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\begin{document}

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\section*{Weyl group}

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

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{lie_theory}{}\paragraph*{{Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{in_lie_theory}{In Lie theory}\dotfill \pageref*{in_lie_theory} \linebreak
\noindent\hyperlink{in_equivariant_homotopy_theory}{In equivariant homotopy theory}\dotfill \pageref*{in_equivariant_homotopy_theory} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\hypertarget{in_lie_theory}{}\subsubsection*{{In Lie theory}}\label{in_lie_theory}

In [[Lie theory]], a \emph{Weyl group} is a [[group]] associated with a [[compact Lie group]] that can either be abstractly defined in terms of a [[root system]] or in terms of a [[maximal torus]]. More generally there are Weyl groups associated with [[symmetric spaces]].

The Weyl group of a [[compact Lie group]] $G$ is equivalently the [[quotient group]] of the [[normalizer]] of any [[maximal torus]] $T$ by that torus.

\begin{displaymath}
W \simeq N_G T / T
  \,.
\end{displaymath}
\hypertarget{in_equivariant_homotopy_theory}{}\subsubsection*{{In equivariant homotopy theory}}\label{in_equivariant_homotopy_theory}

In [[equivariant homotopy theory]] one uses the term \emph{Weyl group} more general for the [[quotient group]]

\begin{displaymath}
W_G H = (N_G H) / H
\end{displaymath}
of the [[normalizer]] of any [[subgroup]] $H \hookrightarrow G$ by that subgroup (e.g. \hyperlink{May96}{May 96, p. 13}).

The relevance of the Weyl group in this sense is that it is the maximal group which canonically [[action|acts]] on $H$-[[fixed points]] of a [[topological G-space]]. (See for instance at \emph{[[tom Dieck splitting]]}.)

Notice that $W_G G = 1$ and $W_G 1 = G$.

 On the other hand, if $H  = N \subset G$ is a [[normal subgroup]], then its [[normalizer]] is $G$ itself, in which case the Weyl group is just the plain [[quotient group]]

\begin{displaymath}
W_G N \;\simeq\; G/N
  \,.
\end{displaymath}
\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Given a [[compact Lie group]] $G$ with chosen [[maximal torus]] $T$, its \textbf{Weyl group} $W(G)=W(G,T)$ is the [[group of automorphisms]] of $T$ which are restrictions of [[inner automorphisms]] of $G$.

This is the [[quotient group]] of the [[normalizer subgroup]] of $T \subset G$ by $T$

\begin{displaymath}
W \simeq N_G(T)/T
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{itemize}%
\item The [[maximal torus]] is of [[finite index subgroup|finite index]] in its [[normalizer]]; the [[quotient]] $N(T)/T$ is [[isomorphism|isomorphic]] to $W(G)$.


\item The [[cardinality]] of $W(G)$ for a compact connected $G$, equals the [[Euler characteristic]] of the [[homogeneous space]] $G/T$ (``[[flag variety]]'').


\item An important approach to the representations of the Weyl groups is the [[Springer theory]].



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Schubert calculus]]

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

\begin{itemize}%
\item [[eom]]: \href{http://eom.springer.de/W/w097710.htm}{Weyl group}; wikipedia \href{http://en.wikipedia.org/wiki/Weyl_group}{Weyl group}


\item N. Chriss, V. Ginzburg, \emph{Representation theory and complex geometry}, Birkh\"a{}user 1997. x+495 pp.


\item Walter Borho, Robert MacPherson, \emph{Repr\'e{}sentations des groupes de Weyl et homologie d'intersection pour les vari\'e{}t\'e{}s nilpotentes}, C. R. Acad. Sci. Paris S\'e{}r. I Math. 292 (1981), no. 15, 707--710 \href{http://www.ams.org/mathscinet-getitem?mr=618892}{MR82f:14002}


\item [[Peter May]], \emph{Equivariant homotopy and cohomology theory} CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (\href{https://web.math.rochester.edu/people/faculty/doug/otherpapers/alaska1.pdf}{pdf}, \href{https://bookstore.ams.org/cbms-91}{cbms-91})



\end{itemize}
[[!redirects Weyl groups]]



\end{document}