Algebra of matrix arithmetic (Q1276984)

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.