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Conjugacy classes in maximal parabolic subgroups of general linear groups

Scott H. Murray
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Abstract

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.

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Scott H. Murray. 2000-01-06. Conjugacy classes in maximal parabolic subgroups of general linear groups. https://arxiv.org/abs/math/0001031

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