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.

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

16 Comments »

  1. [...] from this, we find that the Weyl group of not only acts on itself, but on . Indeed, induces a homomorphism that sends the generator [...]

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

  2. You write that:

    …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.

    It’s clear that this defines a homomorphism from the Weyl group into S_k. But it doesn’t seem trivially obvious that this is injective — a priori, couldn’t there be two funny products of \sigma_\alpha‘s which aren’t equal but agree on elements of \Phi?

    Comment by Peter LeFanu Lumsdaine | January 24, 2010 | Reply

  3. That’s a good question, Peter. But there can’t be, because \Phi spans the whole vector space. So if any two linear transformations agree on \Phi, they must agree on the whole vector space, and thus must be the same transformation.

    On the other hand, you might be thinking that some linear transformation in \mathcal{W} can be written as a product of the \sigma_\alpha in two different ways. This is possible, but it leads to a relation in the Weyl group. Notice that I never said that \mathcal{W} was freely-generated (beyond the obvious fact that each generator is an involution).

    Does that make sense?

    Comment by John Armstrong | January 25, 2010 | Reply

    • Ah! I’d forgotten that \Phi spans the space. OK, it’s obvious :-) (It was indeed your first interpretation that I meant, not your second.)

      In fact, I guess even if we don’t assume that \Phi spans the space, they agree on the span of \Phi as you say, and they also act trivially on the orthogonal complement of \Phi (since each \sigma_\alpha does), so still agree on the whole space.

      Comment by Peter LeFanu Lumsdaine | January 25, 2010 | Reply

  4. [...] are the same as the original reflections, and so they generate the same subgroup of . That is, the Weyl group of is the same as the Weyl group of [...]

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

  5. [...] Weyl group of shuffles Weyl chambers around. Specifically, if and is regular, then [...]

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

  6. [...] our assumption, and . Thus there is some smallest index so that . Then , and we must have . But we know that . In [...]

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

  7. [...] Weyl Chambers With our latest lemmas in hand, we’re ready to describe the action of the Weyl group of a root system on the set of its Weyl chambers. Specifically, the action is “simply [...]

    Pingback by The Action of the Weyl Group on Weyl Chambers « The Unapologetic Mathematician | February 5, 2010 | Reply

  8. [...] from earlier about simple roots and reflections, we can define a few notions that make discussing Weyl groups easier. Any Weyl group element can be written as a composition of simple [...]

    Pingback by Lengths of Weyl Group Elements « The Unapologetic Mathematician | February 8, 2010 | Reply

  9. [...] set — allowing — and show that it’s a fundamental domain for the action of the Weyl group [...]

    Pingback by The Fundamental Weyl Chamber « The Unapologetic Mathematician | February 9, 2010 | Reply

  10. [...] to each one in , then we will find a similar decomposition of . But we know from our study of the Weyl group that every root in can be sent by the Weyl group to some simple root in . So we define to be the [...]

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

  11. [...] is irreducible, then the Weyl group acts irreducibly on . That is, we cannot decompose the representation of on as the direct sum of [...]

    Pingback by Properties of Irreducible Root Systems II « The Unapologetic Mathematician | February 11, 2010 | Reply

  12. [...] that any combination of the serves to permute the coefficients of a given vector. That is, the Weyl group of the system is naturally isomorphic to the symmetric group . Possibly related posts: [...]

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

  13. [...] of all, right when we first talked about the category of root systems, we saw that the Weyl group of is a normal subgroup of . This will give us most of the structure we need, but there may be [...]

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

  14. [...] a collection of reflections, and these reflections generate a group of transformations called the Weyl group of the root system. It’s one of the most useful tools we have at our disposal through the [...]

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

  15. “pronounced “vile”, but we don’t mean that as any sort of value judgement”, lovely really enjoyed reading this blog.

    Comment by James Pearson | April 2, 2013 | Reply


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