Symmetric Functions 2001: Surveys of Developments and Perspectives

281 Pages·2002·12.403 MB·English

by  Mark Haiman (auth.), Sergey Fomin (eds.)

Symmetric Functions 2001:
Surveys of Developments and Perspectives
NATO Science Series
A Series presenting the results of scientific meetings supported under the NATO Science
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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
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recording or otherwise, without written permission from the Publisher, with the exception
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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