Why isn't the Weyl group of a root system defined as the isometry group of that system?

Question

I know that the Weyl group of a root system is a subgroup of its isometry group, but (as in the case of $A_2$) it isn't always the whole isometry group. Why isn't the Weyl group defined as the isometry group?

Answer 1

Indeed, negation of the roots is not always an element of the Weyl group.

For me the most natural reason to restrict ourselves to the group generated by reflections corresponding to the roots comes from representation theory. The formal characters of finite dimensional simple modules are invariant under the Weyl group, but not always under negation of weights. When you negate the weights of the formal character of a f.d. simple module $V$, you get the formal character of the dual module $V^*$. We don't always have $V\simeq V^*$. Basically because sometimes there exists weights $\lambda$ such that the difference $\lambda-(-\lambda)$ is not in the root lattice, and hence both $\lambda$ and $-\lambda$ cannot appear in the same simple module.

It may have also played a role that according to the standard definition the Weyl group is always a Coxeter group. I'm afraid I don't know which appeared earlier in the development of mathematics. The nice theory of Coxeter groups makes a number of arguments simpler, but I don't know which is an extension/special case of the other historically.