×

Permutation notations for the exceptional Weyl group \(F_4\). (English) Zbl 1259.20056

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 \(G_2\).

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
PDFBibTeX XMLCite
Full Text: DOI Link