Editing Maximal torus (section)

== Weyl group ==
Given a torus ''T'' (not necessarily maximal), the [[Weyl group]] of ''G'' with respect to ''T'' can be defined as the [[normalizer]] of ''T'' modulo the [[centralizer]] of ''T''. That is, 
:<math>W(T,G) := N_G(T)/C_G(T).</math> 
Fix a maximal torus <math>T = T_0</math> in ''G;'' then the corresponding Weyl group is called the Weyl group of ''G'' (it depends up to isomorphism on the choice of ''T''). 

The first two major results about the Weyl group are as follows.
* The centralizer of ''T'' in ''G'' is equal to ''T'', so the Weyl group is equal to ''N''(''T'')/''T''.<ref>{{harvnb|Hall|2015}} Theorem 11.36</ref>
* The Weyl group is generated by reflections about the roots of the associated Lie algebra.<ref>{{harvnb|Hall|2015}} Theorem 11.36</ref> Thus, the Weyl group of ''T'' is isomorphic to the [[Weyl group]] of the [[root system]] of the Lie algebra of ''G''.

We now list some consequences of these main results.
* Two elements in ''T'' are conjugate if and only if they are conjugate by an element of ''W''. That is, each conjugacy class of ''G'' intersects ''T'' in exactly one Weyl [[orbit (group theory)|orbit]].<ref>{{harvnb|Hall|2015}} Theorem 11.39</ref> In fact, the space of conjugacy classes in ''G'' is homeomorphic to the [[orbit space]] ''T''/''W''.
* The Weyl group acts by ([[outer automorphism|outer]]) [[automorphism]]s on ''T'' (and its Lie algebra).
* The [[identity component]] of the normalizer of ''T'' is also equal to ''T''. The Weyl group is therefore equal to the [[component group]] of ''N''(''T'').
* The Weyl group is finite.
The [[Compact_group#Representation_theory_of_a_connected_compact_Lie_group|representation theory]] of ''G'' is essentially determined by ''T'' and ''W''.

As an example, consider the case <math>G=SU(n)</math> with <math>T</math> being the diagonal subgroup of <math>G</math>. Then <math>x\in G</math> belongs to <math>N(T)</math> if and only if <math>x</math> maps each standard basis element <math>e_i</math> to a multiple of some other standard basis element <math>e_j</math>, that is, if and only if <math>x</math> permutes the standard basis elements, up to multiplication by some constants. The Weyl group in this case is then the permutation group on <math>n</math> elements.