# The Unapologetic Mathematician

## Transformations of Dynkin Diagrams

Before we continue constructing root systems, we want to stop and observe a couple things about transformations of Dynkin diagrams.

First off, I want to be clear about what kinds of transformations I mean. Given Dynkin diagrams $X$ and $Y$, I want to consider a mapping $\phi$ that sends every vertex of $X$ to a vertex of $Y$. Further, if $\xi_1$ and $\xi_2$ are vertices of $X$ joined by $n$ edges, then $\phi(\xi_1)$ and $\phi(\xi_2)$ should be joined by $n$ edges in $Y$ as well, and the orientation of double and triple edges should be the same.

But remember that $\xi_1$ and $\xi_2$, as vertices, really stand for vectors in some base of a root system, and the number of edges connecting them encodes their Cartan integers. If we slightly abuse notation and write $X$ and $Y$ for these bases, then the mapping $\phi$ defines images of the vectors in $X$, which is a basis of a vector space. Thus $\phi$ extends uniquely to a linear transformation from the vector space spanned by $X$ to that spanned by $Y$. And our assumption about the number of edges joining two vertices means that $\phi$ preserves the Cartan integers of the base $X$.

Now, just like we saw when we showed that the Cartan matrix determines the root system up to isomorphism, we can extend $\phi$ to a map from the root system generated by $X$ to the root system generated by $Y$. That is, a transformation of Dynkin diagrams gives rise to a morphism of root systems.

Unfortunately, the converse doesn’t necessarily hold. Look back at our two-dimensional examples; specifically, consider the $A_2$ and $G_2$ root systems. Even though we haven’t really constructed the latter yet, we can still use what we see. There are linear maps taking the six roots in $A_2$ to either the six long roots or the six short roots in $G_2$. These maps are all morphisms of root systems, but none of them can be given by transformations of Dynkin diagrams. Indeed, the image of any base for $A_2$ would contain either two long roots in $G_2$ or two short roots, but any base of $G_2$ would need to contain both a long and a short root.

However, not all is lost. If we have an isomorphism of root systems, then it must send a base to a base, and thus it can be seen as a transformation of the Dynkin diagrams. Indeed, an isomorphism of root systems gives rise to an isomorphism of Dynkin diagrams.

The other observation we want to make is that duality of root systems is easily expressed in terms of Dynkin diagrams: just reverse all the oriented edges! Indeed, we’ve already seen this in the case of $B_n$ and $C_n$ root systems. When we get to constructing $G_2$ and $F_4$, we will see that they are self-dual, in keeping with the fact that reversing the directed edge in each case doesn’t really change the diagram.

March 5, 2010 - Posted by | Geometry, Root Systems