## Invariant TheoryThis book presents the characteristic zero invariant theory of finite groups acting linearly on polynomial algebras. The author assumes basic knowledge of groups and rings, and introduces more advanced methods from commutative algebra along the way. The theory is illustrated by numerous examples and applications to physics, engineering, numerical analysis, combinatorics, coding theory, and graph theory. A wide selection of exercises and suggestions for further reading makes the book appropriate for an advanced undergraduate or first-year graduate level course. |

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### Contents

ix | |

7 | |

Part 2 Introduction and Göbels Bound | 43 |

Part 3 The First Fundamental Theorem of Invariant Theory and Noethers Bound | 97 |

Part 4 Noethers Theorems | 159 |

Part 5 Advanced Counting Methods and the ShephardToddChevalley Theorem | 223 |

Appendix A Rational Invariants | 287 |

Suggestions for Further Reading | 303 |

305 | |

307 | |

Back Cover | 317 |

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### Common terms and phrases

abelian group algebraically independent assume basis elements called Chapter coefficients commutative ring consists Corollary cosets cyclic group Dade basis defining representation deg(f Denote dihedral group direct sum elementary symmetric functions equation Example Exercise F-algebra faithful representation field extension field F field of fractions finite group ge:G given GL(n graded group G group of order Hence homogeneous homomorphism induced integral domain integral extension integrally closed invariant polynomials Invariant Theory irreducible isomorphism Krull dimension Lemma linear matrix maximal module monomials need to show Noetherian nonzero Note obtain orbit Chern classes orbit sums permutation representation Poincaré series polarized elementary symmetric polynomial f polynomial ring preceding result prime ideal Proof Proposition prove pseudoreflection group quaternion group R-module rational regular representation ring of invariants ring of polynomials root of unity special monomials subgroup subring subspace surjective symmetric group system of parameters Theorem vector space zero