# The Unapologetic Mathematician

## Cartan Matrices

February 16, 2010 - Posted by | Geometry, Root Systems

1. So cool! I admit that I still cannot see the word “intertwine” without think of Ted “hypertext” Nelson’s mantra: “Everything is profoundly intertwingled!”

I just spent two delightful full days on the Caltech campus for the Western States Annual Mathematical Physics Meeting that Barry Simon hosts. I always enjoy being the weakest mathematican or physicist in the room, and seeing deep results from fascinating people.

Comment by Jonathan Vos Post | February 17, 2010 | Reply

2. […] Cartan Matrix to Root System Yesterday, we showed that a Cartan matrix determines its root system up to isomorphism. That is, in principle if we have […]

Pingback by From Cartan Matrix to Root System « The Unapologetic Mathematician | February 17, 2010 | Reply

3. […] Graphs and Dynkin Diagrams We’ve taken our root system and turned it into a Cartan matrix. Now we’re going to take our Cartan matrix and turn it into a pictorial form that we can […]

Pingback by Coxeter Graphs and Dynkin Diagrams « The Unapologetic Mathematician | February 18, 2010 | Reply

4. […] that for each such root system we can construct a connected Dynkin diagram, which determines a Cartan matrix, which determines the root system itself, up to isomorphism. So what he have to find now is a list […]

Pingback by The Classification of (Possible) Root Systems « The Unapologetic Mathematician | February 19, 2010 | Reply

5. […] 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 and for these bases, then the mapping defines images of […]

Pingback by Transformations of Dynkin Diagrams « The Unapologetic Mathematician | March 5, 2010 | Reply

6. […] of and will give us access to . Further, it should be orthogonal to both and , and should have a Cartan integer of with in either order. For this purpose, we pick , which then gives us the last vertex of the […]

Pingback by Construction of the F4 Root System « The Unapologetic Mathematician | March 9, 2010 | Reply

7. […] of the Dynkin diagram of the root system. And for to be an automorphism of , it must preserve the Cartan integers, and thus the numbers of edges between any pair of vertices in the Dynkin diagram. That is, must […]

Pingback by The Automorphism Group of a Root System « The Unapologetic Mathematician | March 11, 2010 | Reply

8. […] classify these, we defined Cartan matrices and verified that we can use it to reconstruct a root system. Then we turned Cartan matrices into […]

Pingback by Root Systems Recap « The Unapologetic Mathematician | March 12, 2010 | Reply