表現理論初探

Abstract

在這篇論文中，我們趁著這次機會整理了一些關於有限群表現理論基本而重要的結果，例如：判定給定的群他的不可分解(indecomposable)表現及不可分解的整數表現(integral representations)，是否只有有限多種。因為分裂的代數圓圈(split algebraic tori)會與有限群的整數表現產生對應，所以在此論文的最後一章，我們也介紹了代數圓圈的可分離分裂體(separable splitting fields)定理。 在第二章中，我們探討模表現(modular representations)，尤其是體(field)的特徵值(characteristic)整除群的元素總個數時。我們介紹了格林對應(Green's correspondence)，在格林對應之後，我們有了相對投射性(relative projectivity)的概念，進而能夠判斷給定的群的不可分解的表現是否有無限多種，同時在模系統(modular system)下我們介紹了格羅滕迪克群(Grothendieck group)及cde三角形。 第三章簡單的介紹了整數表現理論以及判斷不可分解的整數表現的有限性的方式。在第四章，我們整理了一些特定有限群的不可分解整數表現，例如元素個數為質數p的循環群，以及元素個數為2p的二面體群。在最後一章，我們整理了很多代數圓圈會在他的有限可分離體擴張(finite separable field extension)分裂的不同證明，並且推廣了Chow的定理，最後則是給了對於一個代數圓圈，他的分裂體的上限。

In the present thesis, we take the opportunity to discuss several basic and important results in representation theory. More precisely we mainly investigate the criterion of finite groups G that are of finite representation type for both kG-modules or for ZG-lattices, as well as separable splitting fields of algebraic tori. In Chapter 2, we consider the theory of representations of finite groups over a field k. We focus mainly on the case where the characteristic of k divides the order of the group G. This chapter include Green’s correspondence and its the connection to the criterion of kG that is of finite representation. We also discuss the structure and relation of Grothendieck groups RkG and RKG in a modular system setting, namely the cde triangle. In Chapter 3, we give an overview of integral representations based on classical results of Heller and Reiner, which would be useful for further studies. In Chapter 4, we give a description of classification of indecomposable integral representations of cyclic groups of prime order p and dihedral groups of order 2p, based on works of Reiner and of Lee. In the last chapter, we give a connection between algebraic tori and integral representations of finite groups. We give several different proofs of the theorem that any algebraic tori over a field splits over a finite field extension. Besides, we also generalize Chow’s theorem to semi-abelian varieties, and give a sharp bound for the splitting fields of algebraic tori.