Classical invariant theory.

*(English)*Zbl 0971.13004
London Mathematical Society Student Texts 44. Cambridge: Cambridge University Press. xxi, 280 p. (1999).

The book is dedicated to the classical invariant theory, a branch of mathematics which attained its zenith during the heyday of nineteenth century mathematics.

What is classical invariant theory? Let \(Q(x,y) = a_0x^n + a_1x^{n-1}y + \cdots +a_nx^n\) denote a binary form of degree \(n.\) Under a change of coordinates \(x' = \alpha x + \beta y\), \(y' = \gamma x + \delta y\) with non-zero determinant one obtains \(Q'(x',y') = a_0'{x'}^n + a_1'{x'}^{n-1}y' + \cdots + a_n'{x'}^n = Q(x',y').\) A polynomial \(I(\underline a) = I(a_0,\ldots,a_n)\) in the coefficients of \(Q\) is called an invariant of weight \(k\) if \(I(\underline a) = (\alpha\gamma - \beta\delta)^k I(\underline a').\) More generally covariants are considered. A covariant is a binary form whose coefficients are polynomials in the coefficients of \(Q\) which is invariant under unimodular substitutions. An example of an invariant is the discriminant, while an example of a covariant is the Hessian.

The problem of classical invariant theory is to understand all the invariants and covariants for binary forms of given degree. One would like to have a finite set of covariants such that any covariant is a polynomial in those invariants. This was solved by P. Gordan [see J. Reine Angew. Math. 69, 323-354 (1869; JFM 01.0060.01)] for binary forms. It was Hilbert who extended Gordan’s results to more than two variables. In fact he proved his general basis theorem, showing that for a large class of examples a finite basis will exist. Besides of the new direction initiated by Hilbert, in recent years there are applications in new areas of classical invariant theoretic methods [see e.g. V. L. Popov in: Groups, generators, syzygies, and orbits in invariant theory, Transl. Math. Monographs, Vol. 100 (Providence, R.I. 1992; Zbl 0754.13005)]. The book under review serves as a in-depth study of binary forms.

In chapter 1, ‘Prelude – Quadratic polynomials and quadratic forms’, there is a consideration of a single real or complex quadratic polynomial as a ‘warm up’ to the problem.

Chapter 2, ‘Basic invariant theory for binary forms’, is an introduction to the basic definitions and examples as well an review of Hilbert’s basis theorem. The next two chapters, chapter 3, ‘Groups and transformation’, and chapter 4, ‘Representations and invariants’, provide a grounding in the modern mathematical foundations of the subject.

The following three chapters, chapter 5, ‘Transvectants, chapter 6 ‘Symbolic methods’, and chapter 7, ‘Graphical methods’, describe the core of the classical constructive algebraic theory of binary forms. Transvectants of two given covariants are defined by certain differential operators acting on binary forms. This is generalized to partial transvectants, which can be written in the so-called symbolic form, a powerful tool for computing and classifying invariants. At the end of chapter 7, there is an explanation of Gordan’s proof for finding the basis for the ring of covariants of binary forms.

In chapter 8, ‘Lie groups and moving frames’, the author discusses moving frames and differential invariants. He describes the signature curve as a method for deciding whether one binary form can be transformed into another one by a unimodular substitution.

In chapter 9, ‘Infinitesimal methods’, Lie algebras are introduced and their actions on binary forms are studied. There is also a proof of Hilbert’s basis theorem that relies on a particular differential operator converting functions into invariants.

The final chapter 10, ‘Multivariate polynomials’, concerns an orientation to pursue various generalizations of the symbolic method to multivariate polynomials.

The book is self-contained and includes introductions to related subjects as Lie groups, Lie algebras, and representation theory. The aim of the book is to provide an undergraduate student or a graduate student with a firm grounding of classical invariant theory. The book is written in a pleasant style, in a concrete manner and makes fairly low demands on the reader. In addition, a number of new results and the way of the presentation should attract the attention also to mathematicians of different subjects as well as even the most well-seasoned researchers. It is also recommended as complement to the more algebro-geometric point of view of the book by T. Springer, “Invariant theory”, Lect. Notes Math. 585 (1977; Zbl 0346.20020).

