On the Livingstone-Wagner Theorem

Abstract

Let $G$ be a permutation group on the set $\Omega$ and let ${\cal S}$ be a collection of subsets of $\Omega,$ all of size $\geq m$ for some integer $m$. For $s\leq m$ let $n_{s}(G,\,{\cal S})$ be the number of $G$-orbits on the subsets of $\Omega$ which have a representative $y\subseteq x$ with $|y|=s$ and $y\subseteq x$ for some $x\in {\cal S}$. We prove that if $s < t$ with $s+t\leq m$ then $n_{s}(G,\,{\cal S})\leq n_{t}(G,\,{\cal S})$. A special case of this theorem is the Livingstone-Wagner Theorem when ${\cal S}=\{\Omega\}$. We show how the result can be applied to estimate orbit numbers for simplicial complexes, sequences, graphs and amalgamation classes. It is also shown how this theorem can be extended to orbit theorems on more general partially ordered sets.

Permutation Modules associated to the Hyperoctahedron and Group Actions

“…when t+k ≤ n, and this is not difficult to verify from the theorem. In fact, this property holds for the corresponding incidence matrices in any pure simplicial complex, see [7].…”

“…Therefore we have the well-known fact that in the Boolean lattice the incidence maps have maximum rank. Rank maximality is common in many incidence structures, including finite projective spaces, see [7].…”

Top-heavy phenomena for transformations

“…for all f ∈ Q ( Ω k ) and B ∈ Ω ℓ . Following a common approach in establishing homogeneity of permutation groups [15,41] [20, pp. 20-22], we will make use of the ensuing result on the rank of the subset inclusion matrix.…”

Homogeneity and related topics: An extended bibliography

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