Geometry of rectangular block triangular matrices

Abstract

Let D be any division ring, and let \( T_{(m_i ,n_i ,k)} \) be the set of k × k (k ≥ 2) rectangular block triangular matrices over D. For A, B ∈ \( T_{(m_i ,n_i ,k)} \), if rank(A − B) = 1, then A and B are said to be adjacent and denoted by A ∼ B. A map φ: \( T_{(m_i ,n_i ,k)} \to T_{(m_i ,n_i ,k)} \) is said to be an adjacency preserving map in both directions if A ∼ B if and only if φ(A) ∼ φ(B). Let G be the transformation group of all adjacency preserving bijections in both directions on \( T_{(m_i ,n_i ,k)} \). When m1,nk ≥ 2, we characterize the algebraic structure of G, and obtain the fundamental theorem of rectangular block triangular matrices over D.