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Irreducible Characters of GL(n, Z/p^l Z) Open Access


Other title
Type of item
Degree grantor
University of Alberta
Author or creator
Wen, Qianglong
Supervisor and department
Gerald Cliff (Department of Mathematical and Statistical Sciences)
Examining committee member and department
Jochen Kuttler (Department of Mathematical and Statistical Sciences)
David McNeilly (Department of Mathematical and Statistical Sciences)
Piotr Rudnicki (Computing Science)
Allen Herman (University of Regina)
Department of Mathematical and Statistical Sciences

Date accepted
Graduation date
Doctor of Philosopy
Degree level
We first find all the irreducible complex characters of the general linear group GL(2, Z/p^l Z) over the ring Z/p^l Z, where l is an integer >1 and p is an odd prime, and determine all the character values. Our methods rely on Clifford Theory and can be modified easily to get all the irreducible complex characters of GL(2, Z/p^l Z) when p = 2. We deal with irreducible characters which are not inflated from GL(2, Z/p^{l-1} Z). These have three possible degrees. There are characters induced from a Borel subgroup, which have degree (p+1)p^{l-1}; and there are two other families of characters, of degrees (p-1)p^{l-1} and (p^2-1)p^{l-2}. Many results can be extended to the group G=GL(2,R) with R=S/P^l where S is the ring of integers in a local or global field and P is a maximal ideal. If S/P has q elements, we can replace p by $q$ in the degree and number of each degree formulas we find. We study GL(2,Z/p^l Z) in our work not only because it can give us some general results, but also it is simpler when we deal with character values. We also construct irreducible characters of GL(3, Z/p^2 Z) and GL(3, Z/p^3 Z). There are 7 kinds of irreducible characters for each group, and these 7 kinds of irreducible characters also show up for group GL(3, Z/p^l Z) for any l>1. We have all the degrees and the number of characters of each degree for the GL(3, /p^2). Moreover, we find all the irreducible constituents of character Ind_B^G(1) for the two groups GL(3, Z/p^2 Z) and GL(3, Z/p^3 Z), where B is the corresponding Borel subgroup.
License granted by Qianglong Wen ( on 2011-09-28T14:24:10Z (GMT): Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of the above terms. The author reserves all other publication and other rights in association with the copyright in the thesis, and except as herein provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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