TY - JOUR

T1 - Pairs of microweight tori in GL_n

AU - Nesterov, Vladimir

AU - Vavilov, Nilolai

PY - 2020/6/28

Y1 - 2020/6/28

N2 - In the present note we prove a reduction theorem for subgroups of the general linear group GL(n, T) over a skew-field T , generated by a pair of microweight tori of the same type. It turns out, that any pair of such tori of residue m is conjugate to such a pair in GL(3m, T) , and the pairs that cannot be further reduced to GL(3m − 1, T) form a single GL(3m, T) -orbit. For the case m = 1 it leaves us with the analysis of GL(2, T) , which was thoroughly studied some two decades ago by the second author, Cohen, Cuypers and Sterk. For the next case m = 2 this means that the only cases to be considered are GL(4, T) and GL(5, T) . In these cases the problem can be fully resolved by (direct but rather lengthy) matrix calculations, which are relegated to a forthcoming paper by the authors.

AB - In the present note we prove a reduction theorem for subgroups of the general linear group GL(n, T) over a skew-field T , generated by a pair of microweight tori of the same type. It turns out, that any pair of such tori of residue m is conjugate to such a pair in GL(3m, T) , and the pairs that cannot be further reduced to GL(3m − 1, T) form a single GL(3m, T) -orbit. For the case m = 1 it leaves us with the analysis of GL(2, T) , which was thoroughly studied some two decades ago by the second author, Cohen, Cuypers and Sterk. For the next case m = 2 this means that the only cases to be considered are GL(4, T) and GL(5, T) . In these cases the problem can be fully resolved by (direct but rather lengthy) matrix calculations, which are relegated to a forthcoming paper by the authors.

KW - полная линейная группа, унипотентные группы, микровесовые торы, диагональная подгруппа

KW - general linear groups

KW - unipotent root subgroups

KW - semisimple root subgroups

KW - diagonal subgroup

KW - m-tori

U2 - 10.22405/2226-8383-2020-21-3

DO - 10.22405/2226-8383-2020-21-3

M3 - Article

VL - XXI

SP - 256

EP - 265

JO - Chebyshevskii Sbornik

JF - Chebyshevskii Sbornik

SN - 2226-8383

IS - 3(75)

ER -