= Reflection representation of a Weyl group
{title2=$W\longrightarrow O(\operatorname{span}_{\mathbb R}R)$}
The reflection representation is the action of the <Weyl group> on the real span of its <root system>, generated by its <Weyl reflections>. For an <irreducible root system>, it is an <irreducible representation>: if $U$ is an <invariant subspace>, each <root> either lies in $U^\perp$ or, by subtracting a reflected vector of $U$, lies in $U$. These alternatives partition the roots into <orthogonal> sets. Irreducibility and spanning force $U=0$ or the whole space.
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