ON A STUDY OF THE MODULAR REPRESENTATIONS OF SLn(Fq) IN DEFINING CHARACTERISTIC p

Author

Devjani Basu, Southern Illinois University CarbondaleFollow

Date of Award

8-1-2025

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Choiy, Kwangho

Abstract

The modular representations of a finite group G are the representations over a vector space over a field F of nonzero characteristic p, where p divides the order of the group G. The special linear group G=SLn(Fq) where q=p^r is a prime power, is a finite group of Lie type. This thesis aims to provide an explicit description of the modular representations of G, in defining characteristic p. This problem can be restated as the explicit description and classification of the simple F[G]-modules. We bring forward two different approaches. In the first one, we employ the classification of simple modules of an linear algebraic group by their highest weight, and describe the simple F[G]-modules as a restriction of simple modules of the ambient group SLn(F) to that of SLn(Fq). We explicitly show this for the group SL3(Fq) and offer some generalization. Secondly, we outline an modular analog of the Clifford theory, which is an effective way study the represntations of SLn, via the representations of the general linear group GLn, owing to the fact that SLn is a normal subgroup of GLn. We demonstrate this second approach with an example for n=2.