Construction of D-Series Root Systems
The construction is similar to that of the series, but instead of starting with a hyperplane in -dimensional space, we just start with -dimensional space itself with the lattice of integer-coefficient vectors. We again take to be the collection of vectors of squared-length : . Explicitly, this is the collection of vectors for , where we can choose the two signs independently.
Similarly to the case, we define for , but these can only give vectors whose coefficients sum to . To get other vectors, we throw in , which is independent of the others. The linearly independent collection has vectors, and so must be a basis of the -dimensional space.
As before, any vector in of the form for can be written as
while vectors of the form are a little more complicated. We can start with
and from this we can always build for . Then if we can write . This proves that is a base for .
Again, we calculate the Cartan integers. The calculation for and both less than is exactly as before, showing that these vectors form a simple chain in the Dynkin diagram of length . However, when we involve we find
For , this is automatically ; for , we get the value ; and for we again get . This shows that the Dynkin diagram of is .
Finally, we consider the reflections with respect to the . As in the case, we find that swaps the coefficients of and for . But what about ?
This swaps the last two coefficients of and flips their sign. Clearly, this sends the lattice back to itself, showing that is indeed a root system.
Now we can use to flip the signs of coefficients of , two at a time. We use whatever of the we need to get the two coefficients we want into the last two slots, hit it with to flip them, and then invert the first permutation to move everything back where it started from. In fact, this is a lot like what we saw way back with the Rubik’s cube, when dealing with the edge group. We can effect whatever permutation we want on the coefficients, and we can flip any even number of them.
The Weyl group of is then the subgroup of the wreath product consisting of those transformations with an even number of flips coming from the components. Explicitly, we can write as the subgroup of with sum zero. Then we can let act on by permuting the components, and use this to give an action of on , and thus form the semidirect product .