Open AccessTridiagonal pairs of type III with height one
Author(s) -
Xue Li,
Bo Hou,
Suogang Gao
Publication year - 2019
Publication title -
the electronic journal of linear algebra
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.447
H-Index - 31
eISSN - 1537-9582
pISSN - 1081-3810
DOI - 10.13001/ela.2019.5161
Subject(s) - tridiagonal matrix , mathematics , dimension (graph theory) , integer (computer science) , combinatorics , eigenvalues and eigenvectors , algebraically closed field , type (biology) , basis (linear algebra) , vector space , space (punctuation) , field (mathematics) , discrete mathematics , pure mathematics , geometry , physics , ecology , linguistics , philosophy , quantum mechanics , computer science , biology , programming language
Let K denote an algebraically closed field with characteristic 0. Let V denote a vector space over K with finite positive dimension, and let A, A∗ denote a tridiagonal pair on V of diameter d. Let V0, . . . , Vd denote a standard ordering of the eigenspaces of A on V , and let θ0, . . . , θd denote the corresponding eigenvalues of A. It is assumed that d ≥ 3. Let ρi denote the dimension of Vi. The sequence ρ0, ρ1, . . . , ρd is called the shape of the tridiagonal pair. It is known that ρ0 = 1 and there exists a unique integer h (0 ≤ h ≤ d/2) such that ρi−1 ρi for d − h < i ≤ d. The integer h is known as the height of the tridiagonal pair. In this paper, it is showed that the shape of a tridiagonal pair of type III with height one is either 1, 2, 2, . . ., 2, 1 or 1, 3, 3, 1. In each case, an interesting basis is found for V and the actions of A, A∗ on this basis are described.