Paths of unimodular vectors
type: Item
instance of: scholarly article
title: Paths of unimodular vectors
zbMATH Open document ID: 0766.20015
DOI: 10.1016/0021-8693(91)90216-U
author: Edward K. Hinson
published in: Journal of Algebra
publication date: 1992-06-26T00:00:00Z
review text: Let \(R\) be a commutative ring with 1, \(R^ n \) the free \(R\)-module of \(n\)-columns over \(R\), \(Um(n, R)\) the set of all unimodular columns in \(R^ n\). The author introduces a graph structure on \(Um(n, R)\) connecting two columns by an edge if their dot product is 1. His Theorem 1.6 asserts that when \(n \geq \) 3 the orbit \(E(n, R)\alpha\) of any \(\alpha \in R^ n\) under the subgroup \(E(n, R) \subset GL(n, R) \) generated by elementary matrices, consists of the columns \(\beta \in R^ n\) which can be connected with \(\alpha\) by a path of even length. Then the author studies the diameter of the graph. He proves that the diameter is 2 (i.e., as small as possible) if and only if \(R\) is Boolean. For any \(R\), he connects the diameter with the word length in \(E(n, R)\).
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20G35
zbMATH DE Number: 23458
zbMATH Keywords: unimodular columns
zbMATH Keywords: graph
zbMATH Keywords: orbit
zbMATH Keywords: elementary matrices
zbMATH Keywords: word length
reviewed by: L. N. Vaserstein
MaRDI profile type: Publication
cites work: \(K\)-theory and stable algebra
cites work: On a question of M. Newman on the number of commutators
cites work: On vaserstein's power operation on elementary orbits
cites work: Word length in elementary matrices
cites work: Q3836619
cites work: A group structure on certain orbit sets of unimodular rows
cites work: A module structure on certain orbit sets of unimodular rows
cites work: Serre's conjecture
cites work: On Unimodular Rows
cites work: ON STABLY FREE MODULES
cites work: ON THE STRUCTURE OF THE SPECIAL LINEAR GROUP OVER POLYNOMIAL RINGS
cites work: Stable rank of rings and dimensionality of topological spaces
cites work: Operations on orbits of unimodular vectors
description: scientific article; zbMATH DE number 23458