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\begin{document}

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\section*{Brauer induction theorem}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{SnaithExpansion}{Snaith's explicit Brauer induction}\dotfill \pageref*{SnaithExpansion} \linebreak
\noindent\hyperlink{SymondExplicitBrauerInduction}{Symonds' explicit Brauer induction}\dotfill \pageref*{SymondExplicitBrauerInduction} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{Brauer induction theorem} (\hyperlink{Brauer46}{Brauer 46}) states that, for [[ground field]] the [[complex numbers]], [[finite dimensional vector space|finite-dimensional]] [[linear representations]] $V$ of a [[finite group]] $G$ all arise as [[virtual representation|virtual combinations]] of [[induced representations]] $ind_{H}^G \big(W\big)$ of just 1-dimensional representations $W$, $dim_{\mathbb{C}}(W) =1$:

\begin{equation}
[V] 
  \;=\;
  \underset{
    \mathclap{
    {
      H_i \hookrightarrow G
    }
    \atop
    {
      {W_i \in Rep(H_i)\,,}
      \atop
      { dim(W_i) = 1 }
    }
    }
  }{\sum}
  \,
  n_i
  \, 
  \Big[ 
    ind_{H_i}^G \big(W_i\big)
  \Big]
  \,,
  \phantom{AAA}
  a_i \in \mathbb{Z}
  \,.
\label{GenericExampleOfBrauerInduction}\end{equation}
In other words, this says that the [[representation ring]] $R_{\mathbb{C}}(G)$ is generated from [[isomorphism classes]] $\left[ind_{H}^big(W\big)\right]$ of [[induced representations]] $ind_{H}^G \big(W\big)$ of 1-dimensional representations $W$ of [[subgroups]] $H \subset G$.

This may be thought of as (implying) a [[splitting principle]] for [[linear representations]] (\hyperlink{Symonds91}{Symonds 91}), for more on this see at \emph{[[characteristic classes of linear representations]]} the section \emph{\href{characteristic+class+of+a+linear+representation#SplittingPrinciple}{splitting principle}}.

Brauer induction generalizes the immediate statement that [[finite dimensional vector space|finite-dimensional]] [[permutation representations]] are all [[direct sums]] of [[induced representations]] of the \emph{[[trivial representation|trivial]]} 1-dimensional representation; see at \emph{[[induced representation of the trivial representation]]}.

The analogous statement holds true also for [[ground ring]] the [[quaternions]], while for [[ground field]] the [[real numbers]] one has to induce not just from 1-dimensional but also from 2-dimensional representations.

Of course, the expansions \eqref{GenericExampleOfBrauerInduction} are not unique. But one may find [[functor|functorial]] choices that satisfy good extra properties, see below \emph{\hyperlink{SnaithExpansion}{Snaith's explicit Brauer induction}} and \emph{\hyperlink{SymondExplicitBrauerInduction}{Symond's explicit Brauer induction}}.



\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{SnaithExpansion}{}\subsubsection*{{Snaith's explicit Brauer induction}}\label{SnaithExpansion}

(\ldots{})

\hypertarget{SymondExplicitBrauerInduction}{}\subsubsection*{{Symonds' explicit Brauer induction}}\label{SymondExplicitBrauerInduction}

We state an explicit and [[natural transformation|natural]] choice of Brauer induction due to \hyperlink{Symonds91}{Symonds 91} (Prop. \ref{SymondsExplicitBrauerInd}) below. Its main property is good compatibility with the [[total Chern classes of linear representations]] via a certain multiplicative transfer map on [[integral cohomology]] (the latter recalled as Lemma \ref{TransferEvens} below).

\begin{lemma}
\label{TransferEvens}\hypertarget{TransferEvens}{}
\textbf{(Evens' multiplicative transfer)}

