2008 | Book

# Algorithms in Invariant Theory

Author: Dr. Bernd Sturmfels

Publisher: Springer Vienna

Book Series : Texts & Monographs in Symbolic Computation

2008 | Book

Author: Dr. Bernd Sturmfels

Publisher: Springer Vienna

Book Series : Texts & Monographs in Symbolic Computation

J. Kung and G.-C. Rota, in their 1984 paper, write: “Like the Arabian phoenix rising out of its ashes, the theory of invariants, pronounced dead at the turn of the century, is once again at the forefront of mathematics”. The book of Sturmfels is both an easy-to-read textbook for invariant theory and a challenging research monograph that introduces a new approach to the algorithmic side of invariant theory. The Groebner bases method is the main tool by which the central problems in invariant theory become amenable to algorithmic solutions. Students will find the book an easy introduction to this “classical and new” area of mathematics. Researchers in mathematics, symbolic computation, and computer science will get access to a wealth of research ideas, hints for applications, outlines and details of algorithms, worked out examples, and research problems.

Advertisement

Abstract

Invariant theory is both a classical and a new area of mathematics. It played a central role in 19th century algebra and geometry, yet many of its techniques and algorithms were practically forgotten by the middle of the 20th century.

Abstract

Let C [x] denote the ring of polynomials with complex coefficients in n variables x = (x_{1},x_{2},...,xn). We are interested in studying polynomials which remain invariant under the action of a finite matrix group Γ ⊂ GL(C^{n}). The main result of this chapter is a collection of algorithms for finding a finite set I_{1}, I_{2},...,I_{m} of fundamental invariants which generate the invariant subring C[x]^{Γ}. These algorithms make use of the Molien series (Sect. 2.2) and the Cohen-Macaulay property (Sect. 2.3). In Sect. 2.4 we include a discussion of invariants of reflection groups, which is an important classical topic. Sections 2.6 and 2.7 are concerned with applications and special cases.

Abstract

According to the general philosophy outlined in Sect. 1.3, analytic geometry deals with those properties of vectors and matrices which are invariant with respect to some group of linear transformations. Applying this program to projective geometry, one is led in a natural way to the study of the bracket algebra.

Abstract

This chapter deals with methods for computing the invariants of an arbitrary polynomial representation of the general linear group GL(C^{n}). The main algorithm, to be presented in Sect. 4.6, is derived from Hilbert (1893). We will discuss Hilbert’s algorithm from the point of view of Gröbner bases theory. This chapter is less elementary than the previous three. While most of the presentation is self-contained, familiarity with basic notions of commutative algebra and representation theory will be assumed.