On extendability of permutations

TL;DR

This paper characterizes finite subsets of vector spaces and projective spaces where all permutations can be extended to linear automorphisms or projective transformations, linking the results to symmetric group representations.

Contribution

It provides a complete description of subsets allowing permutation extensions to automorphisms or projective transformations, connecting to symmetric group representations.

Findings

Characterization of subsets with permutation extension property

Description of subsets in projective space with permutation extension

Reformulation in terms of symmetric group representations

Abstract

Let $V$ be a left vector space over a division ring and let ${\mathcal P}(V)$ be the associated projective space. We describe all finite subsets $X\subset V$ such that every permutation on $X$ can be extended to a linear automorphism of $V$ and all finite subsets ${\mathcal X}\subset {\mathcal P}(V)$ such that every permutation on ${\mathcal X}$ can be extended to an element of ${\rm PGL}(V)$. Also, we reformulate the results in terms of linear and projective representations of symmetric groups.