Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 1 / 4 / c

Identify the diagonal Cartan subalgebra with ${\mathbb{C}}^n$ through its coordinates $(h_1,\dots ,h_n)$, and identify real roots with coordinate vectors using the invariant Euclidean inner product. A Weyl reflection is $s_{\alpha }(h)=h-2(h,\alpha )\alpha /(\alpha ,\alpha )$. It does not change when $\alpha$ is replaced by $-\alpha$. For the Cn root system, its explicit actions areEvery coordinate not displayed stays fixed. Thus each generating Weyl reflection is a signed permutation, so the Weyl group is contained in the signed symmetric group. Conversely, the first row supplies each independent sign change and the second supplies every coordinate transposition; these generate the entire signed symmetric group. This proves the Cn Weyl group identification, including its size: