Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 3 / 5 / iii

The classification of finite-dimensional sl2 representations identifies $L(n)$ with ${\mathrm{Sym}}^n{\mathbb{C}}^2$. In the defining two-dimensional representation,so $e^2=f^2=0$ andOn the $n$th tensor power, the infinitesimal operator associated with $e$ is the sum of its actions on the individual factors. These actions commute, so exponentiating their sum is the tensor product of the individual exponentials; the same holds for $f$. Restricting to the symmetric power consequently identifies the operator $s$ on $L(n)$ with ${\mathrm{Sym}}^{n}S$. It follows thatIndeed $-I$ contributes one minus sign from each of the $n$ tensor factors. The Weyl reflection lift in an sl2 representation can therefore square to minus the identity, even though its image in the Weyl group is an involution.