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# The irreducible characters of Sylow p-subgroups of split finite groups of Lie type

Paolini, Alessandro (2016)
Ph.D. thesis, University of Birmingham.

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## Abstract

Let $$G$$ be a split finite group of Lie type defined over F$$_q$$, where $$q$$=$$p$$$$^e$$ is a prime power and $$p$$ is not a very bad prime for $$G$$. Let $$U$$ be a Sylow $$p$$-subgroup of $$G$$. In this thesis, we provide a full parametrization of the set Irr($$U$$) of irreducible characters of $$U$$ when $$G$$ is of rank 5 or less. In particular, for every character χ ∈ Irr($$U$$) we determine an abelian subquotient of $$U$$ such that χ is obtained by an inflation, followed by an induction of a linear character of this subquotient.

The characters are given in most cases as the output of algorithm that has been implemented in the computer system GAP, whose validity is proved in this thesis using classical results in representation theory and properties of the root system associated to $$G$$. We also develop a method to determine a parametrization of the remaining irreducible characters, which applies for every split finite group of Lie type of rank at most 5, and lays the groundwork to provide such a parametrization in rank 6 and higher.

Type of Work: Ph.D. thesis. Goodwin, Simon Colleges (2008 onwards) > College of Engineering & Physical Sciences School of Mathematics QA Mathematics University of Birmingham 6716
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