Ebook: Computational Methods for Representations of Groups and Algebras: Euroconference in Essen (Germany), April 1–5, 1977

This book presents material from 3 survey lectures and 14 additional invited lectures given at the Euroconference "Computational Methods for Representations of Groups and Algebras" held at Essen University in April 1997. The purpose of this meeting was to provide a survey of general theoretical and computational methods and recent advances in the representation theory of groups and algebras. The foundations of these research areas were laid in survey articles by P. Dräxler and R. Nörenberg on "Classification problems in the representation theory of finite-dimensional algebras", R. A. Wilson on "Construction of finite matrix groups" and E. Green on "Noncommutative Gröbner bases, and projective resolutions". Furthermore, new applications of the computational methods in linear algebra to the revision of the classification of finite simple sporadic groups are presented. Computational tools (including high-performance computations on supercomputers) have become increasingly important for classification problems. They are also inevitable for the construction of projective resolutions of finitely generated modules over finite-dimensional algebras and the study of group cohomology and rings of invariants. A major part of this book is devoted to a survey of algorithms for computing special examples in the study of Grothendieck groups, quadratic forms and derived categories of finite-dimensional algebras. Open questions on Lie algebras, Bruhat orders, Coxeter groups and Kazhdan Lusztig polynomials are investigated with the aid of computer programs. The contents of this book provide an overview on the present state of the art. Therefore it will be very useful for graduate students and researchers in mathematics, computer science and physics.

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Content:
Front Matter....Pages i-xiv
Front Matter....Pages 1-1
Classification Problems in the Representation Theory of Finite-Dimensional Algebras....Pages 3-28
Noncommutative Gr?bner Bases, and Projective Resolutions....Pages 29-60
Construction of Finite Matrix Groups....Pages 61-83
Front Matter....Pages 85-85
Derived Tubularity: a Computational Approach....Pages 87-106
Problems in the Calculation of Group Cohomology....Pages 107-120
On a Tensor Category for the Exceptional Lie Groups....Pages 121-137
Non-Commutative Gr?bner Bases and Anick’s Resolution....Pages 139-159
A new Existence Proof of Janko’s Simple Group J 4 ....Pages 161-175
The Normalization: a new Algorithm, Implementation and Comparisons....Pages 177-185
A Computer Algebra Approach to sheaves over Weighted Projective Lines....Pages 187-200
Some Open Problems in the Theory of Kazhdan-Lusztig polynomials and Coxeter groups....Pages 201-210
Relative Trace Ideals and Cohen-Macaulay Quotients of Modular Invariant Rings....Pages 211-233
On Sims’ Presentation for Lyons’ Simple Group....Pages 235-240
A Presentation for the Lyons Simple Group....Pages 241-249
Reduction of Weakly Definite Unit Forms....Pages 251-257
Decision Problems in Finitely Presented Groups....Pages 259-265
Some Algorithms in Invariant Theory of Finite Groups....Pages 267-285
Coxeter Transformations associated with Finite Dimensional Algebras....Pages 287-308
Condensation of induced Representations and an Application: the 2-Modular Decomposition Numbers of Co 2 ....Pages 309-321
Bimodule and Matrix Problems....Pages 323-357

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Content:
Front Matter....Pages i-xiv
Front Matter....Pages 1-1
Classification Problems in the Representation Theory of Finite-Dimensional Algebras....Pages 3-28
Noncommutative Gr?bner Bases, and Projective Resolutions....Pages 29-60
Construction of Finite Matrix Groups....Pages 61-83
Front Matter....Pages 85-85
Derived Tubularity: a Computational Approach....Pages 87-106
Problems in the Calculation of Group Cohomology....Pages 107-120
On a Tensor Category for the Exceptional Lie Groups....Pages 121-137
Non-Commutative Gr?bner Bases and Anick’s Resolution....Pages 139-159
A new Existence Proof of Janko’s Simple Group J 4 ....Pages 161-175
The Normalization: a new Algorithm, Implementation and Comparisons....Pages 177-185
A Computer Algebra Approach to sheaves over Weighted Projective Lines....Pages 187-200
Some Open Problems in the Theory of Kazhdan-Lusztig polynomials and Coxeter groups....Pages 201-210
Relative Trace Ideals and Cohen-Macaulay Quotients of Modular Invariant Rings....Pages 211-233
On Sims’ Presentation for Lyons’ Simple Group....Pages 235-240
A Presentation for the Lyons Simple Group....Pages 241-249
Reduction of Weakly Definite Unit Forms....Pages 251-257
Decision Problems in Finitely Presented Groups....Pages 259-265
Some Algorithms in Invariant Theory of Finite Groups....Pages 267-285
Coxeter Transformations associated with Finite Dimensional Algebras....Pages 287-308
Condensation of induced Representations and an Application: the 2-Modular Decomposition Numbers of Co 2 ....Pages 309-321
Bimodule and Matrix Problems....Pages 323-357
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