Double coset decomposition for \(SL(r+s,Z)\) with respect to congruence subgroups
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 \(SL(r+s,\mathbb{Z}))\).
scientific article; zbMATH DE number 931935
scientific article