The history of invariant theory spans nearly a century and a half, with roots in certain problems from number theory, algebra, and geometry appearing in the work of Gauss, Jacobi, Eisenstein, and Hermite. Although the connection between invariants and orbits was essentially discovered in the work of Aronhold and Boole, a clear understanding of this connection had not been achieved until recently, when invariant theory was in fact subsumed by a general theory of algebraic groups.
Written by one of the major leaders in the field, this book provides an excellent, comprehensive exposition of invariant theory. Its point of view is unique in that it combines both modern and classical approaches to the subject. The introductory chapter sets the historical stage for the subject, helping to make the book accessible to nonspecialists.
Notation and terminology
The role of reductive groups in invariant theory
Constructive invariant theory
The degree of the Poincare series of the algebra of invariants and a finiteness theorem for representations with free algebra of invariants
Syzygies in invariant theory
Representations with free modules of covariants
A classification of normal affine quasihomogeneous varieties of SL2
Quasihomogeneous curves, surfaces, and solids
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