Cayley-Sylvester Counting Theorem,Reductive Group And Its Application

The development of invariant theory has a history of more than 200 years.This article focuses on the role of finite groups on polynomial rings.This article is a book report.Many of the conclusions in the article are important results in the history of invariant theory.This article narrates some important results,such as dimension formula,Molien formula,etc.Then We can apply the Cayley-Sylvester counting theorem to calculate the Hilbert series of several classical invariant rings.In the first chapter,We have a brief summary of the simple development of invariant theory in the past 200 years.We also describe the overall structure of the full text.In the second chapter,this article first introduces some basic definitions and concepts in invariant theory,and then introduces the Hilbert series of invariant groups.We can find that we can calculate the Hilbert series of some polynomial invariant groups by using Molien formula and other propositions.In the third chapter of this paper,linearly reductive groups are introduced.We can see reductive groups have an easy structure.We can easily prove the linearly reductivity of the special linear group and the general linear group by the Casimir operator,.We find that the polynomial invariant ring under the action of a finite group is finitely generated from Hilbert theorem.The focus of the fourth chapter of this article is the dimension formula.We can use the dimension formula to calculate the dimension of the finitely generated poly-nomial invariant ring.The Cayley-Sylvester counting theorem is an application of the dimension formula.Finally,this paper uses the Cayley-Sylvester counting theorem to obtain the Hilbert series of several classical invariant rings.