Weyl group
Contents
Definition
Definition with symbols
Let be groups. The Weyl group of
with respect to
can be defined in the following equivalent ways:
- It is the group of those automorphisms of
which extend to inner automorphisms of
- It is the quotient group
where
is the normalizer of
in
and
is the centralizer of
in
.
- it is the image of the natural homomorphism from
to
that sends
to the automorphism of
given via conjugation by
.
Related notions
Relation with subgroup properties
The Weyl group always contains the inner automorphism group of and lies inside the automorphism group of
. This gives two extreme subgroup properties:
- Fully normalized subgroup is a subgroup whose Weyl group is the whole automorphism group
- Central factor of normalizer is a subgroup whose Weyl group is precisely the inner automorphism group
For self-centralizing Abelian subgroups
In the particular case where , the Weyl group of
is simply
. This situation is quite common in the case of linear groups, for instance: each torus (for instance, the subgroup of invertible diagonal matrices) is self-centralizing in the general linear group, and hence its Weyl group is simply the quotient of its normalizer, by itself (this turns out to be the symmetric group).
Weyl groups in algebraic groups
Further information: Weyl group of a maximal torus in a linear algebraic group
In the context of a linear algebraic group, the term Weyl group is typically used to refer to the Weyl group of a maximal torus in the group. If the linear algebraic group is over an algebraically closed field, then the maximal tori are all conjugate, and the Weyl groups are thus all isomorphic. Further, in this case, the normalizer of a maximal torus is actually an internal semidirect product of the maximal torus with another subgroup, and we can treat any of the possible complements as Weyl subgroups of the whole group.