Weyl group of a maximal torus in a linear algebraic group
- 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 .
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 (one of many possible)|
|general linear group of degree , denoted||subgroup of diagonal matrices, isomorphic to direct power of multiplicative group of||symmetric group||embedded as permutation matrices|
|special linear group of degree , denoted||subgroup of diagonal matrices whose determinant is one||symmetric group||tricky, idea is that for odd permutations we make one of the entries -1.|