這篇論文主要我們研究Dynkin diagrams上的一些組合問題. Dynkin diagrams在數學上有許多應用, 首先介紹一種演算法來轉換兩個卡茨圖(Kac diagram),我們定義它們的等價關係, 如果兩個卡茨圖能透過演算法來互相轉換那麼我們說它們等價.
這篇論文的主要問題是研究兩個卡茨圖在相同的群軌道(group orbit)上, 決定它們是否等價. 最後, 我們透過一些觀察現象及結果提出兩個猜測

The thesis studies combinatorial properties of some graphs, known as Kac diagrams, which have important applications in algebra. We introduce an algorithm which converts a diagram to another. Two Kac diagrams are said to be equivalent if we can transform a diagram to another by a sequence of such algorithm. The main problem of this thesis is to study two simple Kac diagrams in the same G-orbit, determine whether they are equivalent. In the end, we have two conjectures based on above work.

Contents
1 Introduction 1
2 Algorithms 4
3 Group action 6
4 Equivalence classes of Kac diagram 7
5 Conjecture 13

[1] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York 1972.
[2] V. G. Kac, Infinite Dimensional Lie Algebras, 3rd. ed, Cambridge Univ. Press, Cambridge 1990.