The Unapologetic Mathematician

Mathematics for the interested outsider

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 | 26 Comments

   

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