Weyl group


Not to be confused with Weil group.


Group Theory

Differential geometry

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from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



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The magic algebraic facts




tangent cohesion

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singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

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Lie theory

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Smooth structure

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Lie theory

∞-Lie groupoids

∞-Lie algebroids

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\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras



In Lie theory

In Lie theory, a 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 GG is equivalently the quotient group of the normalizer of any maximal torus TT by that torus.

WN GT/T. W \simeq N_G T / T \,.

In equivariant homotopy theory

In equivariant homotopy theory one uses the term Weyl group more generally for the quotient group

W GH=(N GH)/H W_G H = (N_G H) / H

of the normalizer of a given subgroup HGH \hookrightarrow G by that subgroup (e.g. May 96, p. 13).

The relevance of the Weyl group in this sense is that it is the maximal group which canonically acts on HH-fixed points of a topological G-space. (See at Change of equivariance group and fixed loci for details and, at, e.g., tom Dieck splitting for applications.)

This may be seen from the fact that the Weyl group of HGH \subset G is the automorphism group of the coset space G/HG/H in the orbit category of GG:

End GOrbits(G/H)=Aut GOrbits(G/H)W G(H). End_{G Orbits} \big( G/H \big) \;\; = Aut_{G Orbits} \big( G/H \big) \;\; \simeq \;\; W_G(H) \,.

Notice that W GG=1W_G G = 1 and W G1=GW_G 1 = G.

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

W GNG/N. W_G N \;\simeq\; G/N \,.


Given a compact Lie group GG with chosen maximal torus TT, its Weyl group W(G)=W(G,T)W(G)=W(G,T) is the group of automorphisms of TT which are restrictions of inner automorphisms of GG.

This is the quotient group of the normalizer subgroup of TGT \subset G by TT

WN G(T)/T. W \simeq N_G(T)/T \,.



  • eom: Weyl group; wikipedia Weyl group

  • N. Chriss, V. Ginzburg, Representation theory and complex geometry, Birkhäuser 1997. x+495 pp.

  • Walter Borho, Robert MacPherson, Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 15, 707–710 MR82f:14002

  • Peter May, 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. (pdf, cbms-91)

Last revised on April 23, 2021 at 01:48:42. See the history of this page for a list of all contributions to it.