Symmetry and Spaces

by Campbell, H. E. A.; Helmonck, Aloysius G.; Kraft, Hanspeter; Wehlau, David

• ISBN13: 9780817648749
• ISBN10: 0817648747
• Edition: 1st
• Format: Hardcover
• Copyright: 2009-11-15
• Publisher: Birkhauser

Summary

This volume includes articles that are a sampling of modern day algebraic geometry with associated group actions from its leading experts. There are three papers examining various aspects of modular invariant theory (Broer, Elmer and Fleischmann, Shank and Wehlau), and seven papers concentrating on characteristic 0 (Brion, Daigle and Freudenberg, Greb and Heinzner, Helminck, Kostant, Kraft and Wallach, Traves).

Table of Contents

Foreword	p. v
Preface	p. ix
Acknowledgements	p. xv
Some Basic Results on Actions of Nonaffine Algebraic Groups	p. 1
Introduction	p. 1
Proof of Theorem 1.1	p. 5
Proof of Theorem 1.2	p. 9
Proof of Theorem 1.3	p. 11
Proof of Theorem 1.4	p. 13
Examples	p. 14
References	p. 20
On Chevalley-Shephard-Todd's Theorem in Positive Characteristic	p. 21
The direct summand property and point-stabilizers	p. 22
Main result and tools for the proof	p. 25
Tools	p. 26
Details	p. 28
Families	p. 29
Exceptional cases	p. 33
References	p. 34
Families of Affine Fibrations	p. 35
Introduction	p. 35
Preliminaries	p. 36
Notation and definitions	p. 36
Some known results on affine nbrations	p. 37
Locally nilpotent derivations	p. 38
A criterion for affine fibrations	p. 39
Dimension four	p. 41
A remark on stable variables	p. 42
References	p. 43
On the Depth of Modular Invariant Rings for the Groups CpxCp	p. 45
Introduction	p. 45
Flatness and strong flatness	p. 47
A sufficient condition for strong flatness	p. 49
Groups of the form CpxCp	p. 52
Decomposable representations	p. 55
References	p. 60
Kählerian Reduction in Steps	p. 63
Introduction	p. 63
Reductive group actions on Kahler spaces	p. 66
Stratifying holomorphic G-spaces	p. 67
Kählerian reduction	p. 68
Reduction in steps	p. 71
Analytic reduction in steps	p. 72
Kählerian reduction in steps	p. 75
Stratifications in steps	p. 77
Projectivity and Kählerian reduction	p. 78
	p. 81
	p. 83
Introduction	p. 83
Preliminaries about symmetric k-varieties	p. 85
Notations	p. 85
Classification of symmetric k-varieties	p. 87
k-Anisotropic symmetric subgroups	p. 88
Orbits of parabolic subgroups acting on symmetric varieties	p. 90
k Algebraically closed and P = B aBorel subgroup	p. 91
k Algebraically closed and P a parabolic subgroup	p. 92
Parabolic subgroups with cr-stable Levi subgroups	p. 93
Some examples	p. 94
k=k and P a minimal parabolic k-subgroup	p. 96
Conjugacy classes of ó-stable tori	p. 98
W-Action on VA and VK	p. 98
Action of the Weyl group on twisted involutions	p. 100
Conjugacy classes of ¿-stable maximal A-split tori	p. 101
tfrConjugacy classes in A¿k for G = SL(2,k)	p. 104
Bruhat order and twisted involution posets	p. 105
Combinatorics of twisted involutions	p. 107
Lifting ¿' to an involution of G	p. 110
a-Stable parabolic subgroups	p. 112
Relation between ¿ and ¿'	p. 113
W-Actionon J¿	p. 114
Orbit closures	p. 114
Bruhat order on Va and VK	p. 116
Combinatorial Bruhat order on Va	p. 117
Admissible sequences in V	p. 118
Bruhat order on J and J¿'	p. 119
Combinatorics of the Richardson-Springer involution poset	p. 122
Other orbit decompositions	p. 123
Orbits of symmetric subgroups	p. 123
Orbits of parahoric subgroups	p. 124
References	p. 125
Root Systems for Levi Factors and Borel-de Siebenthal Theory	p. 129
Introduction	p. 129
	p. 129
Levi Factor Foot System	p. 131
	p. 131
	p. 132
	p. 135
Properties of the t-root System	p. 136
	p. 136
	p. 138
	p. 140
	p. 141
Borel-de Siebenthal Theory, Special Elements, and the Lie Subalgebras they Define	p. 144
	p. 144
	p. 146
	p. 147
	p. 148
Example	p. 150
	p. 150
References	p. 152
Polarizations and Nullcone of Representations of Reductive Groups	p. 153
Linear subspaces of the nullcone	p. 153
Some examples	p. 155
General polarizations	p. 158
Nullcone of several copies of binary forms	p. 159
Generators and system of parameters for the invariants of 3-qubits	p. 162
References	p. 166
Decomposing Symmetric Powers of Certain Modular Representations of Cyclic Groups	p. 169
Introduction	p. 169
Preliminaries	p. 173
Computing F[VP+)z/p2	p. 176
The Noether number of VP+1	p. 183
Decomposing F[Vp+1]	p. 187
Generating functions	p. 191
References	p. 195
Differential Operators on Grassmann Varieties	p. 197
A fundamental theorem	p. 199
The Hilbert series of GrD(R)G and GrD(RG)	p. 203
References	p. 206
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