Polynomial representations of GLn with an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J.A. Green and M. Schocker
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1. Verfasser: | |
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Format: | Buch |
Sprache: | English |
Veröffentlicht: |
Berlin [u.a.]
Springer
2007
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Ausgabe: | 2. corr. and augm. ed. |
Schriftenreihe: | Lecture notes in mathematics
830 |
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Online-Zugang: | Inhaltsverzeichnis |
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Datensatz im Suchindex
DE-BY-TUM_call_number | 0102 MAT 001z 2001 B 999 0104 MAT 001z 2001 B 999 |
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DE-BY-TUM_katkey | 1577453 |
DE-BY-TUM_location | 01 |
DE-BY-TUM_media_number | 040010250158 040010232485 |
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adam_text | CONTENTS POLYNOMIAL REPRESENTATIONS OF GL N 1 INTRODUCTION
............................................... 1 2 POLYNOMIAL
REPRESENTATIONS OF GL N ( K 11 2.1 NOTATION, ETC. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2
THE CATEGORIES M K ( N ), M K ( N, R 2.3 THE SCHUR ALGEBRA S K ( N, R )
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 THE
MAP E : K * * S K ( N, R 2.5 MODULAR THEORY . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 THE MODULE
E * R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 17 2.7 CONTRAVARIANT DUALITY . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 19 2.8 A K ( N, R ) AS K
*-BIMODULE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 21 3 WEIGHTS AND CHARACTERS ................................... 23 3.1
WEIGHTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 23 3.2 WEIGHT SPACES . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3
SOME PROPERTIES OF WEIGHT SPACES . . . . . . . . . . . . . . . . . . . .
. . . . . . 24 3.4 CHARACTERS. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 26 3.5 IRREDUCIBLE
MODULES IN M K ( N, R ) . . . . . . . . . . . . . . . . . . . . . . . .
. . . 28 4 THE MODULES D *,K .........................................
33 4.2 * -TABLEAUX . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 33 4.3 BIDETERMINANTS . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34 4.4 DEFINITION OF D *,K . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 35 4.5 THE BASIS THEOREM FOR D *,K . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.6 THE
CARTER-LUSZTIG LEMMA . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 37 4.7 SOME CONSEQUENCES OF THE BASIS THEOREM . . . . . . .
. . . . . . . . . . . . . 39 4.8 JAMES*S CONSTRUCTION OF D *,K . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 40 VIII CONTENTS 5 THE
CARTER-LUSZTIG MODULES V *,K .......................... 43 5.1
DEFINITION OF V *,K . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 43 5.2 V *,K IS CARTER-LUSZTIG*S *WEYL MODULE*
. . . . . . . . . . . . . . . . . . . . 43 5.3 THE CARTER-LUSZTIG BASIS
FOR V *,K . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.4 SOME
CONSEQUENCES OF THE BASIS THEOREM . . . . . . . . . . . . . . . . . . .
. 47 5.5 CONTRAVARIANT FORMS ON V *,K . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 48 5.6 Z -FORMS OF V *,K . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6
REPRESENTATION THEORY OF THE SYMMETRIC GROUP ............. 53 6.1 THE
FUNCTOR F : M K ( N, R ) * MOD KG ( R )( R * N ) . . . . . . . . . . . .
. 53 6.2 GENERAL THEORY OF THE FUNCTOR F : MOD S * MOD ESE . . . . . . .
. . . 55 6.3 APPLICATION I. SPECHT MODULES AND THEIR DUALS . . . . . . .
. . . . . . . . 57 6.4 APPLICATION II. IRREDUCIBLE KG ( R )-MODULES,
CHAR K = P . . . . . . . 60 6.5 APPLICATION III. THE FUNCTOR F : M K (
N, R ) * M K ( N, R )( N * N ) 65 6.6 APPLICATION IV. SOME THEOREMS ON
DECOMPOSITION NUMBERS . . . 67 APPENDIX: SCHENSTED CORRESPONDENCE AND
LITTELMANN PATHS A INTRODUCTION
............................................... 73 A.1 PREAMBLE . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 73 A.2 THE ROBINSON-SCHENSTED ALGORITHM . . . . . . . . .
. . . . . . . . . . . . . . . 74 A.3 THE OPERATORS * E C , * F C . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 A.4
WHAT IS TO BE DONE . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 78 B THE SCHENSTED PROCESS
.................................... 81 B.1 NOTATIONS FOR TABLEAUX . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 B.2
THE MAP SCH : I ( N, R ) * T ( N, R ) . . . . . . . . . . . . . . . . .
. . . . . . . . . . 81 B.3 INSERTING A LETTER INTO A TABLEAU . . . . . .
. . . . . . . . . . . . . . . . . . . . . 82 B.4 EXAMPLES OF THE
SCHENSTED PROCESS. . . . . . . . . . . . . . . . . . . . . . . . . 85
B.5 PROOF THAT ( µ, U, V ) * X 1 BELONGS TO T ( N, R ) . . . . . . . . .
. . . . . . . 88 B.6 THE INVERSE SCHENSTED PROCESS . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 89 B.7 THE LADDER . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92 C SCHENSTED AND LITTELMANN OPERATORS ....................... 95 C.1
PREAMBLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 95 C.2 UNWINDING A TABLEAU . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 C.3 KNUTH*S
THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 103 C.4 THE *IF* PART OF KNUTH*S THEOREM. . . . . . . .
