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\begin{document}

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\section*{characteristic class of a linear representation}

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

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

[[!include representation theory - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{first_chern_class}{First Chern class}\dotfill \pageref*{first_chern_class} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{SplittingPrinciple}{Splitting principle}\dotfill \pageref*{SplittingPrinciple} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Given a [[linear representation]] of a [[finite group]] $G$ on a [[finite-dimensional vector space]] $V$, this induces an [[associated bundle|associated]] [[vector bundle]] over the [[classifying space]] $B G$. The [[characteristic classes]] of this vector bundle, notably it [[Chern classes]] or [[Pontryagin]], are hence entirely determined by the linear representation, and may be associated with it.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Under the identification of the [[representation ring]] with the [[equivariant K-theory]] of the [[point]] (see \href{equivariant+K-theory#eq:RepresentationRingAsEquivariantKTheoryOfThePoint}{there}) and the [[Atiyah-Segal completion]] map

\begin{displaymath}
R_{\mathbb{C}}(G)
  \simeq
  KU_G^0(\ast)
    \overset{ \widehat{(-)} }{\longrightarrow}
  KU(BG)
\end{displaymath}
one may ask for Chern classes of the K-theory class $\widehat{V} \in KU(B G)$ expressed in terms of the actual [[character]] of the [[representation]] $V$.

(\ldots{})

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{first_chern_class}{}\subsubsection*{{First Chern class}}\label{first_chern_class}

There is a closed formula at least for the [[first Chern class]] (\hyperlink{Atiyah61}{Atiyah 61, appendix}):

For 1-dimensional representations $V$ their [[first Chern class]] $c_1(\widehat{V}) \in H^2(B G, \mathbb{Z})$ is their image under the canonical [[isomorphism]] from 1-[[dimension|dimensional]] characters in $Hom_{Grp}(G,U(1))$ to the [[group cohomology]] $H^2_{grp}(G, \mathbb{Z})$ and further to the [[ordinary cohomology]] $H^2(B G, \mathbb{Z})$ of the [[classifying space]] $B G$:

\begin{displaymath}
c_1\left(\widehat{(-)}\right)
  \;\colon\;
  Hom_{Grp}(G, U(1))
  \overset{\simeq}{\longrightarrow}
  H^2_{grp}(G,\mathbb{Z})
  \overset{\simeq}{\longrightarrow}
  H^2(B G, \mathbb{Z})
  \,.
\end{displaymath}
More generally, for $n$-[[dimension|dimensional]] [[linear representations]] $V$ their [[first Chern class]] $c_1(\widehat V)$ is the previously defined first Chern-class of the [[line bundle]] $\widehat{\wedge^n V}$ corresponding to the $n$-th [[exterior power]] $\wedge^n V$ of $V$. The latter is a 1-dimensional representation, corresponding to the [[determinant line bundle]] $det(\widehat{V}) = \widehat{\wedge^n V}$:

\begin{displaymath}
c_1(\widehat{V}) 
    \;=\; 
  c_1(det(\widehat{V})) 
    \;=\; 
  c_1( \widehat{\wedge^n V} )
  \,.
\end{displaymath}
(\hyperlink{Atiyah61}{Atiyah 61, appendix, item (7)})

More explicitly, via the formula for the [[determinant]] as a [[polynomial]] in [[traces]] of powers (see \href{determinant#eq:DeterminantAsPolynomialInTracesOfPowers}{there}) this means that the first Chern class of the $n$-dimensional representation $V$ is expressed in terms of its [[character]] $\chi_V$ as

\begin{equation}
c_1(V)
  =
  \chi_{\left(\wedge^n V\right)}
  \;\colon\;
  g
  \;\mapsto\;
  \underset{ 
     { k_1,\cdots, k_n \in \mathbb{N} }
     \atop
     { \underoverset{\ell = 1}{n}{\sum} \ell k_\ell = n }
  }{\sum}
  \underoverset{ l = 1 }{ n }{\prod} 
  \frac{ (-1)^{k_l + 1} }{ l^{k_l} k_l !  }
  \left(\chi_V(g^l)\right)^{k_l}
\label{FirstChernClassOfRepresentationInTermsOfTheCharacter}\end{equation}
For example, for a representation of dimension $n = 2$ this reduces to

\begin{displaymath}
c_1(V)
  =
  \chi_{V \wedge V}
  \;\colon\;
  g
  \;\mapsto\;
  \frac{1}{2}
  \left( 
    \left( \chi_V(g)\right)^2
    -
    \chi_V(g^2)
  \right)
\end{displaymath}
(see also e.g. \href{representation+theory#tomDieck09}{tom Dieck 09, p. 45})

\begin{example}
\label{}\hypertarget{}{}
Let $G =\mathbb{Z}_{2n+1}$ be a [[finite group|finite]] [[cyclic group]] of [[odd number|odd]] [[order of a group|order]] and let $k[\mathbb{Z}_{2n+1}]$ be its [[regular representation]]. Then the first Chern class vanishes:

\begin{displaymath}
c_1\big( k[\mathbb{Z}_{2n+1}]\big)
  \;=\;
  0
\end{displaymath}
\end{example}
\begin{proof}
The underlying [[set]] of $\mathbb{Z}_{2n+1}$ constitutes the canonical [[linear basis]] of $k[\mathbb{Z}_{2n+1}]$. Moreover, this carries a canonical [[linear order]] $(e, g_1, g_2, \cdots, g_{2n+1})$. With respect to this ordering, the [[action]] of each group element $g \in \mathbb{Z}_n$ is by a [[cyclic permutation]]. Since for [[odd number]] of elements the [[signature of a permutation|signature]] of a [[cyclic permutation]] is $+1$, it follows that for every group element

\begin{displaymath}
g(e \wedge g_1 \wedge \cdots \wedge g_{2n+1})
  = 
  + e \wedge g_1 \wedge \cdots \wedge g_{2n+1}
  \,.
\end{displaymath}
This shows that the [[character of a linear representation|character]] of $\wedge^{2n+1}k[\mathbb{Z}_{2n+1}]$ equals that of the [[trivial representation]] $\mathbf{1}$

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

\hypertarget{SplittingPrinciple}{}\subsubsection*{{Splitting principle}}\label{SplittingPrinciple}

\begin{quote}%
see at \href{Brauer+induction+theorem#SymondExplicitBrauerInduction}{Symonds' explicit Brauer induction}


\end{quote}
Let $G$ be a [[finite group]], let the [[ground field]] to be the [[complex numbers]].

