, we’ve completely ascertained, in a sense, the identity of and from this we shall show how to compute the irreducible -characters of from the irreducible -characters. From this we’ll derive the result that if denotes the equivalency classes of irreducible -representations then
with equality if and only if is real for every .
Characterization of Irreducible -characters
Let be a finite group, the set of all equivalency classes of -representations of which satisfy the real condition and if their realizer is then the representation space of has no non-trivial proper -invariant subspaces, and the set of all equivalency classes of irreducible -representations.
Theorem: If is an irreducible -character of where then either
Moreover, every character of the above three forms appears in .
Proof: We know from our result concerning the bijection of with that there exists some so that . But, we know that for each one has that is equivalent to where is some real irreducible -representation, for some quaternionic irreducible -representation, or for some complex irreducible -representation. But, since we see that these last three possibilities account for the three possibilities listed above. Moreover, since we know that each possibility is attainable the conclusion follows.
From this we get the two following fascinating corollaries:
(where ambivalent conjugacy classes are defined as before)\
Corollary: Let be a finite group and for each let denote its degree. Then,
And thus with equality if and only if every irrep of is real.
1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.
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