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Tables of Contents for Computational Methods for Representations of Groups and Algebras
Chapter/Section Title
Page #
Page Count
I Introductory Articles
3
85
1 Classification Problems in the Representation Theory of Finite-Dimensional Algebras
3
26
1.1 Introduction
3
2
1.2 Algebras, Aggregates and Modules
5
4
1.3 Quivers and Posets
9
2
1.4 The Auslander-Reiten Quiver
11
3
1.5 Representation Types
14
3
1.6 Strongly simply connected Spectroids of Finite and Tame Representation Type
17
4
1.7 Thin Modules over completely separating Spectroids
21
8
2 Noncommutative Grobner Bases, and Projective Resolutions
29
32
2.1 Overview
29
1
2.2 Grobner Bases
29
7
2.3 Algorithms
36
8
2.4 Computational Aspects
44
4
2.5 Modules, Presentations and Resolutions
48
5
2.6 Applications of Grobner Bases
53
8
3 Construction of Finite Matrix Groups
61
27
3.1 Introduction
61
1
3.2 A Small Example
62
2
3.3 Some Variants
64
2
3.4 Towards a General Method
66
3
3.5 A Generalization
69
2
3.6 Thompson's Theorem
71
2
3.7 The Meataxe
73
1
3.8 Some more Applications
74
2
3.9 The Baby Monster
76
1
3.10 An Analogue for Permutation Groups
76
3
3.11 Some Applications of the Permutation Group Construction
79
1
3.12 The Monster
80
1
3.13 A World-Wide-Web Atlas of Group Representations
81
7
II Keynote Articles
88
4 Derived Tubularity: a Computational Approach
88
19
4.1 Derived Equivalence for Algebras
88
4
4.2 Reduction of Integral Quadratic Forms
92
6
4.3 Derived Tubularity: an Algorithm
98
3
4.4 Examples
101
6
5 Problems in the Calculation of Group Cohomology
107
14
5.1 Introduction
107
2
5.2 Questions and Conjectures
109
4
5.3 Some Problems in the Computations
113
3
5.4 Two Exceptional Cases
116
5
6 On a Tensor Category for the Exceptional Lie Groups
121
19
6.1 Introduction
121
1
6.2 Tensor Categories
122
1
6.3 A Representation Category for the General Linear Groups
123
5
6.4 The Exceptional Series
128
1
6.5 The Category L
129
3
6.6 The Exceptional Categories
132
8
7 Non-Commutative Grobner Bases and Anick's Resolution
140
21
7.1 Main Examples
140
1
7.2 Hilbert Series and global Dimension
140
2
7.3 Normal Words and Grobner Basis
142
2
7.4 Graphs, Languages and Monomial Algebras
144
1
7.5 n--Chains and Poincare series
145
2
7.6 Anick's Resolution
147
1
7.7 Finite State Automata and Lie Algebras
148
1
7.8 Bergman Package under MS-DOS
148
1
7.9 ANICK -- Program for calculating Betti Numbers and Grobner Bases
149
12
8 A new Existence Proof of Janko's Simple Group J(4)
161
16
8.1 Introduction
161
1
8.2 Lempken's Subgroup G = (x, y) of GL(1333) (11)
162
4
8.3 Transformation of G into a Permutation Group
166
1
8.4 Group Structure of the approximate Centralizer H
167
4
8.5 The Order of C(G)(u(1))
171
1
8.6 The Main Result
172
5
9 The Normalization: a new Algorithm, Implementation and Comparisons
177
10
9.1 Introduction
177
1
9.2 Criterion for Normality
178
1
9.3 The Normalization Algorithm
179
2
9.4 Implementation and Comparisons
181
2
9.5 Examples
183
4
10 A Computer Algebra Approach to sheaves over Weighted Projective Lines
187
14
10.1 Introduction
187
3
10.2 Telescoping Functors
190
5
10.3 Computation of Telescoping Functors in K(0) X
195
6
11 Open Problems in the Theory of Kazhdan-Lusztig polynomials
201
10
11.1 Introduction
201
1
11.2 The Basic Definitions
201
2
11.3 Hecke algebras and Kazhdan-Lusztig polynomials
203
1
11.4 Kazhdan-Lusztig polynomials and Bruhat intervals
204
3
11.5 Some Problems in Type A
207
4
12 Relative Trace Ideals and Cohen Macaulay Quotients
211
24
12.1 Introduction
211
2
12.2 Basic Definitions
213
2
12.3 The Trace Map
215
3
12.4 Invariant Rings of Permutation Modules
218
2
12.5 The Regular Representation of Z(p)
220
2
12.6 A canonical Trace Ideal and Cohen-Macaulay Quotients
222
13
13 On Sims' Presentation for Lyons' Simple Group
235
6
13.1 Introduction
235
1
13.2 The Permutation Group of Degree 8835156
236
1
13.3 The Permutation Group of Degree 9606125
237
1
13.4 The final Verification
238
3
14 A Presentation for the Lyons Simple Group
241
10
14.1 Introduction
241
1
14.2 A Presentation for the Lyons Group
242
3
14.3 A computational Proof
245
1
14.4 The Large Coset Enumeration
246
5
15 Reduction of Weakly Definite Unit Forms
251
8
15.1 Introduction
251
1
15.2 Proper Reductions
251
2
15.3 Restricted Reductions
253
1
15.4 Critical Vectors
253
1
15.5 Algorithm for Weak Definiteness
253
6
16 Decision Problems in Finitely Presented Groups
259
8
16.1 Introduction
259
1
16.2 The Word Problem
259
3
16.3 The Conjugacy Problem
262
1
16.4 The Generalized Word Problem
263
4
17 Some Algorithms in Invariant Theory of Finite Groups
267
20
17.1 Introduction
267
1
17.2 Calculating Homogeneous Invariants
268
3
17.3 Constructing Primary Invariants
271
3
17.4 Calculating Secondary Invariants
274
4
17.5 Properties of the Invariant Ring
278
9
18 Coxeter Transformations associated with Finite Dimensional Algebras
287
22
18.1 Introduction
287
3
18.2 Bilinear Lattices
290
1
18.3 Coxeter Polynomials and Growth
291
3
18.4 The K-Theory of Canonical Algebras
294
6
18.5 The K-Theory of Hereditary Algebras
300
3
18.6 Examples, Problems and Questions
303
6
19 The 2-Modular Decomposition Numbers of Co(2)
309
14
19.1 Introduction
309
1
19.2 Condensation
310
2
19.3 Condensation of Induced Modules, theoretically
312
1
19.4 ... and practically
313
2
19.5 An Application: The 2-Modular Character Table of Co(2)
315
8
20 Bimodule and Matrix Problems
323
20.1 Introduction
323
3
20.2 The first Example -- Bimodule Problems
326
3
20.3 Differential (Z-) graded Category (D(Z)GC) and Boxes
329
14
20.4 Reductions
343
8
20.5 Quadratic Forms
351