Irreducible representations of GL2(Fq) and SL2(Fq).

GL2(Fq) 與 SL2(Fq) 的不可約表現

黃振彥(Chen-Yen Huang)

指導教授 ： 潘戍衍

國立清華大學/理學院/數學系/碩士(2010年)

https://doi.org/10.6843/NTHU.2010.00705

摘要

In the first part of of this thesis, we introduce the ideas
by the order that is representations, modules, and characters. Actually, there are very close relations between all of them. That is,
(1) First of all, we have theorem 1.1.7, which states that any two equivalent representations of G arise from the same FG-module.
(2) Next, we have theorem 1.2.13, which states that two representations have the same character if and only if they are equivalent.
(3) Last, we use theorem 1.3.6 to connect each other. If two FG-modules have the same character, then they are isomorphic.
For the rest, we develop some tools to acheive our goal. That is, irreducible representations of GL2(Fq) and SL2(Fq). In order to find out all the irreducible representations of GL2(Fq) and SL2(Fq), we need to know how many we are supposed to find out, but by the theorem 1.3.7, we can exactly know the number. Hence, first, we need to classify the conjugacy classes of GL2(Fq) and SL2(Fq)
respectively. That is why we divide each chapter 2 and chapter 3 into conjugacy classes and irreducible representation.

關鍵字

GL2(Fq) ； SL2(Fq) ； 不可約表現 ； irreducible representations

參考文獻

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[2] Harold Cooper, The conjugacy classes of GL2(Fq), Seminar in Algebra and Number Theory, Handout, (2005).