Cahn, Patricia; Haas, Ruth; Helminck, Aloysius G.; Li, Juan; Schwartz, Jeremy Permutation notations for the exceptional Weyl group \(F_4\). (English) Zbl 1259.20056 Involve 5, No. 1, 81-89 (2012). 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 \(G_2\). Reviewer: Shi Jian-yi (Shanghai) Cited in 2 Documents MSC: 20G15 Linear algebraic groups over arbitrary fields 20F55 Reflection and Coxeter groups (group-theoretic aspects) 20F05 Generators, relations, and presentations of groups 05E15 Combinatorial aspects of groups and algebras (MSC2010) 20G20 Linear algebraic groups over the reals, the complexes, the quaternions Keywords:Weyl groups; Coxeter groups; one-line notations; signed permutations PDFBibTeX XMLCite \textit{P. Cahn} et al., Involve 5, No. 1, 81--89 (2012; Zbl 1259.20056) Full Text: DOI Link