The Unapologetic Mathematician

Some Root Systems and Weyl Orbits

Today, I’d like to show you some examples of two-dimensional root systems, which illustrate a lot of what we talked about last week. I’ve worked them up in a Java application called Geogebra, but I don’t seem to be able to embed the resulting applets into a WordPress post. If someone knows how, I’d be glad to hear it.

Anyhow, in lieu of embedded applets, I’ll post the “.ggb” files, which you can take over to the Geogebra site and load up there. So, with no further ado, I present all four two-dimensional root systems:

Each one of these files illustrates the root system and its Weyl orbit. Each one has two simple roots, labelled $\alpha$ and $beta$ in blue. The rest of the roots are shown in black, and the fundamental domain is marked out by two rays perpendicular to $\alpha$ and $\beta$, respectively.

An arbitrary blue vector $\mu$ is shown, along with its reflected images making up the entire Weyl orbit. You can grab this vector and drag it around, watching how the orbit changes. No matter where you place $\mu$, notice that there is exactly one image in the fundamental domain, as we showed.

The first root system is reducible, but the other three are irreducible. For each of these, we can see that there is a unique maximal root. However, $A_1\amalg A_1$ doesn’t; both $\alpha$ and $beta$ are maximal.

We also see that the Weyl orbit of a root spans the plane in the irreducible cases. But, again, in $A_1\amalg A_1$ the Weyl orbits of $\alpha$ and $\beta$ only span their lines.

Finally, in each of the irreducible cases we see that there are at most two distinct root lengths. And, in each case, the unique maximal root is the long root within the fundamental domain.

February 15, 2010 Posted by | Geometry, Root Systems | 3 Comments