Action of Symmetry Groups
8 Pages Posted: 9 Apr 2009
Date Written: 1996
Abstract
This paper studies maps which are invariant under the action of the symmetry group S,. The problem originates in social choice theory: there arc k individuals each with a space of preferences X, and a social choice map phi:Xk - X which is anonymous i.e. invariant under the action of a group of symmetries. Theorem 1 proves that a full range map Psi: Xk -> X exists which is invariant under the action of S, only if, for all i>1, the elements of the homotopy group IL(X) have orders relatively prime with k. Theorem 2 derives a similar results for actions of subgroups of the group Sk. Theorem 3 proves necessary and sufficient condition for a parafinite CW complex X to admit full range invariant maps for any prime number k:X must be contractible.
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