Polynomial Invariants of Finite Groups (London Mathematical Society Lecture Note Series, Vol 190)

This Is The First Book To Deal With Invariant Theory And The Representations Of Finite Groups. By Restricting Attention To Finite Groups Dr Benson Is Able To Avoid Recourse To The Technical Machinery Of Algebraic Groups, And He Develops The Necessary Results From Commutative Algebra As He Proceeds. Thus The Book Should Be Accessible To Graduate Students. In Detail, The Book Contains An Account Of Invariant Theory For The Action Of A Finite Group On The Ring Of Polynomial Functions On A Linear Representation, Both In Characteristic Zero And Characteristic Pages Special Attention Is Paid To The Role Played By Pseudoreflections, Which Arise Because They Correspond To The Divisors In The Polynomial Ring Which Ramify Over The Invariants. Also Included Is A New Proof By Crawley-boevey And The Author Of The Carlisle-kropholler Conjecture. This Volume Will Appeal To All Algebraists, But Especially Those Working In Representation Theory, Group Theory, And Commutative Or Homological Algebra. 1. Finite Generation Of Invariants -- 1.1. The Basic Object Of Study -- 1.2. Noetherian Rings And Modules -- 1.3. Finite Groups In Arbitrary Characteristic -- 1.4. Krull Dimension And Going Up And Down -- 1.5. Noether's Bound In Characteristic Zero -- 1.6. Linearly Reductive Algebraic Groups -- 2. Poincare Series -- 2.1. The Hilbert-serre Theorem -- 2.2. Noether Normalization -- 2.3. Systems Of Parameters -- 2.4. Degree And [psi] -- 2.5. Molien's Theorem -- 2.6. Reflecting Hyperplanes -- 3. Divisor Classes, Ramification And Hyperplanes -- 3.1. Divisors -- 3.2. Primes Of Height One -- 3.3. Duality -- 3.4. Reflexive Modules -- 3.5. Divisor Classes And Unique Factorization -- 3.6. The Picard Group -- 3.7. The Trace -- 3.8. Ramification -- 3.9. Cl(k[v][superscript G]) -- 3.10. The Different -- 3.11. The Homological Different -- 3.12. A Ramification Formula -- 3.13. The Carlisle-kropholler Conjecture -- 4. Homological Properties Of Invariants -- 4.1. Minimal Resolutions -- 4.2. Hilbert's Syzygy Theorem -- 4.3. Depth And Cohen-macaulay Rings -- 4.4. Homological Characterization Of Depth -- 4.5. The Canonical Module And Gorenstein Rings -- 4.6. Watanabe's Theorem -- 5. Polynomial Tensor Exterior Algebras -- 5.1. Motivation And First Properties -- 5.2. A Variation On Molien's Theorem -- 5.3. The Invariants Are Graded Gorenstein -- 5.4. The Jacobian -- 6. Polynomial Rings And Regular Local Rings -- 6.1. Regular Local Rings -- 6.2. Serre's Converse To Hilbert's Syzygy Theorem -- 6.3. Uniqueness Of Factorization -- 6.4. Reflexive Modules -- 7. Groups Generated By Pseudoreflections -- 7.1. Reflections And Pseudoreflections -- 7.2. The Shephard-todd Theorem -- 7.3. A Theorem Of Solomon -- 8. Modular Invariants -- 8.1. Dickson's Theorem -- 8.2. The Special Linear Group -- 8.3. Symplectic Invariants -- A. Examples Over The Complex Numbers -- B. Examples Over Finite Fields. D.j. Benson. Includes Bibliographical References (p. 109-115) And Index. The process of constructing concepts underpins our capacity to encode information in an efficient and competent manner and also, ultimately, our ability to think in terms of abstract ideas such as justice, love and happiness. But what are the mechanisms which correspond to psychological categorization processes? This book unites many prominent approaches in modelling categorization. Each chapter focuses on a particular formal approach to categorization, presented by the proponent(s) or advocate(s) of that approach, and the authors consider the relation of this approach to other models and the ultimate objectives in their research programmes. The volume evaluates progress that has been made in the field and potential future developments. This is an essential companion to any scientist interested in the formal description of categorization and, more generally, in formal approaches to cognition. It will be the definitive guide to formal approaches in categorization research for years to come.