The Character of an Induced Representation
Point of Post: In this post we discuss the notion of the induced character of an induced representation and provide a formulaic relationship between the induced character and the original character.
As we have already seen given a finite group a subgroup and a representation we can create a representation . Of course though, representations aren’t just the main focus in representation theory, we are also heavily interested in the representations characters since they holds so much information about the representations they come from as well as the underlying group as well. Thus, in this post we derive a relationship between the character of a representation of and the character of the associated induced representation for .
Let be a finite group and . Furthermore, suppose that is a representation and consider the associated induced representation . Let then be the character of . We say then that the character of is induced by the character $ and denote it by when convenient. We now find a formulaic relationship between and . Indeed:
Theorem: Let be a finite group, , and be a representation with character . Let denote the associated induced representation for some choice of transveral of and the associated character. Then, for any one has
where, for every , one has that if and otherwise.
Proof: We begin by fixing an ordered basis for and note that it’s obvious then that if we order the transveral as then is an ordered basis for . We next define then, for every , the matrix elements of as . Then, we note that by definition
We next define to be the matrix and we seek to find the diagonal entries of since their sum will be equal to . To do this we note that by definition
clearly then the diagonal entry associated to will be zero unless . What we claim is that this is true precisely when . Indeed, suppose first that then by definition there exists some such that so that . Conversely, if then since one has that . Thus, the diagonal entry related to will be zero unless . Note then that if this is true we have by the above discussion that . It’s clear to see then that if denotes the diagonal entry of associated to then
Thus, it clearly follows that