Proofs and confirmations: the story of the alternating sign matrix conjecture
"This is an Introduction to Recent Developments in algebraic combinatorics and an illustration of how research in mathematics actually progresses. The author recounts the story of the search for and discovery of a proof of a formula conjectured in the early 1980s: the number of m x n alternatin...
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
1999
|
| Edition: | 1. publ. |
| Series: | Spectrum series
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | "This is an Introduction to Recent Developments in algebraic combinatorics and an illustration of how research in mathematics actually progresses. The author recounts the story of the search for and discovery of a proof of a formula conjectured in the early 1980s: the number of m x n alternating sign matrices, objects that generalize permutation matrices. Although it was soon apparent that the conjecture must be true, the proof was elusive. Researchers became drawn to this problem, making connections to aspects of the invariant theory of Jacobi, Sylvester, Cayley, MacMahon, Schur, and Young, to partitions and plane partitions, to symmetric functions, to hypergeometric and basic hypergeometric series, and, finally, to the six-vertex model of statistical mechanics. All these threads are brought together in Zeilberger's 1995 proof of the original conjecture." "The book is accessible to anyone with a knowledge of linear algebra."--BOOK JACKET. |
| Physical Description: | XV, 274 S. Ill., graph. Darst. |
| ISBN: | 0521661706 0521666465 |
Staff View
MARC
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| 520 | 1 | |a "This is an Introduction to Recent Developments in algebraic combinatorics and an illustration of how research in mathematics actually progresses. The author recounts the story of the search for and discovery of a proof of a formula conjectured in the early 1980s: the number of m x n alternating sign matrices, objects that generalize permutation matrices. Although it was soon apparent that the conjecture must be true, the proof was elusive. Researchers became drawn to this problem, making connections to aspects of the invariant theory of Jacobi, Sylvester, Cayley, MacMahon, Schur, and Young, to partitions and plane partitions, to symmetric functions, to hypergeometric and basic hypergeometric series, and, finally, to the six-vertex model of statistical mechanics. All these threads are brought together in Zeilberger's 1995 proof of the original conjecture." "The book is accessible to anyone with a knowledge of linear algebra."--BOOK JACKET. | |
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| 650 | 7 | |a Bewijs (wetenschap) |2 gtt | |
| 650 | 7 | |a Combinatieleer |2 gtt | |
| 650 | 4 | |a Matrices | |
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| 650 | 4 | |a Combinatorial analysis | |
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Record in the Search Index
| _version_ | 1804127791322497024 |
|---|---|
| adam_text | Contents
Preface page xi
1 The Conjecture 1
1.1 How many are there? 3
1.2 Connections to plane partitions 9
1.3 Descending plane partitions 18
2 Fundamental Structures 33
2.1 Generating functions 35
2.2 Partitions 42
2.3 Recursive formulae 55
2.4 Determinants 63
3 Lattice Paths and Plane Partitions 73
3.1 Lattice paths 74
3.2 Inversion numbers 84
3.3 Plane partitions 92
3.4 Cyclically symmetric plane partitions 101
3.5 Dodgson s algorithm 111
4 Symmetric Functions 119
4.1 Schur functions 120
4.2 Semistandard tableaux 127
4.3 Proof of the MacMahon conjecture 134
5 Hypergeometric Series 151
5.1 Mills, Robbins, and Rumsey s bright idea 151
5.2 Identities for hypergeometric series 160
5.3 Proof of the Macdonald conjecture 177
ix
x Contents
6 Explorations 191
6.1 Charting the territory 191
6.2 Totally symmetric self complementary plane partitions 203
6.3 Proof of the ASM conjecture 215
7 Square Ice 223
7.1 Insights from statistical mechanics 224
7.2 Baxter s triangle to triangle relation 232
7.3 Proof of the refined ASM conjecture 245
7.4 Forward 257
Bibliography 261
Index of Notation 269
General Index 271
|
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| classification_tum | MAT 150f |
| ctrlnum | (OCoLC)40964741 (DE-599)BVBBV013095821 |
| dewey-full | 512.9/434 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.9/434 |
| dewey-search | 512.9/434 |
| dewey-sort | 3512.9 3434 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. |
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| id | DE-604.BV013095821 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:38:58Z |
| institution | BVB |
| isbn | 0521661706 0521666465 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008919775 |
| oclc_num | 40964741 |
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| physical | XV, 274 S. Ill., graph. Darst. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series2 | Spectrum series |
| spelling | Bressoud, David M. 1950- Verfasser (DE-588)136747221 aut Proofs and confirmations the story of the alternating sign matrix conjecture David M. Bressoud 1. publ. Cambridge [u.a.] Cambridge Univ. Press 1999 XV, 274 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Spectrum series "This is an Introduction to Recent Developments in algebraic combinatorics and an illustration of how research in mathematics actually progresses. The author recounts the story of the search for and discovery of a proof of a formula conjectured in the early 1980s: the number of m x n alternating sign matrices, objects that generalize permutation matrices. Although it was soon apparent that the conjecture must be true, the proof was elusive. Researchers became drawn to this problem, making connections to aspects of the invariant theory of Jacobi, Sylvester, Cayley, MacMahon, Schur, and Young, to partitions and plane partitions, to symmetric functions, to hypergeometric and basic hypergeometric series, and, finally, to the six-vertex model of statistical mechanics. All these threads are brought together in Zeilberger's 1995 proof of the original conjecture." "The book is accessible to anyone with a knowledge of linear algebra."--BOOK JACKET. Analyse combinatoire Bewijs (wetenschap) gtt Combinatieleer gtt Matrices Matrices gtt Mécanique statistique Combinatorial analysis Statistical mechanics HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008919775&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bressoud, David M. 1950- Proofs and confirmations the story of the alternating sign matrix conjecture Analyse combinatoire Bewijs (wetenschap) gtt Combinatieleer gtt Matrices Matrices gtt Mécanique statistique Combinatorial analysis Statistical mechanics |
| title | Proofs and confirmations the story of the alternating sign matrix conjecture |
| title_auth | Proofs and confirmations the story of the alternating sign matrix conjecture |
| title_exact_search | Proofs and confirmations the story of the alternating sign matrix conjecture |
| title_full | Proofs and confirmations the story of the alternating sign matrix conjecture David M. Bressoud |
| title_fullStr | Proofs and confirmations the story of the alternating sign matrix conjecture David M. Bressoud |
| title_full_unstemmed | Proofs and confirmations the story of the alternating sign matrix conjecture David M. Bressoud |
| title_short | Proofs and confirmations |
| title_sort | proofs and confirmations the story of the alternating sign matrix conjecture |
| title_sub | the story of the alternating sign matrix conjecture |
| topic | Analyse combinatoire Bewijs (wetenschap) gtt Combinatieleer gtt Matrices Matrices gtt Mécanique statistique Combinatorial analysis Statistical mechanics |
| topic_facet | Analyse combinatoire Bewijs (wetenschap) Combinatieleer Matrices Mécanique statistique Combinatorial analysis Statistical mechanics |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008919775&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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