# nLab Weyl group

Contents

Not to be confused with Weil group.

group theory

### Cohomology and Extensions

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

### 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 $G$ is equivalently the quotient group of the normalizer of any maximal torus $T$ by that torus.

$W \simeq N_G T / T \,.$

### In equivariant homotopy theory

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

$W_G H = (N_G H) / H$

of the normalizer of a given subgroup $H \hookrightarrow G$ 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 $H$-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 $H \subset G$ is the automorphism group of the coset space $G/H$ in the orbit category of $G$:

$End_{G Orbits} \big( G/H \big) \;\; = Aut_{G Orbits} \big( G/H \big) \;\; \simeq \;\; W_G(H) \,.$

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

$W_G N \;\simeq\; G/N \,.$

## Definition

Given a compact Lie group $G$ with chosen maximal torus $T$, its 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 $T$

$W \simeq N_G(T)/T \,.$

## References

• 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.