For $G$ a [[finite group]] and $H \subset G$ a [[subgroup]], there is a [[linear map]]

\begin{displaymath}
\mathcal{N}_H^G
  \;\colon\;
  \underset{k \in \mathbb{N}}{\prod} H^{2k}\big( B H, \mathbb{Z}\big)
  \longrightarrow
  \underset{k \in \mathbb{N}}{\prod} H^{2k}\big( B G, \mathbb{Z}\big)
\end{displaymath}
from the [[cohomology ring]] of the [[classifying space]] of $H$ to that of $G$ which is multiplicative in that its respects the product structure, hence the [[cup product]], on both sides

\begin{displaymath}
\mathcal{N}_H^G\big( \alpha \smile \beta\big)
  \;=\;
  \mathcal{N}_H^G\big( \alpha\big)  \smile \mathcal{N}_H^G\big( \beta\big)
  \,.
\end{displaymath}
\end{lemma}
This is due to \hyperlink{Evens63}{Evens 63}. There, the maps themselves are introduced on the bottom of p. 7, while their multiplicativity is stated as Prop. 4 on p. 10.

\begin{prop}
\label{SymondsExplicitBrauerInd}\hypertarget{SymondsExplicitBrauerInd}{}
\textbf{(Symonds' explicit Brauer induction)}

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

\begin{displaymath}
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]
\end{displaymath}
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

\begin{enumerate}%
\item $L$ is a [[natural transformation]] of [[functors]] $FinGrp^{op} \to Ab$,

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


\item $L$ is a [[section]] of the [[natural transformation]]

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

\begin{displaymath}
\big( \sum ind \big) \circ L(V)
  \coloneqq
  \underset{
    [H \subset G]
  }{
    \sum
  }
  ind_H^G\left[ L(V)_H  \right]
  \;=\;
  V
\end{displaymath}

\item $L$ is compatible with the [[total Chern classes of linear representations]]

\begin{displaymath}
R_{\mathbb{C}}\big( G \big)
  \overset{c}{\longrightarrow}
  \underset{ k \in \mathbb{N} }{\prod}
  H^{2k}\big( B G, \mathbb{Z} \big)
\end{displaymath}
via their multiplicative transfer $\mathcal{N}_H^G$ (Lemma \ref{TransferEvens}) in that

\begin{displaymath}
c
  \big( 
    V
  \big)
  \;=\;
  \underset{ [H \subset G] }{\smile}
  \mathcal{N}_H^G
  \Big(
    c
    \big(
      L(V)_H 
    \big)
  \Big)
  \,,
\end{displaymath}
hence in that the following [[commuting diagram|diagram commutes]]:

\begin{displaymath}
\itexarray{
    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)
  }
\end{displaymath}

\item 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;


\item 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;



\end{enumerate}
\end{prop}
(\hyperlink{Symonds91}{Symonds 91, Prop. 2.1, Theorem 2.2, and Theorem 2.4})

(\ldots{})

\hypertarget{references}{}\subsection*{{References}}\label{references}

Due to

\begin{itemize}%
\item [[Richard Brauer]], \ldots{} (1946)

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Brauer%27s_theorem_on_induced_characters}{Brauer's theorem on induced characters}}

\end{itemize}
In the context of [[characteristic classes of linear representations]]:

\begin{itemize}%
\item [[Peter Symonds]], \emph{A splitting principle for group representations}, Comment. Math. Helv. (1991) 66: 169 (\href{https://eudml.org/doc/140229}{dml:140229}, \href{https://doi.org/10.1007/BF02566643}{doi:10.1007/BF02566643})

\end{itemize}
based on

\begin{itemize}%
\item [[Leonard Evens]], \emph{A Generalization of the Transfer Map in the Cohomology of Groups}, Transactions of the American Mathematical Society Vol. 108, No. 1 (Jul., 1963), pp. 54-65 (\href{https://doi.org/10.1090/S0002-9947-1963-0153725-1}{doi:10.1090/S0002-9947-1963-0153725-1}, \href{https://www.jstor.org/stable/1993825}{jstor:1993825})

\end{itemize}
and

\begin{itemize}%
\item Ove Kroll, \emph{An Algebraic Characterisation of Chern Classes of Finite Group Representations}, Bulletin of the LMS, Volume19, Issue3 May 1987 Pages 245-248 (\href{https://doi.org/10.1112/blms/19.3.245}{doi:10.1112/blms/19.3.245})

\end{itemize}
[[!redirects Brauer's theorem]]

[[!redirects Brauer's theorem on induced characters]]

[[!redirects Brauer induction]] [[!redirects Brauer inductions]]

[[!redirects explicit Brauer induction]] [[!redirects explicit Brauer inductions]]



\end{document}