Weyl group of a maximal torus in a linear algebraic group
From Groupprops
Definition
Suppose is a linear algebraic group over a field
and
is a maximal torus in
. The Weyl group of
, denoted
, is defined as follows:
- As an abstract group, it is the Weyl group of
in
, which can explicitly be defined as the quotient group of the normalizer
by the centralizer
.
- As a subgroup of
, it can be defined as any permutable complement to
in
. (Do such complements always exist?). Any such subgroup is termed a Weyl subgroup.
Note that if is an algebraically closed field, then
is unique up to conjugacy in
, hence
is uniquely determined up to isomorphism by
.
Particular cases
We denote the underlying field by .
Case for group ![]() |
Case for maximal torus ![]() |
Description of Weyl group as abstract group | Description of Weyl group as a subgroup of ![]() |
---|---|---|---|
general linear group of degree ![]() ![]() |
subgroup of diagonal matrices, isomorphic to ![]() ![]() |
symmetric group ![]() |
![]() |
special linear group of degree ![]() ![]() |
subgroup of diagonal matrices whose determinant is one | symmetric group ![]() |
tricky, idea is that for odd permutations we make one of the entries -1. |