Weyl group of a maximal torus in a linear algebraic group

Definition

Suppose $G$ is a linear algebraic group over a field $K$ and $T$ is a maximal torus in $G$. The Weyl group of $T$, denoted $W(T)$, is defined as follows:

• As an abstract group, it is the Weyl group of $T$ in $G$, which can explicitly be defined as the quotient group of the normalizer $N_G(T)$ by the centralizer $C_G(T)$.
• As a subgroup of $G$, it can be defined as any permutable complement to $C_G(T)$ in $N_G(T)$. (Do such complements always exist?). Any such subgroup is termed a Weyl subgroup.

Note that if $K$ is an algebraically closed field, then $T$ is unique up to conjugacy in $G$, hence $W(T)$ is uniquely determined up to isomorphism by $G$.

Particular cases

We denote the underlying field by $K$.

Case for group $G$	Case for maximal torus $T$	Description of Weyl group as abstract group	Description of Weyl group as a subgroup of $G$ (one of many possible)
general linear group of degree $n$, denoted $GL(n,K)$	subgroup of diagonal matrices, isomorphic to $n^{th}$ direct power of multiplicative group of $K$	symmetric group $S_n$	$S_n$ embedded as permutation matrices
special linear group of degree $n$, denoted $SL(n,K)$	subgroup of diagonal matrices whose determinant is one	symmetric group $S_n$	tricky, idea is that for odd permutations we make one of the entries -1.