# The Unapologetic Mathematician

## Root Systems

Okay, now to lay out the actual objects of our current interest. These are basically collections $\Phi$ of vectors in some inner product space $V$, but with each vector comes a reflection and we want these reflections to play nicely with the vectors themselves. In a way, each point acts as both “program” — an operation to be performed — and “data” — an object to which operations can be applied — and the interplay between these two roles leads to some very interesting structure.

First off, only nonzero vectors give rise to reflections, so we don’t really want zero to be in our collection $\Phi$. We also may as well assume that $\Phi$ spans $V$, because it certainly spans some subspace of $V$ and anything that happens off of this subspace is pretty uninteresting as far as $\Phi$ goes. These are the things that would just be silly not to ask for.

Now, the core requirement is that if $\alpha\in\Phi$, then the reflection $\sigma_\alpha$ should leave $\Phi$ invariant. That is, if $\beta$ is any vector in $\Phi$, then $\sigma_\alpha(\beta)=\beta-(\beta\rtimes\alpha)\alpha=\beta-\frac{2\langle\beta,\alpha\rangle}{\langle\alpha,\alpha\rangle}\alpha$

is also a vector in $\Phi$. In particular, this means that we have to have $\sigma_\alpha(\alpha)=-\alpha\in\Phi$. But we don’t want any other scalar multiples of $\alpha$ to be in $\Phi$, because they’d just give the same reflection again and that would be redundant.

Of course, we could just throw in more and more vectors as we need to make $\Phi$ invariant under all of its reflections, and each new vector introduces not only new images under the existing reflections, but whole new reflections we have to handle. We want this process to stop after a while, so we’ll insist that $\Phi$ is a finite collection of vectors. This is probably the biggest constraint on our collections.

We have one last condition to add: we want to ask that for every pair of vectors $\alpha$ and $\beta$ in $\Phi$, we have $\beta\rtimes\alpha\in\mathbb{Z}$. In other words, the length of the projection of $\beta$ onto $\alpha$ must be a half-integral multiple of the length of $\alpha$. This makes it so that the displacement from $\beta$ to $\sigma_\alpha(\beta)$ is some integral multiple of $\alpha$. This provides a certain rigidity to our discussion.

So, let’s recap:

• $\Phi$ is a finite, spanning set of vectors in $V$ which does not contain $0\in V$.
• If $\alpha\in\Phi$ then the only scalar multiples of $\alpha$ in $\Phi$ are $\pm\alpha$.
• If $\alpha\in\Phi$ then the reflection $\sigma_\alpha$ leaves $\Phi$ invariant.
• If $\alpha$ and $\beta$ are in $\Phi$, then $\displaystyle\beta\rtimes\alpha=\frac{2\langle\beta,\alpha\rangle}{\langle\alpha,\alpha\rangle}$ is an integer.

A collection $\Phi$ of vectors satisfying all of these conditions is called a “root system”, and the vectors in $\Phi$ are called “roots” for ABSOLUTELY ARBITRARY REASONS THAT HAVE ABSOLUTELY NOTHING TO DO WITH ANYTHING. As far as we’re concerned for now.

So yeah: “root system”. Just ’cause…

Our lofty goal, for the immediate future, is to classify all the possible root systems.

January 20, 2010 - Posted by | Geometry, Root Systems

1. Oh boy. This should be interesting.

Are you going to do the motivation of, say, classifying complex simple Lie algebras? Comment by Akhil Mathew | January 20, 2010 | Reply

2. What are these “Lie algebras”, of which you speak? I just see a bunch of vectors being reflected around in regular old $n$-dimensional spaces. These “Lie algebras” sound big and scary.

😉 Comment by John Armstrong | January 20, 2010 | Reply

3. […] Weyl Group of a Root System Let’s take a root system in the inner product space . Each vector in gives rise to a reflection in , the group of […]

Pingback by The Weyl Group of a Root System « The Unapologetic Mathematician | January 21, 2010 | Reply

4. […] Category of Root Systems As with so many of the objects we study, root systems form a category. If is a root system in the inner product space , and is a root system in the […]

Pingback by The Category of Root Systems « The Unapologetic Mathematician | January 22, 2010 | Reply

5. […] Root Systems We should also note that the category of root systems has binary (and thus finite) coproducts. They both start the same way: given root systems and in […]

Pingback by Coproduct Root Systems « The Unapologetic Mathematician | January 25, 2010 | Reply

6. […] Root Systems Given a root system , there’s a very interesting related root system , called the “dual” or […]

Pingback by Dual Root Systems « The Unapologetic Mathematician | January 26, 2010 | Reply

7. […] Root Systems Given a root system in an inner product space we may be able to partition it into two collections so that each root […]

