Summary: Communications in Algebraź
, 34: 889891, 2006
Copyright © Taylor & Francis Group, LLC
ISSN: 0092-7872 print/1532-4125 online
UNIPOTENT CONJUGACY IN GENERAL LINEAR GROUPS
J. L. Alperin
Mathematics Department, University of Chicago, Chicago, Illinois, USA
Let U n q be the group of upper uni-triangular matrices in GL n q , the n-
dimensional general linear group over the field of q elements. The number of U n q -
conjugacy classes in GL n q is, as a function of q, for fixed n, a polynomial in q
with integral coefficients.
Key Words: Unipotent conjugacy.
Mathematics Subject Classification: Primary 20C15; Secondary 20D06.
A conjecture for at least forty years states that the number of conjugacy classes
in U n q , the group of upper uni-triangular matrices over a field with q elements, is,
as function of q, with n fixed, given by an integral polyonomial in q (e.g., see Isaacs,
1995; Robinson, 1998). We show here that a similar result is easily established.
Theorem. The number of U n q -conjugacy classes in GL n q is, as a function of q,
for fixed n, given by a polynomial in q with integral coefficients.