Permutation notations for the exceptional Weyl group \(F_4\). (Q444613)

The root system \(\Phi\) of type \(F_4\) contains 24 short roots, say \(\Phi_1=\{\pm\alpha_i\mid 1\leqslant i\leqslant 12\}\). The Weyl group \(W\) of type \(F_4\) faithfully acts on the set \(\Phi_1\). Hence each element of \(W\) can be expressed as a signed permutation on the set \(X=\{1,2,\dots,12\}\). The paper describes the signed permutations on \(X\) corresponding to the elements of \(W\). A number of properties of such a presentation are discussed. A similar presentation is given for the Weyl group of type \(G_2\).