Maximal antipodal sets of \(E_6\) and some compact symmetric spaces
Given a compact Riemannian symmetric space \(M\), a set is called antipodal if its elements are fixed by a given geodesic symmetry. It is called great if it is maximal and if its cardinality is also maximal. The cardinality is then called the 2-number of \(M\). Antipodal sets were introduced in [\textit{B.-Y. Chen} and \textit{T. Nagano}, Trans. Am. Math. Soc. 308, No. 1, 273--297 (1988; Zbl 0656.53049)]. The author considers the classification and constructions of congruent classes of maximal antipodal sets of the exceptional compact Lie group \(E_6\) and compact symmetric spaces of type EI, \dots, EIV. The authors continue the works of [\textit{R. L. Griess jun.}, Geom. Dedicata 39, No. 3, 253--305 (1991; Zbl 0733.20023)]. In addition to classifying them, the author constructs them. The author introduces a strategy to classify congruent classes of maximal antipodal sets of \(E_6\) and type EI, \dots, EIV. It is shown that the number of congruent classes of maximal antipodal sets of \(E_6\) is 2 and the cardinalities of maximal antipodal sets contains in each congruent classes are 64 and 32. For the type EII, it is shown that the number of congruent classes of maximal antipodal sets of type EII is 2 and the cardinalities of maximal antipodal sets contained in each congruent classes are 36 and 27. A realization of type EII is given and maximal antipodal sets are constructed. Similar results are given for types EIII, EIV and EI.
scientific article; zbMATH DE number 7622614