Tree Matrices and a Matrix Reduction Algorithm of Belitskii

Article type: Research Article

Authors: Grzecza, Marcin | Kasjan, Stanisław | Mróz, Andrzej

Affiliations: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland, grzeczny@mat.umk.pl; skasjan@mat.umk.pl; amroz@mat.umk.pl

Note: [] Address for correspondence: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

Note: [] The second and the third author are supported by the research grant N N201 542138 of the Polish Ministry of Science and Higher Education.

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.

DOI: 10.3233/FI-2012-713

Journal: Fundamenta Informaticae, vol. 118, no. 3, pp. 253-279, 2012