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That second point is surely wrong as stated. $\mathbb{C}$ has characteristic $0$.
The reference is talking about rational valued characters.
Thanks. Looks like the coffee stopped working half way through the sentence.
Have expanded as intended now:
if the ground field $k$ has characteristic zero, then a character with values in the rational numbers in fact already takes values in the integers
in particular if the ground field $k =\mathbb{Q}$ is the rational numbers, then all characters take values in the actual integers.
So here is why I wanted to record this:
The computations I mentioned show that in all examples of interest, $\beta$ is surjective onto $R_{\mathbb{R}}^{int}(G) \subset R_{\mathbb{R}}(G)$, the sublattice of the representation/character ring over the real numbers on those that are integer valued.
This makes good sense for the physics application that I am after, since the values of the character at conjugacy class $[g]$ are supposed to be the RR-charge of the given D-brane in the $g$-twisted sector. And it seems weird for a charge to be irrational (we just saw that if its not integer, then its already irrational). In fact, there was once a little commotion in the field when people seemed to find irrational RR-charges (see the references here).
But I need to get a better grasps on how the actual values of a character on a conjugacy class translate into charge, hence into Chern character, under the isomorphism $R_{\mathbb{R}}(G) \simeq KO^0_G(\ast)$.
I was hoping to find this made explicit in
but maybe not.
I was hoping to find this made explicit in
- Michael F. Atiyah, Characters and cohomology of finite groups (PMIHES_1961__9__23_0)
but maybe not.
Oh, now I see that it (= the algebraic expression of the Chern character of a K-class on $BG$ in terms of the character of a corresponding representation in $R(G)$) is mentioned in Atiyah’s old article: First paragraph of the appendix, where it says that … this is an open problem(!)
I have forwarded this question to MO: here
Is there a way to derive such a formula from what is presented in the 7 page introduction to this thesis?
Characters are reformulated in terms of the cohomology of the loop space of $B G$. The Chern character appears via rationalized K-theory (p. 6).
Thanks for the pointer. That’s a nice account of transchromatic characters. Not sure yet if this is of use for what I am after, since I can neither restrict to $p$-groups nor to $p$-completions.
Another thing I noticed is that it was a bit pointless of me to speak of the Chern character instead of the Chern classes. On classifying spaces of finite groups all cohomology groups tend to be torsion, so that the actual Chern character, in the sense of rationalization, will just vanish away from degree 0.
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