Tree Matrices and a Matrix Reduction Algorithm of Belitskii

Abstract

Inspired by the bimodule matrix problem technique and various classification problems in poset representation theory, finite groups and algebras, we study the action of Belitskii algorithm on a class of square n by n block matrices M with coefficients in a field K. One of the main aims is to reduce M to its special canonical form M∞ with respect to the conjugation by elementary transformations defined by a class of matrices chosen in a subalgebra of the full matrix algebra $\mathbb{M}_n$(K). The algorithm can be successfully applied in the study of indecomposable linear representations of finite posets by a computer search using numeric and symbolic computation. We mainly study the case when the di-graph (quiver) associated to the output matrix M∞ of the algorithm is a disjoint union of trees. We show that exceptional representations of any finite poset are determined by tree matrices. This generalizes a theorem of C.M. Ringel proved for linear representations of di-graphs.