Theory of Association Schemes

by Zieschang, Paul-Hermann

• ISBN13: 9783540261360
• ISBN10: 3540261362
• Format: Hardcover
• Copyright: 2005-12-16
• Publisher: Springer Nature

Summary

Theory of Association Schemes is the first concept-oriented treatment of the structure theory of association schemes. It contains several recent results which appear for the first time in book form. The generalization of Sylow's group theoretic theorems to scheme theory arises as a consequence of arithmetical considerations about quotient schemes. The theory of Coxeter schemes (equivalent to the theory of buildings) emerges naturally and yields a purely algebraic proof of Tits' main theorem on buildings of spherical type. Also a scheme-theoretic characterization of Glauberman's Z*-involutions is included. The text is self-contained and accessible for advanced undergraduate students.

Author Biography

Paul-Hermann Zieschang received a Doctor of Natural Sciences and the Habilitation in Mathematics from the Christian-Albrechts-Universit+ñt zu Kiel. He is also Extraordinary Professor of the Christian-Albrechts-Universit+ñt zu Kiel. Presently, he holds the position of an Associate Professor at the University of Texas at Brownsville. He held visiting positions at Kansas State University and at Kyushu University in Fukuoka.

Table of Contents

1 Basic Facts	1	(16)
1.1 Structure Constants	1	(4)
1.2 Symmetric Elements	5	(2)
1.3 The Complex Product	7	(5)
1.4 Complex Products and Valencies	12	(2)
1.5 Complex Products of Subsets of Cardinality 1	14	(3)
2 Closed Subsets	17	(22)
2.1 Basic Facts	17	(5)
2.2 Dedekind Identities	22	(1)
2.3 Structure Constants	23	(5)
2.4 Maximal Closed Subsets	28	(3)
2.5 Normalizer and Strong Normalizer	31	(4)
2.6 Conjugates of Closed Subsets	35	(4)
3 Generating Subsets	39	(24)
3.1 Basic Facts	40	(5)
3.2 The Thin Residue	45	(3)
3.3 Elements of Valency 2	48	(3)
3.4 Closed Subsets Generated by Involutions	51	(4)
3.5 Basic Results on Constrained Sets of Involutions	55	(2)
3.6 Basic Results on Coxeter Sets	57	(6)
4 Quotient Schemes	63	(20)
4.1 Basic Definitions	64	(5)
4.2 General Facts	69	(5)
4.3 Valencies	74	(3)
4.4 Hall Subsets	77	(1)
4.5 Sylow Subsets	78	(5)
5 Morphisms	83	(20)
5.1 Basic Facts	84	(4)
5.2 Isomorphisms	88	(2)
5.3 The Isomorphism Theorems	90	(3)
5.4 Composition Series	93	(2)
5.5 The Group Correspondence	95	(3)
5.6 Residually Thin Schemes	98	(5)
6 Faithful Maps	103	(30)
6.1 Basic Facts	104	(3)
6.2 Faithfully Embedded Closed Subsets	107	(5)
6.3 The Schur Group of a Closed Subset	112	(4)
6.4 Elements of Valency 2	116	(6)
6.5 More About Elements of Valency 2	122	(3)
6.6 Constrained Sets of Involutions	125	(3)
6.7 Thin Thin Residues	128	(5)
7 Products	133	(20)
7.1 Direct Products of Closed Subsets	133	(5)
7.2 Quasidirect Products of Schemes	138	(6)
7.3 Semidirect Products	144	(5)
7.4 A Characterization of Semidirect Products	149	(4)
8 From Thin Schemes to Modules	153	(30)
8.1 Rings and Modules	154	(7)
8.2 Integrality in Associative Rings with 1	161	(3)
8.3 Completely Reducibility	164	(4)
8.4 Irreducible Modules over Associative Rings with 1	168	(4)
8.5 Semisimple Associative Rings with 1	172	(3)
8.6 Characters of Associative Rings with 1	175	(4)
8.7 Roots of Unity in Integral Domains	179	(4)
9 Scheme Rings	183	(26)
9.1 Basic Facts	183	(9)
9.2 Algebraically Closed Base Fields	192	(3)
9.3 Scheme Rings over the Field of Complex Numbers	195	(4)
9.4 Closed Subsets	199	(5)
9.5 Schemes with at most Five Elements	204	(3)
9.6 Constrained Sets of Involutions	207	(2)
10 Dihedral Closed Subsets	209	(28)
10.1 General Remarks	210	(3)
10.2 The Spherical Case	213	(4)
10.3 Arithmetic of the Length Function	217	(3)
10.1 Two Characteristic Subsets	220	(6)
10.5 The Constrained Spherical Case	226	(2)
10.6 Dihedral Closed Subsets of Finite Valency	228	(9)
11 Coxeter Sets	237	(12)
11.1 Parabolic Subsets	238	(2)
11.2 Direct Products	240	(3)
11.3 Faithful Maps	243	(1)
11.4 The Extension Theorem	244	(5)
12 Spherical Coxeter Sets	249	(28)
12.1 Elements of Maximal Length	250	(3)
12.2 Faithful Maps	253	(4)
12.3 The Main Theorem	257	(2)
12.4 Coxeter Schemes of Finite Valency and Rank 2	259	(8)
12.5 Valencies and Multiplicities	267	(4)
12.6 Polarities	271	(6)
References	277	(4)
Index	281