Series of Lectures: Invariantentheorie

Context and aims
Hilbert’s 1927 Göttingen lectures on invariant theory revisit the classical subject of invariants and covariants of polynomial forms with the tools of modern algebra that his own work had helped inaugurate. The course contrasts Gordan’s constructive, symbolic calculus with a structural, ideal-theoretic approach. It positions invariant theory as the study of the graded algebra of polynomial functions on a representation space under a linear group action, focusing on finite generation, dimension, and the asymptotic behavior of graded pieces, and it integrates connections to algebraic geometry through the Nullstellensatz.

Algebraic framework
The lectures set up polynomial rings S = k[x1,…,xm] with a rational action of a linear algebraic group G (notably GLn and SLn), and define the invariant subring S^G and the module of covariants. Grading by degree, homogeneity, and weight under tori inside G provides a decomposition that links classical concomitants to representation-theoretic weight spaces. The viewpoint centers on ideals, homomorphisms, and graded modules, emphasizing structural properties shared by rings of invariants across examples.

Finite generation and methods
A central theme is finite generation: under mild hypotheses in characteristic zero, the invariant algebra S^G is finitely generated. Hilbert’s basis theorem guarantees Noetherianity of polynomial rings and their quotients, and his finiteness proof for invariants replaces constructive syzygy-chasing with existence arguments via continuity and generic specialization. For finite groups, averaging (the Reynolds operator) projects onto S^G and yields separation of orbits by invariants; for classical continuous groups, integration and representation-theoretic decomposition underpin analogous conclusions. Noether normalization is used to exhibit a polynomial subalgebra over which S^G is module-finite, consolidating dimension statements and enabling reductions.

Hilbert functions and syzygies
The graded algebraic perspective brings in the Hilbert function H(n) = dimk(S/I)n and its eventual polynomial behavior. The lectures develop Hilbert series, rationality of generating functions for graded modules, and the extraction of numerical invariants such as dimension and degree. Syzygies appear through free resolutions and the finiteness of relations among generators; the syzygy theorem bounds the length of resolutions over polynomial rings, giving a controlled framework for relations among invariants and covariants without descending into the full symbolic calculus.

Classical examples and computations
Canonical families of representations anchor the general theory. Binary and ternary forms under SL2 and SL3 serve to illustrate generators, relations, and covariants such as discriminants, Hessians, and transvectants, now interpreted through decomposition of symmetric powers and Clebsch–Gordan rules. Fundamental invariants of low-degree forms, invariants of point configurations in the projective line, and concomitants of quadrics and cubics are reconsidered via graded algebra. Elimination theory and resultants connect with invariants that detect singularity and orbit closure.

Geometry of invariants
Algebra meets geometry through the Nullstellensatz, tying radical ideals to algebraic sets, and through the projective viewpoint on homogeneous invariants. The null cone, the set of points where all positive-degree invariants vanish, is identified as the algebraic locus capturing instability of orbits and the boundary of classification by invariants. Although quotient varieties are not yet constructed in modern form, the lectures emphasize how invariants separate generic orbits, how vanishing patterns carve stratifications, and how dimension counts control moduli questions for forms.

Influence and outlook
The 1927 presentation consolidates invariant theory as a chapter of commutative algebra and representation theory. It clarifies why existence theorems and graded methods supplant algorithmic symbolics, sets numerical invariants via Hilbert functions alongside generators and relations, and anticipates later developments: Weyl’s systematic representation theory of classical groups, Noether’s abstract algebra, and the geometric invariant theory of Mumford. The course thus frames classical calculations within a durable conceptual architecture that remains foundational.