The Unapologetic Mathematician

Mathematics for the interested outsider

The Weyl Group of a Root System

Let’s take a root system \Phi in the inner product space V. Each vector \alpha in \Phi gives rise to a reflection in \sigma_\alpha\in\mathrm{O}(V), the group of transformations preserving the inner product on V. So what sorts of transformations can we build up from these reflections? The subgroup of \mathrm{O}(V) generated by the reflections \sigma_\alpha for all \alpha\in\Phi is called the Weyl group \mathcal{W} of the root system. It’s pronounced “vile”, but we don’t mean that as any sort of value judgement.

Anyway, we can also realize \mathcal{W} as a subgroup of the group of permutations on the vectors in \Phi. Indeed, by definition each \sigma_\alpha sends each vector in \Phi back to another vector in \Phi, and so shuffles them around. So if \Phi has k vectors, the Weyl group can be realized as a subgroup of S_k.

In particular, \mathcal{W} is a finite group, as a subgroup of another finite group. In fact, we even know that the number of transformations in \mathcal{W} divides k!. It may well (and usually does) have elements which are not of the form \sigma_\alpha, but there are still only a finite number of them.

The first thing we want to take note of is how certain transformations in \mathrm{GL}(V) act on \mathcal{W} by conjugation. Specifically, if \tau leaves \Phi invariant, then it induces an automorphism on \mathcal{W} that sends the generator \sigma_\alpha to \tau\sigma_\alpha\tau^{-1} — which (it turns out) is the generator \sigma_{\tau(\alpha)} — for all \alpha\in\Phi. Further, it turns out that \beta\rtimes\alpha=\tau(\beta)\rtimes\tau(\alpha) for all \alpha,\beta\in\Phi.

Indeed, we can calculate

\displaystyle\left[\tau\sigma_\alpha\tau^{-1}\right](\tau(\beta))=\tau(\sigma_\alpha(\beta))=\tau(\beta-(\beta\rtimes\alpha)\alpha)=\tau(\beta)-(\beta\rtimes\alpha)\tau(\alpha)

Now, every vector in \Phi is of the form \tau(\beta) for some \beta, and so \tau\sigma_\alpha\tau^{-1} sends it to the vector \tau(\sigma_\alpha(\beta)), which is again in \Phi, so it leaves \Phi invariant. The transformation \tau\sigma_\alpha\tau^{-1} also fixes every vector in the hyperplane \tau(P_\alpha), for if \beta is orthogonal to \alpha, then the above formula shows that \tau(\beta) is left unchanged by the transformation. Finally, \tau\sigma_\alpha\tau^{-1} sends \tau(\alpha) to -\tau(\alpha).

This is all the data we need to invoke our lemma, and conclude that \tau\sigma_\alpha\tau^{-1} is actually equal to \sigma_{\tau(\alpha)}. Specifying the action on the generators of \mathcal{W} is enough to determine the whole automorphism. Of course, we can also just let \sigma act on each element of \mathcal{W} by conjugation, but it’s useful to know that the generating reflections are sent to each other exactly as their corresponding vectors are.

Now we can calculate from the definition of a reflection

\displaystyle\left[\tau\sigma_\alpha\tau^{-1}\right](\tau(\beta))=\sigma_{\tau(\alpha)}(\tau(\beta))=\tau(\beta)-(\tau(\beta)\rtimes\tau(\alpha))\tau(\alpha)

Comparing this with the equation above, we find that \tau(\beta)\rtimes\tau(\alpha)=\beta\rtimes\alpha, as asserted.

January 21, 2010 Posted by | Algebra, Geometry, Group Actions, Group theory, Root Systems | 16 Comments

   

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