Singular loci of Schubert varieties:
"Singular Loci of Schubert Varieties is a work at the crossroads of representation theory, algebraic geometry, and combinatorics. In this work, Billey and Lakshmibai have recreated and restructured the various theories and approaches of those articles and present a clearer understanding of this...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
2000
|
| Series: | Progress in mathematics
182 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | "Singular Loci of Schubert Varieties is a work at the crossroads of representation theory, algebraic geometry, and combinatorics. In this work, Billey and Lakshmibai have recreated and restructured the various theories and approaches of those articles and present a clearer understanding of this important subdiscipline of Schubert varieties - namely singular loci."--BOOK JACKET. |
| Physical Description: | XII, 251 S. |
| ISBN: | 3764340924 0817640924 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Contents
Preface xi
Chapter 1. Introduction 1
Chapter 2. Generalities on G/B and G/Q 7
2.1. Abstract root systems 7
2.2. Root systems of algebraic groups 9
2.3. Root subgroups 10
2.4. Parabolic subgroups 11
2.5. The Weyl group of a parabolic subgroup 12
2.6. Schubert varieties 12
2.7. The Bruhat Chevalley order 13
2.8. Line bundles on G/Q 13
2.9. Geometric properties of Schubert varieties 15
2.10. Equations defining a Schubert variety 16
2.11. Representations of semisimple algebraic groups 17
Chapter 3. Specifics for the Classical Groups 23
3.1. The Grassmannian variety Gd,n 23
3.2. The special linear group SL(n) 26
3.3. The symplectic group Sp(2n) 29
3.4. The odd orthogonal group SO(2ra + 1) 31
3.5. The even orthogonal group SO(2n) 33
Chapter 4. The Tangent Space and Smoothness 37
4.1. The Zariski tangent space 37
4.2. Smooth and singular points 37
4.3. The space T(w, r) 38
4.4. A canonical affine neighborhood of a T fixed point 39
4.5. Tangent cone and Jacobian criteria for smoothness 40
4.6. Discussion of smoothness at a T fixed point 41
4.7. Multiplicity at a point P on a variety X 42
4.8. Degree of X(w) 44
4.9. Summary of smoothness criteria 46
Chapter 5. Root System Description of T(w, r) 47
5.1. Polo s results 47
viii CONTENTS
5.2. Bases Bx, B*x for VK(X) and H°(G/B, Lx) 49
5.3. Description of T(w, id) 52
5.4. Description of T(w, r) 56
5.5. Tangent space and certain weight multiplicities 63
5.6. The B module T(w, id) 67
5.7. Two smoothness criteria of Carrell Kuttler 68
Chapter 6. Rational Smoothness and Kazhdan Lusztig Theory 71
6.1. Kazhdan Lusztig polynomials 72
6.2. Carrell Peterson s criteria 77
6.3. Combinatorial formulas for Kazhdan Lusztig polynomials 81
Chapter 7. Nil Hecke Ring and the Singular Locus of X(w) 91
7.1. The nil Hecke ring 91
7.2. Criteria for smoothness and rational smoothness 94
7.3. Representation theoretic results on the tangent cone 97
7.4. Proof of smoothness criterion 98
Chapter 8. Patterns, Smoothness and Rational Smoothness 103
8.1. Type A: criterion in terms of patterns 103
8.2. Conjecture in type A 104
8.3. Types B, C, D: criterion in terms of patterns 106
8.4. Type C results of Lakshmibai Song using permutations 115
Chapter 9. Minuscule and cominuscule G/P 119
9.1. Results on small resolutions 122
9.2. Brion Polo results 131
9.3. Irreducible components of SingX(u;) in special cases 138
9.4. Multiplicity at a singular point 144
9.5. The symplectic Grassmannian Sp(2n)/Pn 155
Chapter 10. Rank Two Results 159
10.1. Kumar s method 159
10.2. Tangent space computations 161
Chapter 11. Related Combinatorial Results 169
11.1. Factoring the Poincare polynomial of a Schubert variety 169
11.2. Structure of Bruhat intervals 170
11.3. Generating function for smooth permutations 172
11.4. Bona s results 172
Chapter 12. Related Varieties 175
12.1. Opposite cells in Schubert varieties in SL{n)/B 175
12.2. Determinantal varieties 180
12.3. Ladder determinantal varieties 185
12.4. Quiver varieties 201
CONTENTS ix
12.5. Variety of complexes 203
Chapter 13. Addendum 207
13.1. Dynkin Diagrams 207
13.2. Summary of Smoothness Criteria 208
13.3. Table of Minimal Bad Patterns 210
13.4. Singular loci of A5, B4, C4, D4 212
Bibliography 239
Index 247
|
| any_adam_object | 1 |
| author | Billey, Sara Lakshmibai, Venkatramani |
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| ctrlnum | (OCoLC)44750779 (DE-599)BVBBV013352208 |
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| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/53 |
| dewey-search | 516.3/53 |
| dewey-sort | 3516.3 253 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV013352208 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:44:19Z |
| institution | BVB |
| isbn | 3764340924 0817640924 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009106416 |
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| physical | XII, 251 S. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Progress in mathematics |
| series2 | Progress in mathematics |
| spelling | Billey, Sara Verfasser aut Singular loci of Schubert varieties Sara Billey ; V. Lakshmibai Boston [u.a.] Birkhäuser 2000 XII, 251 S. txt rdacontent n rdamedia nc rdacarrier Progress in mathematics 182 "Singular Loci of Schubert Varieties is a work at the crossroads of representation theory, algebraic geometry, and combinatorics. In this work, Billey and Lakshmibai have recreated and restructured the various theories and approaches of those articles and present a clearer understanding of this important subdiscipline of Schubert varieties - namely singular loci."--BOOK JACKET. Schubert, Variétés de Variëteiten (wiskunde) gtt Schubert varieties Schubert-Mannigfaltigkeit (DE-588)4512043-2 gnd rswk-swf Singularität Mathematik (DE-588)4077459-4 gnd rswk-swf Schubert-Mannigfaltigkeit (DE-588)4512043-2 s Singularität Mathematik (DE-588)4077459-4 s DE-604 Lakshmibai, Venkatramani Verfasser (DE-588)122537416 aut Progress in mathematics 182 (DE-604)BV000004120 182 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009106416&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Billey, Sara Lakshmibai, Venkatramani Singular loci of Schubert varieties Progress in mathematics Schubert, Variétés de Variëteiten (wiskunde) gtt Schubert varieties Schubert-Mannigfaltigkeit (DE-588)4512043-2 gnd Singularität Mathematik (DE-588)4077459-4 gnd |
| subject_GND | (DE-588)4512043-2 (DE-588)4077459-4 |
| title | Singular loci of Schubert varieties |
| title_auth | Singular loci of Schubert varieties |
| title_exact_search | Singular loci of Schubert varieties |
| title_full | Singular loci of Schubert varieties Sara Billey ; V. Lakshmibai |
| title_fullStr | Singular loci of Schubert varieties Sara Billey ; V. Lakshmibai |
| title_full_unstemmed | Singular loci of Schubert varieties Sara Billey ; V. Lakshmibai |
| title_short | Singular loci of Schubert varieties |
| title_sort | singular loci of schubert varieties |
| topic | Schubert, Variétés de Variëteiten (wiskunde) gtt Schubert varieties Schubert-Mannigfaltigkeit (DE-588)4512043-2 gnd Singularität Mathematik (DE-588)4077459-4 gnd |
| topic_facet | Schubert, Variétés de Variëteiten (wiskunde) Schubert varieties Schubert-Mannigfaltigkeit Singularität Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009106416&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000004120 |
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