The Algebra of Invariants

Abstract

Invariant theory is a subject within abstract algebra that studies polynomial functions which do not change under transformations from a linear group. John Hilton Grace (1873–1958) was a research mathematician specialising in algebra and geometry. He was elected a Fellow of the Royal Society in 1908. His co-author Dr Alfred Young (1873–1940) was also a research mathematician before being ordained in 1908; in 1934 he too was elected a Fellow of the Royal Society. Abstract algebra was one of the new fields of study within mathematics which developed out of geometry during the nineteenth century. It became a major area of research in the late nineteenth and early twentieth centuries. First published in 1903, this book introduced the work on invariant theory of the German mathematicians Alfred Clebsch and Paul Gordan into British mathematics. It was considered the standard work on the subject.

On the rationality of the moduli space of Lüroth quartics

“…The book [8] is a historical account of invariant theory which also contains a readable account of the symbolic method. A modern treatment of the symbolic method can be found in [7].…”

“…The original paper is [1]. Morley proves that there are generically 8 Bateman configurations leading to a Lüroth quartic. It is shown below that also the moduli space of Bateman seven-tuples of points is rational.…”

Rationality of the moduli spaces of plane curves of sufficiently large degree

“…Proof Regardless of whether d = 3n + 1 or d = 3n + 2, the conditions that the monomials in (14) or (16) are invariant under the actions of the one-parameter subgroups λ H 1 resp. λ H 2 read…”

Algebraic and Differential Invariants

“…Both algebraic and differential invariant theories have become in recent years the subject of computational mathematics [13,14,17,40,60]. Algebraic invariant theory studies polynomial or rational invariants of algebraic group actions [18,22,23,54]. A typical example is the discriminant of a quadratic binary form as an invariant of an action of the special linear group.…”