What fraction of an $S_n$-orbit can lie on a hyperplane?

Abstract

Consider the $S_n$-action on ${\mathbb{R}}^n$ given by permuting coordinates. This paper addresses the following problem: compute ${\max }_{v,H}|H\cap S_{n}v|$ as $H\subset {\mathbb{R}}^n$ ranges over all hyperplanes through the origin and $v\in {\mathbb{R}}^n$ ranges over all vectors with distinct coordinates that are not contained in the hyperplane $\sum x_i=0$. We conjecture that for $n\ge 3$, the answer is $(n-1)!$ for odd $n$, and $n(n-2)!$ for even $n$. We prove that if $p$ is the largest prime with $p\le n$, then ${\max }_{v,H}|H\cap S_{n}v|\le \frac{n!}{p}$. In particular, this proves the conjecture when $n$ or $n-1$ is prime.

16 pages

Cited by in corpus (1)

• The fraction of an $S_n$-orbit on a hyperplane