The Unapologetic Mathematician

Mathematics for the interested outsider

Properties of Irreducible Root Systems II

We continue with our series of lemmas on irreducible root systems.

If \Phi is irreducible, then the Weyl group \mathcal{W} acts irreducibly on V. That is, we cannot decompose the representation of \mathcal{W} on V as the direct sum of two other representations. Even more explicitly, we cannot write V=W\oplus W' for two nontrivial subspaces W and W' with each one of these subspaces invariant under \mathcal{W}. If W is an invariant subspace, then the orthogonal complement W' will also be invariant. This is a basic fact about the representation theory of finite groups, which I will simply quote for now, since I haven’t covered that in detail. Thus my assertion is that if W is an invariant subspace under \mathcal{W}, then it is either trivial or the whole of V.

For any root \alpha\in\Phi, either \alpha\in W or W\subseteq P_\alpha. Indeed, since \sigma_\alpha\in\mathcal{W}, we must have \sigma_\alpha(W)=W. As a reflection, \sigma_\alpha breaks V into a one-dimensional eigenspace with eigenvalue -1 and another complementary eigenspace with eigenvalue {1}. If W contains the -1-eigenspace, then \alpha\in W. If not, then \alpha is perpendicular to W or W couldn’t be invariant under \sigma_\alpha, and in this case W\subseteq P_\alpha.

So then if \alpha isn’t in W then it must be in the orthogonal complement W'. Thus every root is either in W or in W', and this gives us an orthogonal decomposition of the root system. But since \Phi is irreducible, one or the other of these collections must be empty, and thus W must be either trivial or the whole of V.

Even better, the span of the \mathcal{W}-orbit of any root \alpha\in\Phi spans V. Indeed, the subspace spanned by roots of the form \sigma(\alpha) is invariant under the action of \mathcal{W}, and so since V is irreducible it must be either trivial (clearly impossible) or the whole of V.

February 11, 2010 Posted by | Geometry, Root Systems | 4 Comments

   

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