Symmetries of voltage operations on polytopes, maps and maniplexes

Abstract

Voltage operations extend traditional geometric and combinatorial operations (such as medial, truncation, prism, and pyramid over a polytope) to operations on maniplexes, maps, polytopes, and hypertopes. In classical operations, the symmetries of the original object remain in the resulting one, but sometimes additional symmetries are created; the same situation arises with voltage operations. We characterise the automorphisms of the new object that are derived from the original one and use this to bound the number of flag orbits (under the action of its automorphism group) of the new object in terms of the original one. The conditions under which the automorphism group of the original object is the same as the automorphism group of the resulting object are given. We also look at the cases where there is additional symmetry which can be accurately described due to the symmetries of the operation itself.