Symmetric Functions 2001: Surveys of Developments and Perspectives PDF
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Symmetric Functions 2001: Surveys of Developments and Perspectives NATO Science Series A Series presenting the results of scientific meetings supported under the NATO Science Programme. The Series is published by lOS Press, Amsterdam, and Kluwer Academic Publishers in conjunction with the NATO Scientific Affairs Division Sub-Series I. Life and Behavioural Sciences lOS Press II. Mathematics, Physics and Chemistry Kluwer Academic Publishers III. Computer and Systems Science lOS Press IV. Earth and Environmental Sciences Kluwer Academic Publishers V. Science and Technology Policy lOS Press The NATO Science Series continues the series of books published formerly as the NATO ASI Series. The NATO Science Programme offers support for collaboration in civil science between scientists of countries of the Euro-Atlantic Partnership Council. The types of scientific meeting generally supported are "Advanced Study Institutes" and "Advanced Research Workshops", although other types of meeting are supported from time to time. The NATO Science Series collects together the results of these meetings. The meetings are co-organized bij scientists from NATO countries and scientists from NATO's Partner countries - countries of the CIS and Central and Eastern Europe. Advanced Study Institutes are high-level tutorial courses offering in-depth study of latest advances in a field. Advanced Research Workshops are expert meetings aimed at critical assessment of a field, and identification of directions for future action. As a consequence of the restructuring of the NATO Science Programme in 1999, the NATO Science Series has been re-organised and there are currently Five Sub-series as noted above. Please consult the following web sites for information on previous volumes published in the Series, as well as details of earlier Sub-series. http://www.nato.intiscience http://www.wkap.nl http://www.iospress.nl http://www.wtv-books.de/nato-pco.htm Series II: Mathematics, Physics and Chemistry - Vol. 74 Symmetric Functions 2001: Surveys of Developments and Perspectives edited by Sergey Fomin University of Michigan, Department of Mathematics, Ann Arbor, Michigan, U.S.A. Springer-Science+Business Media, B.V. Proceedings of the NAra Advanced Study Instutute on Symmetric Functions 2001 : Surveys of Oevelopments and Perspectives Cambridge, U.K. 25 June-6 July 2001 A C.I. P. Catalogue record for this book is available from the Library of Congress. ISBN 978-1-4020-0774-3 ISBN 978-94-010-0524-1 (eBook) DOI 10.1007/978-94-010-0524-1 Printed an acid-free paper AII Rights Reserved © 2002 Springer Science+Business Media Oordrecht Originally published by Kluwer Academic Publishers in 2002 Softcover reprint of the hardcover 1 st edition 2002 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Contents Preface v List of contributors VB List of invited NATO ASI participants IX M. Haiman/ Notes on Macdonald polynomials and the geometry of Hilbert schemes 1 P. Hanlon/ The Laplacian method 65 V. Ivanov and G. Olshanski/ Kerov's central limit theorem for the Plancherel measure on Young diagrams 93 B. Leclerc/ Symmetric functions and the Fock space 153 M. Noumi/ An introduction to birational Weyl group actions 179 A. Okounkov / Symmetric functions and random partitions 223 A. Zelevinsky / From Littlewood-Richardson coefficients to cluster algebras in three lectures 253 PREFACE This volume contains the proceedings of the NATO Advanced Study Institute "Symmetric Functions 2001: Surveys of Developments and Per spectives", held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK, during the two weeks 25 June - 6 July 2001. The objective of the ASI was to survey recent developments and outline research perspectives in various fields, for which the fundamental questions can be stated in the language of symmetric functions (along the way emphasizing interdisciplinary connections). The instructional goals of the event determined its format: the ASI consisted of about a dozen mini-courses. Seven of them served as a basis for the papers comprising the current volume. The ASI lecturers were: Persi Diaconis, William Fulton, Mark Haiman, Phil Hanlon, Alexander Klyachko, Bernard Leclerc, Ian G. Macdonald, Masatoshi Noumi, Andrei Okounkov, Grigori Olshanski, Eric Opdam, Ana toly Vershik, and Andrei Zelevinsky. The organizing committee consisted of Phil Hanlon, Ian Macdonald, Andrei 0 kounkov, G rigori 