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: Green, James A. (VerfasserIn)
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Sprache:English
Veröffentlicht: Berlin [u.a.] Springer 2007
Ausgabe:2. corr. and augm. ed.
Schriftenreihe:Lecture notes in mathematics 830
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Datensatz im Suchindex

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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
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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
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Darstellung Mathematik (DE-588)4128289-9 gnd
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(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