. . . . . . . . . . . . . . . . . . 107 C.5 LITTELMANN OPERATORS ON
TABLEAUX. . . . . . . . . . . . . . . . . . . . . . . . . . 114 C.6 THE
PROOF OF PROPOSITION B . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 116 D THEOREM A AND SOME OF ITS CONSEQUENCES
.................. 121 D.1 INGREDIENTS FOR THE PROOF OF THEOREM A . . .
. . . . . . . . . . . . . . . . . . 121 D.2 PROOF OF THEOREM A . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 D.3
PROPERTIES OF THE OPERATOR C . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 127 CONTENTS IX D.4 THE LITTELMANN ALGEBRA L ( N, R ). .
. . . . . . . . . . . . . . . . . . . . . . . . . . 129 D.5 THE MODULES
M Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 131 D.6 THE * -RECTANGLE . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 134 D.7 CANONICAL MAPS . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 135 D.8 THE ALGEBRA STRUCTURE OF L ( N, R ) . . . . . . . . . . .
. . . . . . . . . . . . . . . . 137 D.9 THE CHARACTER OF M * . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 D.10
THE LITTLEWOOD*RICHARDSON RULE . . . . . . . . . . . . . . . . . . . . .
. . . . . 140 D.11 LASCOUX, LECLERC AND THIBON . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 143 E TABLES
..................................................... 147 E.1
SCHENSTED*S DECOMPOSITION OF I (3 , 3) . . . . . . . . . . . . . . . . .
. . . . . . 147 E.2 THE LITTELMANN GRAPH I (3 , 3) . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 148 INDEX OF SYMBOLS
.............................................. 151 REFERENCES
..................................................... 155 INDEX
.......................................................... 159 PPN:
259778249 TITEL: POLYNOMIAL REPRESENTATIONS OF GLN / WITH AN APPENDIX ON
SCHENSTED CORRESPONDENCE AND LITTELMANN PATHS BY K. ERDMANN, J. A. GREEN
AND M. SCHOCKER ; J. A. GREEN. - . - SPRINGER BERLIN 2007 ISBN:
978-3-540-46944-5; 3-540-46944-3 BIBLIOGRAPHISCHER DATENSATZ IM
SWB-VERBUND
|
any_adam_object | 1 |
author | Green, James A. |
author_facet | Green, James A. |
author_role | aut |
author_sort | Green, James A. |
author_variant | j a g ja jag |
building | Verbundindex |
bvnumber | BV022193496 |
classification_rvk | SI 850 |
classification_tum | MAT 202f |
ctrlnum | (OCoLC)315748983 (DE-599)BVBBV022193496 |
discipline | Mathematik |
edition | 2. corr. and augm. ed. |
format | Book |
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id | DE-604.BV022193496 |
illustrated | Illustrated |
indexdate | 2024-12-23T19:54:28Z |
institution | BVB |
isbn | 9783540469445 3540469443 |
language | English |
oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015405033 |
oclc_num | 315748983 |
open_access_boolean | |
owner | DE-91G DE-BY-TUM DE-824 DE-83 DE-11 DE-188 |
owner_facet | DE-91G DE-BY-TUM DE-824 DE-83 DE-11 DE-188 |
physical | IX, 161 S. graph. Darst. |
publishDate | 2007 |
publishDateSearch | 2007 |
publishDateSort | 2007 |
publisher | Springer |
record_format | marc |
series | Lecture notes in mathematics |
series2 | Lecture notes in mathematics |
spellingShingle | Green, James A. Polynomial representations of GLn with an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J.A. Green and M. Schocker Lecture notes in mathematics Allgemeine lineare Gruppe (DE-588)4284587-7 gnd Schur-Algebra (DE-588)4180242-1 gnd Darstellung (DE-588)4200624-7 gnd Polynom (DE-588)4046711-9 gnd Lineare Gruppe (DE-588)4138778-8 gnd Darstellung Mathematik (DE-588)4128289-9 gnd |
subject_GND | (DE-588)4284587-7 (DE-588)4180242-1 (DE-588)4200624-7 (DE-588)4046711-9 (DE-588)4138778-8 (DE-588)4128289-9 |
title | Polynomial representations of GLn with an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J.A. Green and M. Schocker |
title_auth | Polynomial representations of GLn with an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J.A. Green and M. Schocker |
title_exact_search | Polynomial representations of GLn with an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J.A. Green and M. Schocker |
title_full | Polynomial representations of GLn with an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J.A. Green and M. Schocker J. A. Green |
title_fullStr | Polynomial representations of GLn with an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J.A. Green and M. Schocker J. A. Green |
title_full_unstemmed | Polynomial representations of GLn with an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J.A. Green and M. Schocker J. A. Green |
title_short | Polynomial representations of GLn |
title_sort | polynomial representations of gln with an appendix on schensted correspondence and littelmann paths by k erdmann j a green and m schocker |
title_sub | with an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J.A. Green and M. Schocker |
topic | Allgemeine lineare Gruppe (DE-588)4284587-7 gnd Schur-Algebra (DE-588)4180242-1 gnd Darstellung (DE-588)4200624-7 gnd Polynom (DE-588)4046711-9 gnd Lineare Gruppe (DE-588)4138778-8 gnd Darstellung Mathematik (DE-588)4128289-9 gnd |
topic_facet | Allgemeine lineare Gruppe Schur-Algebra Darstellung Polynom Lineare Gruppe Darstellung Mathematik |
url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015405033&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
volume_link | (DE-604)BV000676446 |
work_keys_str_mv | AT greenjamesa polynomialrepresentationsofglnwithanappendixonschenstedcorrespondenceandlittelmannpathsbykerdmannjagreenandmschocker |