TY - JOUR

T1 - Subgroups Generated by a Pair of 2-Tori in $\operatorname{GL}(4,K)$, II

AU - Нестеров, Владимир Викторович

AU - Чжан, Мэйлин

PY - 2025/9/18

Y1 - 2025/9/18

N2 - The present paper is another work of major cycle 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 from it 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 present manuscript we describe spans in $\operatorname{GL}(4,K)$, corresponding to degenerate orbits.

AB - The present paper is another work of major cycle 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 from it 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 present manuscript we describe spans in $\operatorname{GL}(4,K)$, corresponding to degenerate orbits.

UR - https://www.mendeley.com/catalogue/38b1547b-50b7-3836-8dc4-fddc0724326e/

U2 - 10.46698/t9254-6010-7867-w

DO - 10.46698/t9254-6010-7867-w

M3 - Article

VL - 27

SP - 101

EP - 119

JO - ВЛАДИКАВКАЗСКИЙ МАТЕМАТИЧЕСКИЙ ЖУРНАЛ

JF - ВЛАДИКАВКАЗСКИЙ МАТЕМАТИЧЕСКИЙ ЖУРНАЛ

SN - 1683-3414

IS - 3

ER -