Algorithms in Invariant Theory

By: Peter Paule, Bernd Sturmfels

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Paperback | 28 April 2008 | Edition Number 2

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Paperback
208 Pages

Edition Type
Revised

Dimensions(cm)
24.41 x 16.99 x 1.12

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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.

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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
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Algorithms in Invariant Theory

At a Glance

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