0 lshanski (co-director), and myself ( co-director). The original ASI co-director Sergei Kerov, who was instrumental in determining the format and scope of the event, selection of speakers, and drafting the initial grant proposal, died in July 2000. Kerov's mathemat ical ideas strongly influenced the field, and were presented at length in a number of ASI lectures. A special afternoon session on Monday, July 2, was dedicated to his memory. This NATO ASI rounded off the half-year-long program "Symmetric Functions and Macdonald Polynomials" at the Newton Institute, led by Phil Hanlon, Ian Macdonald, and Alun Morris. The dedicated staff at the Newton Institute provided excellent logistical support for the meeting. The generous sponsorship by NATO's Scientific and Environmental Affairs Division is gratefully acknowledged. Last but not least, I would like to thank all the ASI participants for making this an unforgettable event. Sergey Fomin vii LIST OF CONTRIBUTORS Mark Haiman Department of Mathematics, 970 Evans Hall University of California Berkeley, CA 94720-3840 USA Phil Hanlon Department of Mathematics University of Michigan Ann Arbor, MI48109-1109 USA Vladimir Ivanov Chair of Algebra Faculty of Mathematics and Mechanics Moscow State University Vorob'evy Gory, Moscow 119992, GSP-2 Russia Bernard Leclerc Departement de MatMmatiques U niversite de Caen, Campus II Blvd. Marechal Juin, BP 5186, 14032 Caen Cedex France Masatoshi Noumi Department of Mathematics Kobe University Rokko, Kobe 657-8501 Japan ix x Grigori Olshanski Dobrushin Mathematics Laboratory Institute for Information Transmission Problems Bolshoy Karetny 19 Moscow 101447, GSP-4 Russia Andrei Okounkov Department of Mathematics, 970 Evans Hall University of California Berkeley, CA 94720-3840 USA Andrei Zelevinsky Department of Mathematics Northeastern University Boston, MA 02115 USA LIST OF INVITED NATO ASI PARTICIPANTS Adin, Ron Bar-Han University Anisimov, Nikita Moscow State University Azenhas, Olga University of Coimbra Bagno, Eli Bar-Han University Bazlov, Yuri Weizmann Institute Bessenrodt, Christine Universitiit Magdeburg Bills, Morgen University of Michigan Blitzstein, Joseph Stanford University Borodin, Alexei University of Pennsylvania Buch, Anders Massachusetts Institute of Technology Burman, Yuri Independent University of Moscow Carter, Roger Warwick University Chapoton, Frederic U niversite Paris 6 Chalykh, Oleg Loughborough University Chervov, Alexander Universite d' Angers Corran, Ruth University of Leicester Denham, Graham University of Oregon Diaconis, Persi Stanford University Donin, Joseph Bar-Han University Dotsenko, Vladimir Independent University of Moscow Dunkl, Charles University of Virginia Feigin, Mikhail Loughborough University Fishel, Susanna Mental Images Fomin, Sergey University of Michigan Fukuda, Kaori Kobe University Fulton, William University of Michigan Fulman, Jason Stanford University Gamburd, Alexander Mathematical Sciences Research Institute Gekhtman, Michael University of Notre Dame Gessel, Ira Brandeis University Xl XlI Grojnowski, Ian University of Cambridge Guterman, Alexander Moscow State University Haiman, Mark University of California at San Diego Hanlon, Phil University of Michigan Ion, Bogdan Princeton University Kajihara, Yasushi Kobe University Katriel, Jacob Technion Khoroshkin, Anton Independent University of Moscow Kleber, Mark Massachusetts Institute of Technology Klimyk, Anatoli Institute for Theoretical Physics Klyachko, A Bilkent University Kumar, Shrawan University of North Carolina Leclerc, Bernard Universite de Caen Leidwanger, Severine Universite Paris VII Lenart, Cristian State University of New York at Albany Loktev, Sergei Independent University of Moscow Macdonald, Ian G. QMW, University of London Maeno, Toshiaki Kyoto University Marsh, Robert University of Leicester Melnikov, Nikolai Moscow State University Miller, Ezra Massachusetts Institute of Technology Morris, Alun University of Wales Nakajima, Tatsuhiro Meikei University N oumi, Masatoshi Kobe University Oblomkov, Alexei Moscow State University Okada, Soichi Nagoya University Okounkov, Andrei University of California at Berkeley Olshanski, Grigori Dobrushin Lab, IPPI, Moscow Opdam, Eric University of Amsterdam Papadopoulo, Georges Universitiit Basel Pasquale, Angela TU- Clausthal Popovych, Stanislav Kiev Taras Shevchenko University Pragacz, Piotr Max-Planck-Institut Bonn Proskurin, Daniil Kiev Taras Shevchenko University
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