Symmetry Groups of Cephoids. Part 1: Square Cephoids

We continue the analysis of Cephoids. A Cephoid is a Minkowski sum of finitely many standardized simplices ("deGua simplices''). Pareto faces of a Cephoid are maximal polyhedra facing "outward''. A Pareto face consists of an algebraic sum of subsimplices of the deGua Simplices involved, the dimension of which is well specified. As a consequence, a Pareto face is characterized by its reference system, i.e., a system of subsets of the coordinates indicating the various subsimplices involved. Within this first presentation we focus on Cephoids in $n$ dimensions with $K=n$ deGua simplices involved ("square'' Cephoids). A reordering of the deGua simplices involved in a Cephoid is effected by a permutation. Permutations act also on the Pareto faces of the Cephoid, more preciseley, on the representing reference systems. The combined actions of a permutation allow for the definition of symmetries. In good tradition (Felix Klein) symmetries of a geometrical body (in our case a Cephoid) amount to subgroups of permutations which leave the body unchanged. Here we focus on the full group of all permutations (the Symmetric Group) in order to specify symmetric Cephoids.