Introduction

Summary

Invariant theory is a subject with a long history, a subject which in the words of Dieudonné and Carrell [32] “has already been pronounced dead several times, and like the Phoenix it has been again and again arising from the ashes.”

The starting point is a linear representation of a linear algebraic group on a vector space, which then induces an action on the ring of polynomial functions on the vector space, and one looks at the ring consisting of those polynomials which are invariant under the group action. The reason for restricting ones attention to linear algebraic group representations, or equivalently to Zariski closed subgroups of the general linear group on the vector space, is that the polynomial invariants of an arbitrary subgroup of the general linear group are the same as the invariants of its Zariski closure.

In the nineteenth century, attention focused on proving finite generation of the algebra of invariants by finding generators for the invariants in a number of concrete examples. One of the high points was the proof of finite generation for the invariants of SL₂(ℂ) acting on a symmetric power of the natural representation, by Gordan [39] (1868). The subject generated a language all its own, partly because of the influence of Sylvester, who was fond of inventing words to describe rather specialized concepts.

In the late nineteenth and early twentieth century, the work of David Hilbert and Emmy Noether in Göttingen clarified the subject considerably with the introduction of abstract algebraic machinery for addressing questions like finite generation, syzygies, and so on.