Abstract:
The present paper is the next in a large series of works devoted to the geometry of microweight tori in the Chevalley groups. Namely, we describe the subgroups generated by a pair of $2$-tori in $\operatorname{GL}(4,K)$. Recall that $2$-tori in $\operatorname{GL}(n,K)$ are the subgroups conjugate to the diagonal subgroup of the following form $\operatorname{diag}(\varepsilon, \varepsilon, 1,\dots, 1)$. In one of the previous work we proved the reduction theorem for the pairs of $m$-tori. It follows that any pair of $2$-tori can be embedded in $\operatorname{GL}(6,K)$ by simultaneous conjugation. The orbit of a pair of $2$-tori $(X,Y)$ is called the orbit in $\operatorname{GL}(n,K)$, if the pair $(X,Y)$ is embedded in $\operatorname{GL}(n,K)$ by simultaneous conjugation and it can not be embedded in $\operatorname{GL}(n-1,K)$. Here $n$ can take values $3, 4, 5$ and $6$. The most difficult and general case is the case of $\operatorname{GL}(4,K)$. In the article we describe spans in $\operatorname{GL}(4,K)$, corresponding to degenerate orbits.
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