Double coset decomposition for \(SL(r+s,Z)\) with respect to congruence subgroups
type: Item
instance of: scholarly article
title: Double coset decomposition for \(SL(r+s,Z)\) with respect to congruence subgroups
zbMATH Open document ID: 0864.15006
DOI: 10.1016/0024-3795(94)00201-0
author: Shu-Chu Chang
published in: Linear Algebra and its Applications
publication date: 1997-06-22T00:00:00Z
review text: Let \(M_n(R)\) be the set of all \(n\times n\) matrices with entries in the principal-ideal ring \(R\). The two-sided equivalence [\textit{M. Newman}, Integral matrices, (1972; Zbl 0254.15009)] on \(M_n(R)\) is well known: \(B\) is equivalent to \(A\) \([A,B\in M_n(R)]\) if there exist two unimodular matrices \(U,V\) in \(M_n(R)\) such that \(B=UAV\). In this equivalence relation, each matrix \(A\) in \(M_n(R)\) is equivalent to a unique diagonal matrix \(S(A)= \text{diag} (s_1,\dots, s_r,0, \dots,0)\), \(r=\text{rank} A\), \(s_1,\dots,s_r\)-nonzero elements of \(R\) and \(s_i|s_{i+1}\), \(i=1,\dots,r-1\). \(S(A)\) is called the Smith normal form of \(A\). M. Newman [loc. cit.] proposed a generalization of the two-sided equivalence on \(M_{r+s} (\mathbb{Z})\): \(B\) is equivalent to \(A\) with respect to the congruence subgroup \(\Gamma_0 (n;r,s)\) if there exist \(U,V\in\Gamma_0(n;r,s)\) such that \(B=UAV\). \(\Gamma_0 (n;r,s)\) consists of matrices of the form \(\left[\begin{smallmatrix} A & B \\ C & D \end{smallmatrix}\right]\), \(A\) and \(D\) are square matrices of dimensions \(r\) and \(s\), respectively, and \(C\equiv 0\bmod n\) entrywise. In this paper the author gives a canonical form of the generalized two-sided equivalence classes of matrices of \(M_{r+s}(\mathbb{Z})\) of determinant 1 (i.e. of \(SL(r+s,\mathbb{Z}))\).
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 931935
zbMATH Keywords: double coset decomposition
zbMATH Keywords: Smith normal form
zbMATH Keywords: congruence subgroup
zbMATH Keywords: canonical form
zbMATH Keywords: generalized two-sided equivalence
reviewed by: Jan Kubarski
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)00201-0
OpenAlex ID: W1984435174
cites work: Sous-groupes d'indice fini dans $SL\left( {n,Z} \right)$
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cites work: Finite factor groups of the unimodular group
cites work: Q5668937
cites work: Multiplicativity of the Smith Normal Form
cites work: The Lattice Point Covering Theorem for Rectangles
cites work: Q3720731
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description: scientific article; zbMATH DE number 931935