Introduction to the Theory of Standard Monomials: Second Edition PDF
Preview Introduction to the Theory of Standard Monomials: Second Edition
Texts and Readings in Mathematics 46 C.S. Seshadri Introduction to the Theory of Standard Monomials Second Edition Texts and Readings in Mathematics Volume 46 Advisory Editor C.S. Seshadri, Chennai Mathematical Institute, Chennai Managing Editor Rajendra Bhatia, Indian Statistical Institute, New Delhi Editor Manindra Agrawal, Indian Institute of Technology Kanpur, Kanpur V. Balaji, Chennai Mathematical Institute, Chennai R.B. Bapat, Indian Statistical Institute, New Delhi V.S. Borkar, Indian Institute of Technology Bombay, Mumbai T.R. Ramadas, Chennai Mathematical Institute, Chennai V. Srinivas, Tata Institute of Fundamental Research, Mumbai The Texts and Readings in Mathematics series publishes high-quality textbooks, research-level monographs, lecture notes and contributed volumes. Undergraduate and graduate students of mathematics, research scholars, and teachers would find this book series useful. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. The books in this series are co-published with Hindustan Book Agency, New Delhi, India. More information about this series at http://www.springer.com/series/15141 C.S. Seshadri Introduction to the Theory of Standard Monomials Second Edition 123 C.S.Seshadri Chennai MathematicalInstitute Chennai India Thisworkisaco-publicationwithHindustanBookAgency,NewDelhi,licensedforsaleinall countriesinelectronicformonly.SoldanddistributedinprintacrosstheworldbyHindustan BookAgency,P-19GreenParkExtension,NewDelhi110016,India.ISBN:978-93-80250-58-8 ©HindustanBookAgency2015. ISSN 2366-8725 (electronic) TextsandReadings inMathematics ISBN978-981-10-1813-8 (eBook) DOI 10.1007/978-981-10-1813-8 LibraryofCongressControlNumber:2016944375 ©SpringerScience+BusinessMediaSingapore2016andHindustanBookAgency2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublishers,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Thepublishers,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthis book are believed to be true and accurate at the date of publication. Neither the publishers nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerScience+BusinessMediaSingaporePteLtd. D E D I C A T I O N I would like to dedicate this book to the memory of C. Musili my former student, friend and collaborator in the subject of these lectures. His sudden death in October, 2005, came as a shock to his friends. Contents Preface to the second edition ix Preface to the first edition xi Introduction xiii About the Author xv 1 Schubert Varieties in the Grassmannian 1 1.1 Plu¨cker coordinates . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Schubert varieties . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Standard monomials . . . . . . . . . . . . . . . . . . . . . 12 1.4 Some Applications . . . . . . . . . . . . . . . . . . . . . . 19 1.5 Degeneration of Schubert varieties . . . . . . . . . . . . . 32 1.6 Connection with determinantal varieties and invariant theory . . . . . . . . . . . . . . . . . . . . . 38 2 Standard monomial theory on SL (k)/Q 55 n 2.1 Some facts about G/Q . . . . . . . . . . . . . . . . . . . . 55 2.2 Young diagrams and standard monomials . . . . . . . . . 59 2.3 Linear independence of standard monomials . . . . . . . . 61 2.4 Some facts about the partial order on W/W . . . . . . 63 Qi 2.5 Preparation for the main theorem . . . . . . . . . . . . . 65 2.6 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.7 Another proof for generation by standard monomials . . . 74 3 Applications 81 3.1 Singularities of Schubert varieties . . . . . . . . . . . . . . 81 3.2 Vanishing theorem . . . . . . . . . . . . . . . . . . . . . . 85 3.3 Character formula . . . . . . . . . . . . . . . . . . . . . . 89 3.4 Ideal theory of Schubert varieties . . . . . . . . . . . . . . 93 3.5 The variety of complexes . . . . . . . . . . . . . . . . . . . 97 vii viii CONTENTS 4 Schubert varieties in G/Q 107 4.1 Some remarks on linear algebraic groups . . . . . . . . . . 107 4.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . 110 4.3 Reduced decompositions . . . . . . . . . . . . . . . . . . . 118 4.4 The normalization map . . . . . . . . . . . . . . . . . . . 126 4.5 Chevalley’s multiplicity formula . . . . . . . . . . . . . . . 129 4.6 Deodhar’s Lemma . . . . . . . . . . . . . . . . . . . . . . 134 Appendix A. Cohen-Macaulay Properties 139 Appendix B. Normality of Schubert varieties 157 Appendix C. Standard Monomial Theory 165 Bibliography 213 Notation 217 Index 219 Symbols 221 Texts and Readings in Mathematics 223 Preface to the second edition Intheintroductiontothefirstedition,therewasamentionofconjec- tures of a Standard Monomial Theory (SMT) for a general semi-simple (simply-connected) algebraic group. These conjectures, due to Laksh- mibai,appearedinapaperintheproceedingsofaconferenceheldatthe University of Hyderabad in 1989. This paper is added as Appendix C in this second edition, keeping the same list of references as in the paper. The conjectures were later proved by P. Littelmann by using ideas from quantum groups and marked a significant progress in SMT. ManytypographicalerrorshavebeencorrectedandtheBibliography has been revised. I wish to thank my colleague Manoj Kummini for his meticulous reading of the revised manuscript. ix Preface to the first edition This book is a reproduction of the Brandeis Lectures Notes 4 with corrections of typographical errors and bringing up to date some refer- ences. I wish to thank my colleague K.V. Subrahmanyam for his help in this regard. xi
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