Then by the [[Brauer induction theorem]] every [[virtual representation]]

\begin{displaymath}
[V] \in R_\mathbb{C}(G)
\end{displaymath}
has a presentation as a virtual combination of [[induced representations]] of 1-[[dimension|dimensional]] representations:

\begin{displaymath}
[V]
 \;=\;
 \underset{
     \mathclap{
         {H_i \hookrightarrow G} 
           \atop 
         { 
           { W_i \in Rep(H_i) \,, }
           \atop
           { dim(W_i) =1 }
         }
     }
   }{\sum}
   n_i
  \left[
     ind_{H_i}^G W_i
  \right]
\end{displaymath}
Of course this expansion is not unique.

According to \hyperlink{Symonds91}{Symonds 91, p. 4 \& Prop. 2.4}, there is a natural choice for this expansion, and for this there holds a [[splitting principle]] for the corresponding [[Chern classes]] summarized in the [[total Chern class]] (formal sum of all [[Chern classes]])

\begin{displaymath}
c(V)
  \;\coloneqq\;
  1 + c_2(V) + c_2(V) + \cdots
  \;\in\;
  \underset{k}{\prod} H^{2 k}\big( B G , \mathbb{Z}\big)
\end{displaymath}
as follows:

\begin{equation}
ch
  \left(
     V
  \right)
  \;=\;
  \underset{ 
     \mathclap{
         {H_i \hookrightarrow G} 
           \atop 
         { 
           { W_i \in Rep(H_i)\,, } 
           \atop
           { dim(W_i) = 1 } 
         }
     }
   }{\prod}
   \mathcal{N}_{H_i}^G
   \Big(
     \overset{
        \mathclap{
          = (1 + c_1(W_i))
        }
      }{
      \overbrace{
        ch\left(W_i\right)
      }
     }
     {}^{\alpha(W_i)}
   \Big)
   \;\;
   \in 
   \underset{ k \in \mathbb{N} }{\prod}
   H^{2k}\big(B G, \mathbb{Z} \big)
\label{SplittingFormulaForChernCharacterOfLinearRepresentation}\end{equation}
where

\begin{enumerate}%
\item the transfer maps

\begin{equation}
\mathcal{N}_H^G
  \;\colon\;
  H^\bullet(B H, \mathbb{Z})
    \longrightarrow
  H^\bullet( B G, \mathbb{Z} )
\label{MultiplicativeTransfer}\end{equation}
are from \hyperlink{Evens63}{Evens 63, bottom of p. 7},


\item the $\alpha(W_i)$-s are the [[Euler characteristics]] of certain [[CW-complexes]], described in \hyperlink{Symonds91}{Symonds 91, p. 3}.



\end{enumerate}
Here over the brace we used that the $W_i$ are 1-dimensional, so that at most their [[first Chern class]] may be non-vanishing.

Notice that the transfer maps \eqref{MultiplicativeTransfer} are multiplicative under [[cup product]] (\hyperlink{Evens63}{Evens 63, prop. 4}), whence \hyperlink{Symonds91}{Symonds 91} refers to them as the ``mutliplicative transfer''.

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

\begin{itemize}%
\item [[Michael Atiyah]], Appendix of \emph{Characters and cohomology of finite groups}, Publications Mathématiques de l'IHÉS, Volume 9 (1961) , p. 23-64 (\href{http://www.numdam.org/item?id=PMIHES_1961__9__23_0}{numdam})


\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})


\item [[Leonard Evens]], \emph{On the Chern Classes of Representations of Finite Groups}, Transactions of the American Mathematical Society, Vol. 115 (Mar., 1965), pp. 180-193 (\href{https://www.jstor.org/stable/1994264}{doi:10.2307/1994264})


\item F. Kamber, [[Philippe Tondeur]], \emph{Flat Bundles and Characteristic Classes of Group-Representations}, American Journal of Mathematics, Vol. 89, No. 4 (Oct., 1967), pp. 857-886 (\href{https://www.jstor.org/stable/2373408}{doi:10.2307/2373408})


\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})


\item J. Gunarwardena, B. Kahn, C. Thomas, \emph{Stiefel-Whitney classes of real representations of finite groups}, Journal of Algebra Volume 126, Issue 2, 1 November 1989, Pages 327-347 ()


\item Arnaud Beauville, \emph{Chern classes for representations of reductive groups} (\href{https://arxiv.org/abs/math/0104031}{arXiv:math/0104031})



\end{itemize}
[[splitting principle]]:

\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}
[[!redirects characteristic classes of a linear representation]] [[!redirects characteristic classes of linear representations]]

[[!redirects Chern class of a linear representation]] [[!redirects Chern classes of a linear representation]] [[!redirects Chern classes of linear representations]]

[[!redirects total Chern class of a linear representation]] [[!redirects total Chern classes of a linear representation]] [[!redirects total Chern classes of linear representations]]



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