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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 28th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexam starting something here <a href="https://ncatlab.org/nlab/revision/Brauer+induction+theorem/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Brauer+induction+theorem">current</a>

   am starting something here

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 29th 2019
   • (edited Jan 29th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have tried to record the result of [Symonds 91](https://ncatlab.org/nlab/show/Brauer+induction+theorem#Symonds91) ("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 $L\big( f^\ast V\big) = f^\ast( L(V) )$; 1. $L$ 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 [[induced representation|induction]] and then sums everything up, in that the [[composition]] $\big( \sum ind \big) \circ L$ is the [[identity morphism|identity]]: $$ \big( \sum ind \big) \circ L(V) \coloneqq \underset{ [H \subset G] }{ \sum } ind_H^G\left[ L(V)_H \right] \;=\; V $$ 1. $L$ 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 [[commuting diagram|diagram commutes]]: $$ \array{ R_{\mathbb{C}}\big( G\big) &\overset{L}{\longrightarrow}& \underset{ [H \subset G] }{\prod} \mathbb{Z} \left[ 1dRep_{\mathbb{C}}\big( H \big)_{/\sim} \right] \\ {}^{\mathllap{c}}\Big\downarrow && \Big\downarrow {}^{ \underset{ [H \subset G] }{\prod} \left( c \circ \mathcal{N}_H^G \right) } \\ \underset{ k \in \mathbb{N} }{\prod} H^{2k}\big( B G, \mathbb{Z} \big) & \underset{ \underset{ [H \subset G] }{\smile} }{\longleftarrow} & \underset{ [H \subset G] }{\prod} \underset{ k \in \mathbb{Z} } {\Prod} H^{2k}\big( B G, \mathbb{Z}\big) } $$ 1. 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; 1. in contrast, if $V \in Rep_{\mathbb{C}}\big( G \big)_{/\sim}$ has no 1-dimensional [[direct sum|direct summand]], then the $G$-compnents of $L(V)$ is zero; <a href="https://ncatlab.org/nlab/revision/diff/Brauer+induction+theorem/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Brauer+induction+theorem/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Brauer+induction+theorem">current</a>

   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;

   diff, v3, current