## Cartan Matrices

As we move towards our goal of classifying root systems, we find new ways of encoding the information contained in a root system . First comes the Cartan matrix.

Pick a base of the root system. Since is finite-dimensional and is a basis of , must be finite, and so there’s no difficulty in picking some fixed order on the simple roots. That is, we write where . Now we can define the “Cartan matrix” as the matrix whose entry in the th row and th column is . These entries are called “Cartan integers”.

The matrix we get, depends on the particular ordering of the base we chose, of course, so the Cartan matrix isn’t *quite* uniquely determined by the root system. This is relatively unimportant, actually. More to the point is the other direction: the Cartan matrix determines the root system up to isomorphism!

That is, let’s say is another root system in another vector space with another identified base . Further, assume that for all we have , so the Cartan matrix determined by is equal to the Cartan matrix determined by . I say that the bijection extends to an isomorphism that sends onto and satisfies for all roots .

The unique extension to is trivial. Indeed, since is a basis for all we have to do is specify all the images and there is a unique linear transformation extending the mapping on basis vectors. And it’s an isomorphism, since the image of our basis of is itself a basis of , so we can turn around and reverse everything.

Now our hypothesis that the bases give rise to the same Cartan matrix allows us to calculate for simple roots :

That is, intertwines the actions of each of the simple reflections . But we know that the simple reflections with respect to any given base generate the Weyl group!

And so must intertwine the actions of the Weyl groups and . That is, the mapping is an isomorphism which sends to for all .

We can go further. Each root is in the -orbit of some simple root . Say for . Then we find

And so must send to . A straightforward calculation (unwinding the one before) shows that must then preserve the Cartan integers for any roots and .