Induced Class Functions and the Space of Integral Class Functions (Pt. II)
Point of Post: This is a continuation of this post.
What we now noteis that the map is linear. Indeed:
Theorem: Let be a finite group and . Then the map is linear.
Proof: This follows immediately from the obvious fact that for any two and any one has
There’s no hope that will be a ring morphism since a quick check shows that for any .
That said, we clearly get the result that an isomorphic copy of the additive group sits inside .
Space of Integral Class Functions
We define the space as the space of integral class functions and denote it . It’s evident that is a subring of . The first thing we note about is its relation to . Namely:
Theorem: Let be a finite group and . Then, if is defined as usual it’s true that .
Proof: Since is linear we have that thus it suffices to prove that the containment holds. But, we know that is a character of for each and so from a previous theorem we have that . The conclusion follows.
Corollary: Let be some ordering of the basis and respectively. Then, (where are the number of conjugacy classes in and respectively).
Thus, we may restrict to get a mapping . We denote this map by .
Remark: The notation is chosen for two reasons: 1) it saves me a lot of typing (and writing at the white board) and 2) it suggests the ‘go upedness’ of the map.
Dual Notions of Restriction
There is a mapping which is dual, in a sense soon to be made precise, to . Namely, we are speaking the restriction mapping which is literally just restricting an element of to thus producing an element of . Accordingly we have the dualized map of , denoted , which maps given by the restriction of to . The interplay of and shall be the focus of our next few posts.
1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Math. Soc., 1996. Print.
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