Construction of B- and C-Series Root Systems
As we did for the series, we start out with an dimensional space with the lattice of integer-coefficient vectors. This time, though, we let be the collection of vectors of squared-length or : either or . Explicitly, this is the collection of vectors for (signs chosen independently) from the root system, plus all the vectors .
Similarly to the series, and exactly as in the series, we define for . This time, though, to get vectors whose coefficients don’t sum to zero we can just define , which is independent of the other vectors. Since it has vectors, the independent set is a basis for our vector space.
As in the and cases, any vector with can be written
This time, any of the can be written
Thus any vector can be written as the sum of two of these vectors. And so is a base for .
We calculate the Cartan integers. For and less than , we again have the same calculation as in the case, which gives a simple chain of length vertices. But when we involve things are a little different.
If , then both of these are zero. On the other hand, if , then the first is and the second is . Thus we get a double edge from to , and is the longer root. And so we obtain the Dynkin diagram.
Considering the reflections with respect to the , we find that swaps the coefficients of and for . But what about ? We calculate
which flips the sign of the last coefficient of . As we did in the case, we can use this to flip the signs of whichever coefficients we want. Since these transformations send the lattice back into itself, they send to itself and we do have a root system.
Finally, since we don’t have any restrictions on how many signs we can flip, the Weyl group for is exactly the wreath product .
So, 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. The Weyl group is the same — — but now the short root in the base is the long root, and so we flip the direction of the double arrow in the Dynkin diagram, giving the diagram.