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

  • Author / Creator
    Wen, Qianglong
  • 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.

  • Subjects / Keywords
  • Graduation date
    Fall 2011
  • Type of Item
  • Degree
    Doctor of Philosopy
  • DOI
  • License
    This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.