Starting from our setup, we construct root systems corresponding to the 
Dynkin diagrams (for 
).
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 
.
March 3, 2010
Posted by John Armstrong |
Geometry, Root Systems |
2 Comments
