Given a root system , there’s a very interesting related root system , called the “dual” or “inverse” root system. It’s made up of the “duals” , defined by
This is the vector that represents the linear functional . That is, .
The dual root is proportional to , and so . The dual reflections are the same as the original reflections, and so they generate the same subgroup of . That is, the Weyl group of is the same as the Weyl group of .
As we should hope, dualizing twice gives back the original root system. That is, . We can even show that . Indeed, we calculate
It turns out that passing to duals reverses the roles of roots, in a way, just as we might expect from a dualization. Specifically, . Indeed, we calculate