Algorithms in Invariant Theory
by Sturmfels, BerndEdition: 2nd
Format: Paperback
Pub. Date: 2008-06-25
Publisher(s): Springer Verlag
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Summary
J. Kung and G.-C. Rota, in their 1984 paper, write: 'œLike 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 'œclassical 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.
Table of Contents
| Introduction | p. 1 |
| Symmetric polynomials | p. 2 |
| Grobner bases | p. 7 |
| What is invariant theory? | p. 14 |
| Torus invariants and integer programming | p. 19 |
| Invariant theory of finite groups | p. 25 |
| Finiteness and degree bounds | p. 25 |
| Counting the number of invariants | p. 29 |
| The Cohen-Macaulay property | p. 37 |
| Reflection groups | p. 44 |
| Algorithms for computing fundamental invariants | p. 50 |
| Grobner bases under finite group action | p. 58 |
| Abelian groups and permutation groups | p. 64 |
| Bracket algebra and projective geometry | p. 77 |
| The straightening algorithm | p. 77 |
| The first fundamental theorem | p. 84 |
| The Grassmann-Cayley algebra | p. 94 |
| Applications to projective geometry | p. 100 |
| Cayley factorization | p. 110 |
| Invariants and covariants of binary forms | p. 117 |
| Gordan's finiteness theorem | p. 129 |
| Invariants of the general linear group | p. 137 |
| Representation theory of the general linear group | p. 137 |
| Binary forms revisited | p. 147 |
| Cayley's [Omega]-process and Hilbert finiteness theorem | p. 155 |
| Invariants and covariants of forms | p. 161 |
| Lie algebra action and the symbolic method | p. 169 |
| Hilbert's algorithm | p. 177 |
| Degree bounds | p. 185 |
| References | p. 191 |
| Subject index | p. 196 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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