What fraction of an S-orbit can lie on a hyperplane?
| Autor: | Matthew Satriano, Jiahui Huang, David McKinnon |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Numerical Analysis
Algebra and Number Theory Conjecture 010102 general mathematics 01 natural sciences Action (physics) Prime (order theory) Combinatorics Hyperplane 0103 physical sciences Discrete Mathematics and Combinatorics Fraction (mathematics) 010307 mathematical physics Geometry and Topology 0101 mathematics Orbit (control theory) Mathematics |
| Zdroj: | Linear Algebra and its Applications. 613:1-23 |
| ISSN: | 0024-3795 |
| Popis: | Consider the S n -action on R n given by permuting coordinates. This paper addresses the following problem: compute max v , H | H ∩ S n v | as H ⊂ R n ranges over all hyperplanes through the origin and v ∈ R n ranges over all vectors with distinct coordinates that are not contained in the hyperplane ∑ x i = 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 max v , H | H ∩ S n v | ≤ n ! p . In particular, this proves the conjecture when n or n − 1 is prime. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|
Full Text from ScienceDirect