Pingback by Irreducible Root Systems « The Unapologetic Mathematician | January 27, 2010 | Reply

8. […] of Roots When we look at a root system, the integrality condition puts strong restrictions on the relationship between any two vectors in […]

Pingback by Pairs of Roots « The Unapologetic Mathematician | January 28, 2010 | Reply

9. […] First, a lemma: given a root system , let and be nonproportional roots. That is, . Then if — if the angle between the vectors […]

Pingback by Root Strings « The Unapologetic Mathematician | January 29, 2010 | Reply

10. […] for Root Systems We don’t always want to deal with a whole root system . Indeed, that’s sort of like using a whole group when all the information is contained in […]

Pingback by Bases for Root Systems « The Unapologetic Mathematician | February 1, 2010 | Reply

11. […] Existence of Bases for Root Systems We’ve defined what a base for a root system is, but we haven’t provided any evidence yet that they even exist. Today we’ll not only […]

Pingback by The Existence of Bases for Root Systems « The Unapologetic Mathematician | February 2, 2010 | Reply

12. […] A very useful concept in our study of root systems will be that of a Weyl chamber. As we showed at the beginning of last time, the hyperplanes for […]

Pingback by Weyl Chambers « The Unapologetic Mathematician | February 3, 2010 | Reply

13. […] Lemmas on Simple Roots If is some fixed base of a root system , we call the roots “simple”. Simple roots have a number of nice properties, some of […]

Pingback by Some Lemmas on Simple Roots « The Unapologetic Mathematician | February 4, 2010 | Reply

14. […] of Irreducible Root Systems I Now we can turn towards the project of classifying irreducible root systems up to isomorphism. And we start with some properties of irreducible root […]

Pingback by Properties of Irreducible Root Systems I « The Unapologetic Mathematician | February 10, 2010 | Reply

15. […] 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 […]

Pingback by Cartan Matrices « The Unapologetic Mathematician | February 16, 2010 | Reply

16. […] 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 […]

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

17. […] of (Possible) Root Systems At long last, we can state the classification of irreducible root systems up to isomorphism. We’ve shown that for each such root system we can construct a connected […]

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

18. […] the Classification Theorem I This week, we will prove the classification theorem for root systems. The proof consist of a long series of steps, and we’ll split it up over a number of […]

Pingback by Proving the Classification Theorem I « The Unapologetic Mathematician | February 22, 2010 | Reply

19. […] so we must construct some actual root systems. For this task, we let stand for a finite-dimensional real vector space for various , equipped […]

Pingback by Construction of Root Systems (setup) « The Unapologetic Mathematician | March 1, 2010 | Reply

20. […] of A-Series Root Systems Starting from our setup, we construct root systems corresponding to the Dynkin […]

Pingback by Construction of A-Series Root Systems « The Unapologetic Mathematician | March 2, 2010 | Reply

21. […] of D-Series Root Systems Starting from our setup, we construct root systems corresponding to the Dynkin diagrams (for […]

Pingback by Construction of D-Series Root Systems « The Unapologetic Mathematician | March 3, 2010 | Reply

22. […] of B- and C-Series Root Systems Starting from our setup, we construct root systems corresponding to the (for ) and (for ) Dynkin diagrams. First will be the […]

Pingback by Construction of B- and C-Series Root Systems « The Unapologetic Mathematician | March 4, 2010 | Reply

23. […] of the G2 Root System We’ve actually already seen the root system, back when we saw a bunch of two-dimensional root system. But let’s examine how we can […]

Pingback by Construction of the G2 Root System « The Unapologetic Mathematician | March 8, 2010 | Reply

24. […] of E-Series Root Systems Today we construct the last of our root systems, following our setup. These correspond to the Dynkin diagrams , , and . But there are […]

Pingback by Construction of E-Series Root Systems « The Unapologetic Mathematician | March 10, 2010 | Reply

25. […] laying down some definitions on reflections, we defined a root system as a collection of vectors with certain properties. Specifically, each vector is a point in a […]

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

26. […] Lie algebras, and/or algebraic groups; the former are very important in understanding the latter. John Armstrong over at the Unapologetic Mathematician has a series on root systems. In addition, for a […]

Pingback by Coxeter groups « Annoying Precision | June 27, 2010 | Reply

27. both program and data

I like to say “both verb and noun” — keep it closer to elementary school level when possible 😉 Comment by isomorphismes | March 11, 2015 | Reply

• “program and data”, on the other hand, keeps it closer to the topics in Gödel Escher Bach, which a lot of math people are also fans of. Comment by John Armstrong | March 11, 2015 | Reply

28. Reblogged this on Human Mathematics. Comment by isomorphismes | May 15, 2015 | Reply