Thin Hessenberg Pairs and Double Vandermonde Matrices

TL;DR

This paper introduces and explores the concept of thin Hessenberg pairs, establishing a connection with double Vandermonde matrices and providing bijections among various algebraic structures related to these pairs.

Contribution

It defines TH pairs and systems, and establishes bijections between their isomorphism classes, parameter arrays, and Vandermonde matrices, advancing the algebraic understanding of these structures.

Findings

Established a bijection between TH systems and double Vandermonde matrices.

Connected isomorphism classes of TH systems with parameter arrays.

Provided a classification framework for TH pairs using Vandermonde matrices.

Abstract

A square matrix is called {\it Hessenberg} whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let $V$ denote a nonzero finite-dimensional vector space over a field $\fld$. We consider an ordered pair of linear transformations $A: V \rightarrow V$ and $A^*: V \rightarrow V$ which satisfy both (i), (ii) below. (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is Hessenberg and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is Hessenberg. \noindent We call such a pair a {\it thin Hessenberg pair} (or {\it TH pair}). By the {\it diameter} of the pair we mean the dimension of $V$ minus one. There is an "oriented" version of a TH pair called a TH system. In this paper we investigate a connection between TH systems and double Vandermonde matrices. We give a bijection between any two of the following three sets: \cdot The set of isomorphism classes of TH systems over $\K$ of diameter $d$. \cdot The set of normalized west-south Vandermonde systems in $\Mdf$. \cdot The set of parameter arrays over $\K$ of diameter $d$. We give a bijection between any two of the following five sets: \cdot The set of affine isomorphism classes of TH systems over $\K$ of diameter $d$. \cdot The set of isomorphism classes of RTH systems over $\K$ of diameter $d$. \cdot The set of affine classes of normalized west-south Vandermonde systems in $\Mdf$. \cdot The set of normalized west-south Vandermonde matrices in $\Mdf$. \cdot The set of reduced parameter arrays over $\K$ of diameter $d$.