Conjugacy classes in maximal parabolic subgroups of general linear groups

[摘要] We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a matrix problem. Such problems involve finding normal forms for matrices under a specified set of row and column operations. We solve the relevant matrix problem in small dimensional cases. This gives us all conjugacy classes in maximal parabolic subgroups over a perfect field when one of the two blocks has dimension less than 6. In particular, this includes every maximal parabolic subgroup of GL(n)(k) for n < 12 and k a perfect field. If our field is finite of size q, we also show that the number of conjugacy classes, and so the number of characters, of these groups is a polynomial in q with integral coefficients, (C) 2000 Academic press.

[发布日期] 2000-11-01

[关键词] conjugacy classes;parabolic subgroup;general linear group