University of Illinois Urbana-Champaign

On the Projective Characters of the Finite Chevalley Groups

Holmes, Randall Reed

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Description

Title
On the Projective Characters of the Finite Chevalley Groups
Author(s)
Holmes, Randall Reed
Issue Date
1987
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Abstract
  • Let p be a prime number, let m be a positive integer and let G be a universal Chevalley group constructed over a field of order ${\rm p\sp{m}}.$ We consider the modular representations (in characteristic p) of the group G. More precisely, we study the Brauer characters of G, concentrating on the projective indecomposable characters (those characters afforded by the projective indecomposable modules). The main tool used in the paper is Steinberg's Tensor Product Theorem.
  • To begin with, we develop a systematic method, using the language of graphs, to decompose a product of irreducible characters into a sum of its irreducible constituents. The main result is an expression for the multiplicity of a given irreducible as a constituent in a product of irreducibles.
  • This leads to a recursion formula for the projective indecomposable characters which generalizes a formula obtained by Chastkofsky and Feit in their work on the special linear group ${\rm SL(3,2\sp{m}}).$ This formula is then used to factor certain projective indecomposable characters into products of "twisted" characters, the twists being given by the Frobenius automorphism of G as in Steinberg's Theorem.
  • Using our findings, we compute the degrees of some projective indecomposable characters of some specific groups. We recover degree formulas obtained earlier by Srinivasan, by Chastkofsky and Feit, and by Cheng for the groups ${\rm SL(2,p\sp{m}})$ (p arbitrary), ${\rm SL(3,2\sp{m}}),$ and ${\rm G\sb2(p\sp{m}})$ (p = 2,3), respectively. In addition, we obtain new degree formulas for the groups ${\rm SL(3,3\sp{m}})$ and ${\rm SL(4,2\sp{m}}).$
  • Finally, we develop an algorithm, based on the recursion formula, for the computation of the projective indecomposable characters of the group ${\rm SL(4,2\sp{m}}).$ We provide a computer program which carries out the algorithm and, by way of illustration, we use the program to generate tables of degrees for m = 1, 2, 3 and 4.
Graduation Semester
1987
Type of Resource
text
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http://hdl.handle.net/2142/71256

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