Gap conditions for representations of symmetric groups
In the process of constructing smooth group actions with exotic fixed point behaviour, one usually needs linear representations satisfying gap conditions between dimensions of fixed point sets for certain pairs of subgroups of the group. Standard methods need so-called admissible representations, while a much more complicated method due to \textit{A. Bak} and \textit{M. Morimoto} [Forum Math. 8, No. 3, 267-302 (1996; Zbl 0860.57028)] can do with almost admissible representations. In a 1993 preprint, Laitinen and Morimoto studied a specific representation giving rise to almost admissible representations. For every Oliver group, Bak-Morimoto surgery then shows existence of one fixed point actions on spheres, in particular for \(G=S(n), n\geq 5\), the symmetric group of order \(n\). Using intricate group theoretic arguments and calculations with the software package GAP, the authors modify this approach to yield admissible representations for many groups. This makes standard surgery applicable to produce one fixed point actions, in particular for \(G=S(n), n\geq 6\). (It is known that admissible representations do not exist for \(S(5)\).) Moreover, for Oliver groups with these admissible representations, the authors produce -- modifying ideas of Pawałowski -- smooth actions on spheres whose fixed point set is any given closed smooth stably parallelizable manifold all of whose components have the same dimension.
scientific article; zbMATH DE number 1038608
scientific article