循环 Hessenberg 对与三对角关系
Circular Hessenberg pairs and the tridiagonal relations
AI总结：
本文证明了 2022 年李在镐的推测，即循环 Hessenberg 对满足三对角关系，证明方式并非初等，核心在于对循环 Hessenberg 对相关性质的研究及对该推测的论证。
AI中文摘要：
当方阵次对角线以下的每个元素为零且次对角线上的每个元素非零时，该方阵称为 Hessenberg 矩阵。当右上角元素非零且超对角线以上的所有其他元素为零时，Hessenberg 矩阵称为循环矩阵。循环 Hessenberg 对由非零有限维向量空间上的两个可对角化线性映射组成，它们以循环 Hessenberg 方式作用于彼此的特征基。2022 年，李在镐推测循环 Hessenberg 对满足两个称为三对角关系的关系。在本文中，我们证明了李的推测。我们的证明不是初等的。
英文摘要：
A square matrix is said to be Hessenberg whenever each entry below the subdiagonal is zero, and each entry on the subdiagonal is nonzero. A Hessenberg matrix is called circular whenever the top-right corner entry is nonzero, and every other entry above the superdiagonal is zero. A circular Hessenberg pair consists of two diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on an eigenbasis of the other one in a circular Hessenberg fashion. In 2022, Jae-ho Lee conjectured that a circular Hessenberg pair satisfies two relations called the tridiagonal relations. In the present paper, we prove Lee's conjecture. Our proof is not elementary.
