A Note on Orbits of Subgroups of the Permutation Groups

Extract

In [2] we studied Milgram's Complex, C(n – 1), which was first defined in [3], in the following manner. Let S_n, the permutation group of n symbols, act on R_n in the obvious manner; put α(x) = (y), where y_i = x_α–1_(i). Let s = (1, 2, …, n), then C(n – 1) is the convex hull of the points α(s), α ∊ S_n. Here we shall generalise this construction as follows. Let G be a subgroup of S_n, and let v ∊ Rⁿ. Then C(G, v) is the convex hull of α(v), α ∊ G. We prove invariance over v subject to certain restrictions, give counter-examples to shew lack of invariance if we alter G, discuss how we may describe C(G, v), shew that the only “nice” case is essentially when G is S_n, and lastly give some examples.