Applied Finite Group Actions

Front Cover
Springer Science & Business Media, Aug 18, 1999 - Mathematics - 454 pages
0 Reviews
Also the present second edition of this book is an introduction to the theory of clas sification, enumeration, construction and generation of finite unlabeled structures in mathematics and sciences. Since the publication of the first edition in 1991 the constructive theory of un labeled finite structures has made remarkable progress. For example, the first- designs with moderate parameters were constructed, in Bayreuth, by the end of 1994 ([9]). The crucial steps were - the prescription of a suitable group of automorphisms, i. e. a stabilizer, and the corresponding use of Kramer-Mesner matrices, together with - an implementation of an improved version of the LLL-algorithm that allowed to find 0-1-solutions of a system of linear equations with the Kramer-Mesner matrix as its matrix of coefficients. of matrices of the The Kramer-Mesner matrices can be considered as submatrices form A" (see the chapter on group actions on posets, semigroups and lattices). They are associated with the action of the prescribed group G which is a permutation group on a set X of points induced on the power set of X. Hence the discovery of the first 7-designs with small parameters is due to an application of finite group actions. This method used by A. Betten, R. Laue, A. Wassermann and the present author is described in a section that was added to the manuscript of the first edi tion.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

1 Unlabeled Structures
21
12 Orbits Cosets and Double Cosets
29
13 Symmetry Classes of Mappings
36
14 Invariant Relations
45
15 Hidden Symmetries
50
2 Enumeration of Unlabeled Structures
53
22 Enumeration of Symmetry Classes
59
23 Application to Incidence Structures
67
8 Permutations
275
82 Root Number Functions
284
83 Equations in Groups
293
84 UpDown Sequences
297
85 Foulkes Characters
303
86 Schubert Polynomials
307
9 Construction and Generation
317
91 Orbit Evaluation
318

24 Special Symmetry Classes
73
3 Enumeration by Weight
85
32 Cycle Indicator Polynomials
91
33 Sums of Cycle Indicators Recursive Methods
100
34 Applications to Chemistry
103
35 A Generalization
109
36 The Decomposition Theorem
115
4 Enumeration by Stabilizer Class
121
42 Asymmetric Orbits Lyndon Words the Cyclotomic Identity
126
43 Tables of Marks and Burnside Matrices
131
44 Weighted Enumeration by Stabilizer Class
137
5 Poset and Semigroup Actions
141
52 Examples
150
53 Application to Combinatorial Designs
157
54 The Burnside Ring
162
6 Representations
169
62 Tableaux and Matrices
181
63 The Determinantal Form
187
64 Standard Bideterminants
194
7 Further Applications
213
72 Symmetric Polynomials
219
73 The Diagram Lattice
223
74 Unimodality
228
75 The LittlewoodRichardson Rule
236
76 The MurnaghanNakayama Rule
247
77 Symmetrization and Permutrization
255
78 Plethysm of Representations
260
79 Actions on Chains
267
92 Transversals of Symmetry Classes
321
93 Orbits of Centralizers
326
94 The Homomorphism Principle
329
95 Orderly Generation
334
96 Generating Orbit Representatives
337
97 Symmetry Adapted Bases
341
98 Applications of Symmetry Adapted Bases
346
10 Tables
353
1011 Cyclic Groups
354
1012 Dihedral Groups
358
1013 Alternating Groups
363
1014 Symmetric Groups
366
102 Characters of Symmetric Groups
369
1022 Foulkes Tables
378
1023 Character Polynomials
380
103 Schubert Polynomials
392
11 Appendix
397
112 Finite Symmetric Groups
399
113 Rothe Diagram and Lehmer Code
406
114 Linear Representations
412
115 Ordinary Characters of Finite Groups
418
116 The Mobius Inversion
424
12 Comments and References
429
122 Further Comments
432
123 Suggestions for Further Reading
433
References
437
Index
445
Copyright

Other editions - View all

Common terms and phrases

Popular passages

Page 438 - Sloane, NJA 1979 Self-dual codes and lattices. In Relations between combinatorics and other parts of mathematics (Proc. Symp. Pure Math., vol. XXXIV), pp. 273-308. Providence, Rhode Island: American Mathematical Society.
Page ix - ... sciences. The structures in question are those which can be defined as equivalence classes on finite sets and in particular on finite sets of mappings. Prominent examples are graphs, switching functions, physical states and chemical isomers.

References to this book

All Book Search results »

Bibliographic information