Subgroups generatred by a pair of $2$-tori in $\mathrm{GL}(5,K)$

N. Vavilov ; Vl. Nesterov

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 39, Tome 522 (2023), pp. 8-45

Résumé

A goal of the paper is to describe the orbits of the general linear group $\mathrm{GL}(n,K)$ over a field $K$, acting by simultaneous conjugation on pairs of $2$-tori, i.e., subgroups conjugate to the diagonal subgroup $\big\{\mathrm{diag}(\varepsilon,\varepsilon,1,{\ldots},1), \varepsilon\in K^*\big\}$, and identify their spans. For the easier case of $1$-tori similar results were previously obtained by the first author, A. Cohen, H. Vuypers and H. Sterk. The present paper is the second one pertaining to this case, in the first one a reduction theorem is proved establishing that a pair of $2$-tori is conjugate to such a pair in $\mathrm{GL}(6,K)$, and a classification of such pairs that cannot be embedded in $\mathrm{GL}(5,K)$ is given. Here, the orbits and spans of $2$-tori in $\mathrm{GL}(5,K)$, that cannot be embedded in $\mathrm{GL}(4,K)$ are described. A typical qualitatove corollary of the obtained results asserts that for $|K|\ge 7$ every non-diagonalisable subgroup spanned by $2$-tori contains unipotent elements of residue $1$ or $2$.