= Weyl reflection lift in an sl2 representation
{c}
{title2=$s=\exp(e)\exp(-f)\exp(e)$}
In a finite-dimensional <sl2 Lie algebra> representation, raising and lowering operators are <nilpotent operators>, so their exponentials are finite sums. In the defining two-dimensional representation, the displayed product is $\begin{pmatrix}0&1\\-1&0\end{pmatrix}$. It sends $h$ to $-h$ by conjugation, lifting the nontrivial <Weyl group> reflection. Its square acts on the irreducible highest-weight module $L(n)=\operatorname{Sym}^n\mathbb C^2$ by $(-1)^n$. Thus a Weyl reflection can have a representative of order four on a representation.
// Target: linear-operator-theory.bigb
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