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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 28th 2019

am starting something here

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJan 29th 2019
• (edited Jan 29th 2019)

I have tried to record the result of Symonds 91 (“Symonds’ explicit Brauer induction”) in somewhat more modern form:

For $G \in$ FinGrp there is a linear map (homomorphism of abelian groups)

$R_{\mathbb{C}}\big( G \big) \overset {L} {\longrightarrow} \underset{ [H \subset G] }{\prod} \mathbb{Z} \left[ 1dRep_{\mathbb{C}}\big( H \big)_{/\sim} \right]$

from the underlying abelian group of the representation ring to the product group of the free abelian groups that are spanned by the isomorphism classes of 1-dimensional representations over all conjugacy classes of subgroup $H \subset G$,

such that

1. $L$ is a natural transformation of functors $FinGrp^{op} \to Ab$,

hence $L\big( f^\ast V\big) = f^\ast( L(V) )$;

2. $L$ is a section of the natural transformation

$\underset{ [H \subset G] }{\prod} R^{1d}_{\mathbb{Z}}\big( H\big) \overset {\sum ind} {\longrightarrow} R_{\mathbb{C}}\big( G \big)$

which applies induction and then sums everything up, in that the composition $\big( \sum ind \big) \circ L$ is the identity:

$\big( \sum ind \big) \circ L(V) \coloneqq \underset{ [H \subset G] }{ \sum } ind_H^G\left[ L(V)_H \right] \;=\; V$
3. $L$ is compatible with the total Chern classes of linear representations

$R_{\mathbb{C}}\big( G \big) \overset{c}{\longrightarrow} \underset{ k \in \mathbb{N} }{\prod} H^{2k}\big( B G, \mathbb{Z} \big)$

via their multiplicative transfer $\mathcal{N}_H^G$ (Lemma \ref{TransferEvens}) in that

$c \big( V \big) \;=\; \underset{ [H \subset G] }{\smile} \mathcal{N}_H^G \Big( c \big( L(V)_H \big) \Big) \,,$

hence in that the following diagram commutes:

$\array{ R_{\mathbb{C}}\big( G\big) &\overset{L}{\longrightarrow}& \underset{ [H \subset G] }{\prod} \mathbb{Z} \left[ 1dRep_{\mathbb{C}}\big( H \big)_{/\sim} \right] \\ {}^{\mathllap{c}}\Big\downarrow && \Big\downarrow {}^{ \underset{ [H \subset G] }{\prod} \left( c \circ \mathcal{N}_H^G \right) } \\ \underset{ k \in \mathbb{N} }{\prod} H^{2k}\big( B G, \mathbb{Z} \big) & \underset{ \underset{ [H \subset G] }{\smile} }{\longleftarrow} & \underset{ [H \subset G] }{\prod} \underset{ k \in \mathbb{Z} } {\Prod} H^{2k}\big( B G, \mathbb{Z}\big) }$
4. a 1-dimensional representation $W \in 1dRep\big(G\big)_{/\sim} \subset R_{\mathbb{C}}\big(G\big)$ is sent to the tuple $L(W) = (W,0,0, \cdots)$ whose component over $G \subset G$ is $V$ itself, and all whose other components vanish;

5. in contrast, if $V \in Rep_{\mathbb{C}}\big( G \big)_{/\sim}$ has no 1-dimensional direct summand, then the $G$-compnents of $L(V)$ is zero;