## Dual Root Systems

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

[...] what about ? This is just the dual root system to ! The roots of squared-length are left unchanged, but the roots of squared-length are doubled. [...]

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[...] other observation we want to make is that duality of root systems is easily expressed in terms of Dynkin diagrams: just reverse all the oriented [...]

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[...] we can see that this root system is isomorphic to its own dual. Indeed, if is a short root, then the dual root is [...]

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[...] like the case, the root system is isomorphic to its own dual. The long roots stay the same length when dualized, while the short roots double in length and [...]

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