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

Consider the Sn-action on Rn given by permuting coordinates. This paper addresses the following problem: compute maxv,H⁡|H∩Snv| as H⊂Rn ranges over all hyperplanes through the origin and v∈Rn ranges over all vectors with distinct coordinates that are not contained in the hyperplane ∑xi=0. We conject...

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Bibliographic Details
Published inLinear algebra and its applications Vol. 613; pp. 1 - 23
Main Authors Huang, Jiahui, McKinnon, David, Satriano, Matthew
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier Inc 15.03.2021
American Elsevier Company, Inc
Subjects
Hyperplane
Hyperplanes
Linear algebra
Permutation
Symmetric group
secondary
Permutation
Symmetric group
Hyperplane
primary
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ISSN0024-3795
1873-1856
DOI10.1016/j.laa.2020.12.011

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Summary:Consider the Sn-action on Rn given by permuting coordinates. This paper addresses the following problem: compute maxv,H⁡|H∩Snv| as H⊂Rn ranges over all hyperplanes through the origin and v∈Rn ranges over all vectors with distinct coordinates that are not contained in the hyperplane ∑xi=0. We conjecture that for n≥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≤n, then maxv,H⁡|H∩Snv|≤n!p. In particular, this proves the conjecture when n or n−1 is prime.
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ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2020.12.011