Algebra of matrix arithmetic
type: Item
instance of: scholarly article
title: Algebra of matrix arithmetic
zbMATH Open document ID: 0921.15007
author: Gautami Bhowmik
author: Olivier Ramaré
published in: Journal of Algebra
publication date: 1999-09-15T00:00:00Z
review text: Let \(M\) be an invertible \(r\times r\) matrix with integer coefficients and identify it with the linear mapping \(Z^r\to Z^r\) induced by it. Two integer matrices \(A,B\) are called equivalent, if \(\det(AB^{-1})=\pm 1\). Denote by \(LD(M)\) the set of equivalence classes of left divisors of \(M\) ordered by divisibility, and let \(G(M)\) be the cokernel of \(M\), i.e. the factor group \(Z^r/M(Z^r)\). The authors show that there is an order-preserving one-to-one map between \(LD(M)\) and the set of all subgroups of \(G(M)\). This permits them to give simple proofs of certain results concerning integral matrices, obtained earlier by \textit{R. C. Thompson} [Linear Multilinear Algebra 19, 287-295 (1986; Zbl 0589.15010); Proc. Am. Math. Soc. 86, 9-11 (1982; Zbl 0503.15007)]. This approach leads also to interesting links between integral matrices, partitions of integers and subgroups of finite abelian \(p\)-groups with a prescribed factorization into cyclic factors. Finally, functions of matrices are considered. The authors call a function \(f(M)\) arithmetical if its value depends only on the Smith normal form of \(M\) and show that the Dirichlet convolution of such functions can be used to define the product in a related Hecke algebra. They show also that for \(r=1,2,3,\dots\) the \(C\)-algebras of all arithmetical functions on \(r\times r\) integer matrices are all isomorphic to the algebra of formal power series in countably many variables.
Mathematics Subject Classification ID: 15B36
Mathematics Subject Classification ID: 11C20
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 20K01
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 11A25
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 1247573
zbMATH Keywords: matrix arithmetic
zbMATH Keywords: lattices
zbMATH Keywords: Hall algebra
zbMATH Keywords: subgroups of finite abelian groups
zbMATH Keywords: integer matrices
zbMATH Keywords: arithmetical functions of matrices
zbMATH Keywords: partitions of integers
zbMATH Keywords: cyclic factors
zbMATH Keywords: Smith normal form
zbMATH Keywords: Dirichlet convolution
zbMATH Keywords: Hecke algebra
zbMATH Keywords: \(C\)-algebras
zbMATH Keywords: algebra of formal power series
reviewed by: Władysław Narkiewicz
MaRDI profile type: Publication
full work available at URL: jabr.1998.9999
OpenAlex ID: W2092299235
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DOI: 10.1006/JABR.1998.9999
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description: scientific article; zbMATH DE number 1247573