## Applied Finite Group ActionsAlso 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. |

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### 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 |

437 | |

445 | |

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### Common terms and phrases

action of G application assume bijection Burnside matrix called canonic Cauchy-Frobenius Lemma character polynomial coefficients column conjugacy classes consider consisting contains Corollary corresponding cycle indicator polynomials cyclic factors cyclic group decomposition defined denote diagram double cosets elements entries enumeration equal equation equivalence evaluate example Exercises Exercise finite action finite group fixed points g e G GF(q graphs group actions group G H x G Hence homomorphism identity incidence structures induced injective inverse irreducible characters isomorphic lattice left cosets length linear linear codes matrix representation Mobius function monomial multiplication natural action natural numbers node number of orbits obtain orbits of G ordinary irreducible particular plethysm poset Proof proper partition prove recursion Schubert polynomials Schur polynomials shows species stabilizer structures subsets subspace summand symmetric group symmetry classes table of marks Theorem transitive transversal unimodal vector space vertices wreath product yields

### 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.