A criterion for occurrence in a subgroup generated by two-dimensional elementary matrices
In this very interesting paper, the author describes two remarkable algorithms for \(2\times 2\) matrices -- the one dealing with matrices over \({\mathbb{Z}}\) and the other with matrices over the polynomial ring \(R=K[x_ 1,...,x_ t]\), where \(K\) is a (commutative) field. The first algorithm decides when an element \(x\) of \(SL_ 2({\mathbb{Z}})\) lies in the subgroup \(H_{m,n}\) generated by \(a=\left( \begin{matrix} 1\quad m\\ 0\quad 1\end{matrix} \right)\) and \(b=\left( \begin{matrix} 1\quad 0\\ n\quad 1\end{matrix} \right)\), where \(m,n\) are given integers \(\geq 2\). A simple necessary condition is that \(x\) should lie in the subgroup \(G_{m,n}\) of \(SL_ 2({\mathbb{Z}})\) formed by the elements \((x_{ij})\) satisfying \(x_{11}\equiv x_{22}\equiv 1(mod mn),\quad x_{12}\equiv 0(mod m),\quad x_{21}\equiv 0(mod n)\). For each \(x\in G_{m,n}\), the author defines a corresponding integer \(s\geq 0\) and function \(\phi\) such that \(x\in H_{m,n}\) if, and only if, \(\phi\) (k)\(\leq 0\) for \(k=1,2,...,s+1.\) The method not only shows that \(H_{m,n}\) is freely generated by \(a,b\) but even produces the indices in the expression of an element of \(H_{m,n}\) as a product of powers of \(a,b\). The author's results confirm those of \textit{I. N. Sanov} [Dokl. Akad. Nauk SSSR 57, 657-659 (1947; Zbl 0029.00404)] for the case \(m=n=2\) but disprove a (later retracted) claim of \textit{J. L. Brenner} [C. R. Acad. Sci., Paris 241, 1689-1691 (1955; Zbl 0065.254)] concerning the general case \(m=n\geq 2.\) With \(R=K[x_ 1,...,x_ t]\), as above, let \(GE_ 2(R)\) denote the subgroup of \(GL_ 2(R)\) generated by the elements of the forms \(\left( \begin{matrix} 1\quad\alpha \\ 0\quad 1\end{matrix} \right)\), \(\left( \begin{matrix} 1\quad 0\\ \beta\quad 1\end{matrix} \right)\), \(\left( \begin{matrix} \gamma_ 1\quad 0\\ 0\quad\gamma_ 2\end{matrix} \right)\). The second algorithm, which has many points of resemblance with the first, decides when an element of \(GL_ 2(R)\) lies in \(GE_ 2(R)\). For \(t=1\), it is well known that \(GE_ 2(R)=GL_ 2(R)\); but for each \(t\geq 2\), the author constructs a non-cyclic free subgroup of \(GL_ 2(R)/GE_ 2(R)\).
elementary matrices over polynomial ring
scientific article; zbMATH DE number 3871651