Overgroups of $E(m,R)\otimes E(n,R)$ I. Levels and normalizers

Abstract:

The study of subgroups $H$ such that \[ E(m,R)\otimes E(n,R)\le H\le G=\operatorname {GL}(mn,R) \] is started, provided that the ring $R$ is commutative and $m,n\ge 3$. The principal results of this part can be summarized as follows. The group $\operatorname {GL}_m\otimes \operatorname {GL}_n$ is described by equations, and it is proved that the elementary subgroup $E(m,R)\otimes E(n,R)$ is normal in $(\operatorname {GL}_m\otimes \operatorname {GL}_n)(R)$. Moreover, when $m\neq n$, the normalizers of all three subgroups $E(m,R)\otimes e$, $e\otimes E(n,R)$, and $E(m,R)\otimes E(n,R)$ in $\operatorname {GL}(mn,R)$ coincide with $(\operatorname {GL}_m\otimes \operatorname {GL}_n)(R)$. With each such intermediate subgroup $H$, a uniquely defined level $(A,B,C)$ is associated where $A,B,C$ are ideals in $R$ such that $mA,A^2\le B\le A$ and $nA,A^2\le C\le A$. Conversely, a level determines a perfect intermediate subgroup $\operatorname {EE}(m,n,R,A,B,C)$. It is shown that each intermediate subgroup contains a unique largest subgroup of this type. Next, the normalizer $N_G(\operatorname {EE}(m,n,R,A))$ of these perfect intermediate subgroups is calculated completely in the crucial case, where $A=B=C$. The standard answer to the above problem can now be stated as follows. Every such intermediate subgroup $H$ is contained in the normalizer $N_G(\operatorname {EE}(m,n,R,A,B,C))$. In the special case where $n\ge m+2$, such a standard description will be established in the second part of the present work.