Polynomial Invariant Theory of the Classical Groups

TL;DR

This paper investigates the generators of invariant algebras for classical groups, establishing finiteness results and presenting fundamental theorems that explicitly describe these invariants.

Contribution

It proves the finiteness of basic invariants for reductive groups and states the First Fundamental Theorems for classical groups, advancing invariant theory.

Findings

Finiteness of basic invariants for reductive groups

Explicit listing of invariants for classical groups

Discussion of applications and historical context

Abstract

The goal of invariant theory is to find all the generators for the algebra of representations of a group that leave the group invariant. Such generators will be called \emph{basic invariants}. In particular, we set out to find the set of basic invariants for the classical groups GL$(V)$, O$(n)$, and Sp$(n)$ for $n$ even. In the first half of the paper we set up relevant definitions and theorems for our search for the set of basic invariants, starting with linear algebraic groups and then discussing associative algebras. We then state and prove a monumental theorem that will allow us to proceed with hope: it says that the set of basic invariants is finite if $G$ is reductive. Finally we state without proof the First Fundamental Theorems, which aim to list explicitly the relevant sets of basic invariants, for the classical groups above. We end by commenting on some applications of invariant theory, on the history of its development, and stating a useful theorem in the appendix whose proof lies beyond the scope of this work.