Induced Class Functions and the Space of Integral Class Functions (Pt. I)
Point of Post: In this post we set up the frame work to say the Frobenius reciprocity theorem intelligently. Namely, we discuss the space of all linear characters of a group and describe the natural maps and .
The beginnings of the theory in this post can be traced back to our first derivation for the induced character of an induced representation when one noticed that by extending every (irreducible) character on to a character (thus a class function) we’ve automatically defined a map by extending the map by linearity (recalling that the irreducible characters form a basis for ). But, what we’ll note that is if we restrict to the space we get that . Thus when we restrict to we get a map . The subject of this post will be to explore theses topics and their dual concepts–namely the obviously defined map and .
Induced Class Functions
Let be a finite group, , and . Recall that we could then define the induced character on which has explicit formula
where is any transveral of . Since is a character and thus trivially a class function we now have a mapping thus we definitively have a map by extending by linearity. But this is very pedantic since we know that since we know is an (orthonormal) basis for . Moreover, we know that since we have proven that is a subalgebra of the group algebra . So, explicitly we have a map given by
We denote the map more formally as . While fairly obvious we prove the following fact: namely, if define by if and otherwise. Then,
Theorem: Let the map be as above, then for any transveral of and any one has
Proof: We merely note that by definition
for some . Thus, by definition
Note though that if that
and if then
the result then follows from .
1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Math. Soc., 1996. Print.