What is classical invariant theory? Let \(Q(x,y) = a_0x^n + a_1x^{n-1}y + \cdots +a_nx^n\) denote a binary form of degree \(n.\) Under a change of coordinates \(x' = \alpha x + \beta y\), \(y' = \gamma x + \delta y\) with non-zero determinant one obtains \(Q'(x',y') = a_0'{x'}^n + a_1'{x'}^{n-1}y' + \cdots + a_n'{x'}^n = Q(x',y').\) A polynomial \(I(\underline a) = I(a_0,\ldots,a_n)\) in the coefficients of \(Q\) is called an invariant of weight \(k\) if \(I(\underline a) = (\alpha\gamma - \beta\delta)^k I(\underline a').\) More generally covariants are considered. A covariant is a binary form whose coefficients are polynomials in the coefficients of \(Q\) which is invariant under unimodular substitutions. An example of an invariant is the discriminant, while an example of a covariant is the Hessian.

The problem of classical invariant theory is to understand all the invariants and covariants for binary forms of given degree. One would like to have a finite set of covariants such that any covariant is a polynomial in those invariants. This was solved by P. Gordan [see J. Reine Angew. Math. 69, 323-354 (1869; JFM 01.0060.01)] for binary forms. It was Hilbert who extended Gordan’s results to more than two variables. In fact he proved his general basis theorem, showing that for a large class of examples a finite basis will exist. Besides of the new direction initiated by Hilbert, in recent years there are applications in new areas of classical invariant theoretic methods [see e.g. V. L. Popov in: Groups, generators, syzygies, and orbits in invariant theory, Transl. Math. Monographs, Vol. 100 (Providence, R.I. 1992; Zbl 0754.13005)]. The book under review serves as a in-depth study of binary forms.

In chapter 1, ‘Prelude – Quadratic polynomials and quadratic forms’, there is a consideration of a single real or complex quadratic polynomial as a ‘warm up’ to the problem.

Chapter 2, ‘Basic invariant theory for binary forms’, is an introduction to the basic definitions and examples as well an review of Hilbert’s basis theorem. The next two chapters, chapter 3, ‘Groups and transformation’, and chapter 4, ‘Representations and invariants’, provide a grounding in the modern mathematical foundations of the subject.

The following three chapters, chapter 5, ‘Transvectants, chapter 6 ‘Symbolic methods’, and chapter 7, ‘Graphical methods’, describe the core of the classical constructive algebraic theory of binary forms. Transvectants of two given covariants are defined by certain differential operators acting on binary forms. This is generalized to partial transvectants, which can be written in the so-called symbolic form, a powerful tool for computing and classifying invariants. At the end of chapter 7, there is an explanation of Gordan’s proof for finding the basis for the ring of covariants of binary forms.

In chapter 8, ‘Lie groups and moving frames’, the author discusses moving frames and differential invariants. He describes the signature curve as a method for deciding whether one binary form can be transformed into another one by a unimodular substitution.

In chapter 9, ‘Infinitesimal methods’, Lie algebras are introduced and their actions on binary forms are studied. There is also a proof of Hilbert’s basis theorem that relies on a particular differential operator converting functions into invariants.

The final chapter 10, ‘Multivariate polynomials’, concerns an orientation to pursue various generalizations of the symbolic method to multivariate polynomials.

The book is self-contained and includes introductions to related subjects as Lie groups, Lie algebras, and representation theory. The aim of the book is to provide an undergraduate student or a graduate student with a firm grounding of classical invariant theory. The book is written in a pleasant style, in a concrete manner and makes fairly low demands on the reader. In addition, a number of new results and the way of the presentation should attract the attention also to mathematicians of different subjects as well as even the most well-seasoned researchers. It is also recommended as complement to the more algebro-geometric point of view of the book by T. Springer, “Invariant theory”, Lect. Notes Math. 585 (1977; Zbl 0346.20020).

Reviewer: Peter Schenzel (Halle)

##### MSC:

13A50 | Actions of groups on commutative rings; invariant theory |

13-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

15-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra |

15A72 | Vector and tensor algebra, theory of invariants |

11Exx | Forms and linear algebraic groups |