這篇論文中,我們主要是對不變量環感到興趣.古典不變量在十九世紀和二十世紀初是一個熱門的議題.我們研究多項式在有限矩陣群G的作用下保持不變.這個結論是找出有限基本不變量的集合{I1, …,In}生成不變量環 .

在第四節我們介紹Molien series, Molien series 對於不變量子環的計算有很大的幫助. 同時在第五節介紹Choen-Macaulay性質.

第六節我們證明了廣義對稱函數環在 上是Choen-Macaulay . 最重要的結果在第七節.當m=2時,我們可以很明確的找到廣義對稱函數環在 上的基底.

In this thesis, we are interested in ring of invariant. Classical invariant theory was a hot topic in the 19th century and in the beginning of the 20th century. We study polynomials which remain invariant under the action of finite matrix group G. The result is a collection of algorithms for finding a finite set {I1, …,In} of fundamental invariants which generate the invariant subring  .
We introduce Molien series in section 4, to aid in the calculation of invariant subring and introduce the Choen-Macaulay properties in section 5.
In section 6, we prove that the ring of generalized symmetric polynomials is Choen-Macaulay over . The most important result lies in section 7. When m=2, we find an explicit basis of ring of generalized symmetric polynomials over  .

contents
 1Introduction 2

 2SymmetricPolynomials 3

 3RingofInvariants 5

 4MolienSeries 10

 5TheCohen¬MacaulayProperty 13

 6GeneralizedSymmetricPolynomials 15

 7MainTheorem 17

 References 25

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