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Symmetric Functions and Hall Polynomials OXFORD EliV"1'III 11V1 ICAI. AIU\OGR;APIIs Symmetric Functions and Hall Polynomials SECOND EDITION I. G. MACDONALD OXFORD SCIENCE PUBLICATIONS OXFORD MATHEMATICAL MONOGRAPHS Series Editors J. M. BallE. M. FriedlanderI. G. Macdonald L. NirenbergR. PenroseJ. T. Stuart OXFORD MATHEMATICAL MONOGRAPHS A. Belleni-Moranti: Applied semigroups and evolution equations A. M. Arthurs: Complementary variational principles 2nd edition M. Rosenblum and J. Rovnyak: Hardy classes and operator theory J. W. P. Hirschfeld: Finite projective spaces of three dimensions A. Pressley and G. Segal: Loop groups D. E. Edmunds and W. D. Evans: Spectral theory and differential operators Wang Jianhua: The theory of games S. Omatu and J. H. Seinfeld: Distributed parameter systems: theory and applications J. Hilgert, K. H. Hofmann, and J. D. Lawson: Lie groups, convex cones, and semigroups S. Dineen: The Schwarz lemma S. K. Donaldson and P. B. Kronheimer The geometry offour-manifolds D. W. Robinson: Elliptic operators and Lie groups A. G. Werschulz: The computational complexity of differential and integral equations L. Evens: Cohomology of groups G. Effinger and D. R. Hayes: Additive number theory of polynomials J. W. P. Hirschfeld and J. A. Thas: General Galois geometries P. N. Hoffman and J. F. Humphreys: Projective representations of the symmetric groups 1. GyOri and G. Ladas: The oscillation theory of delay differential equations J. Heinonen, T. Kilpelainen, and O. Martio: Non-linear potential theory B. Amberg, S. Franciosi, and F. de Giovanni: Products of groups M. E. Gurtin: Thermomechanics of evolving phase boundaries in the plane I. lonescu and M. F. Sofonea: Functional and numerical methods in viscoplasticity N. Woodhouse: Geometric quantization 2nd edition U. Grenander: General pattern theory J. Faraut and A. Koranyi: Analysis on symmetric cones 1. G. Macdonald: Symmetric functions and Hall polynomials 2nd edition B. L. R. Shawyer and B. B. Watson: Borel's methods of summability D. McDuff and D. Salamon: Introduction to symplectic topology M. Holschneider: Wavelets: an analysis tool Jacques Thdvenaz: G-algebras and modular representation theory Haas-Joachim Baues: Homotopy type and homology P. D. D'Eati: Black holes: gravitational interactions R. Lowen: Approach spaces: the missing link in the topology-uniformity-metric triad Nguyen Dinh Cong: Topological dynamics of random dynamical systems J. W. P. Hirschfeld: Projective geometries over finite fields 2nd edition K. Matsuzaki and M. Taniguchi: Hyperbolic manifolds and Kleinian groups David E. Evans and Yasuyuki Kawahigashi: Quantum symmetries on operator algebras Norbert Klingen: Arithmetical similarities: prime decomposition and finite group theory Isabelle Catto, Claude Le Bris, and Pierre-Louis Lions: The mathematical theory of thermodynamic limits: Thomas-Fermi type models Symmetric Functions and Hall Polynomials Second Edition I. G. MACDONALD Queen Mary and Westfield College University of London Oxford University Press, Walton Street, Oxford OX2 6DP Oxford New York Athens Auckland Bangkok Bombay CalcuttaCape Town Dares SalaamDelhi Florence Hong Kong IstanbulKarachi KualaLumpur Madras Madrid Melbourne Mexico CityNairobiParisSingapore TaipeiTokyoToronto and associated companies in BerlinIbadan Oxford is a trade mark of Oxford University Press Published in the United States by Oxford University Press Inc., New York ©I. G. Macdonald,1979, 1995 First edition published 1979 Second edition published 1995 Reprinted 1995, First printed in paperback, 1998 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press. Within the UI, exceptions are allowed in respect of any fair dealing for the purpose of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms and in other countries should be sent to the Rights Department, Oxford University Press, at the address above. This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data Macdonald, I. G. (Ian Grant) Symmetric functions and Hall polynomials / I. G. Macdonald. - 2nd ed. (Oxford mathematical monographs) Includes bibliographical references and index. 1. Abelian groups.2. Finite groups.3. Hall polynomials. 4. Symmetric functions.L Title.11. Series. QA180.M33 1995 512'.2-dc2O 94-27392CIP ISBN 0 19 853489 2 h/b 0 19 850450 0 p/b Typeset by Technical Typesetting Ireland Printed in Great Britain by Bookcraft (Bath) Ltd., MidsomerNorton, Avon Preface to the second edition The first edition of this book was translated into Russian by A. Zelevinsky in 1984, and for the Russian version both the translator and the author furnished additional material, both text and examples. Thus the original purpose of this second edition was to make this additional material accessible to Western readers. However, in the intervening years other developments in this area of mathematics have occurred, some of which I have attempted to take account of: the result, I am afraid, is a much longer book than its predecessor. Much of this extra bulk is due to two new chapters (VI and VII) about which I shall say something below. For readers acquainted with the first edition, it may be of use to indicate briefly the main additions and new features of this second edition. The text of Chapter I remains largely unchanged, except for a discussion of transi- tion matrices involving the power-sums in §6, and of the internal (or inner) product of symmetric functions in §7. On the other hand, there are more examples at the ends of the various sections than there were before. To the appendix on polynomial functors I have added an account of the related theory of polynomial representations of the general linear groups (always in characteristic zero), partly for its own sake and partly with the aim of rendering the account of zonal polynomials in Chapter VII self- contained. I have also included, as Appendix B to Chapter I, an account, following Specht's thesis, of the characters of wreath products G - S (G any finite group), along the same lines as the account of the characters of the symmetric groups in Chapter I, §7: this may serve the reader as a sort of preparation for the more difficult Chapter IV on the characters of the finite general linear groups. In Chapter II, one new feature is that the formula for the Hall polynomial (or, more precisely, for the polynomial gs(t) (4.1)) is now made completely explicit in (4.11). The chapter is also enhanced by the appendix, written by A. Zelevinsky for the Russian edition. The main addition to Chapter III is a section (§8) on Schur's Q- functions, which are the case t= -1 of the Hall-Littlewood symmetric functions. In this context I have stopped short of Schur's theory of the projective representations of the symmetric groups, for which he intro- duced these symmetric functions, since (a) there are now several recent accounts of this theory available, among them the monograph of P. Hoffman and J. F. Humphreys in this series, and (b) this book is already long enough. vi PREFACE TO THE SECOND EDITION Chapters IV and V are unchanged, and require no comment. Chapter VI is new, and contains an extended account of a family of symmetric functions P,(x; q, t), indexed as usual by partitionsA, and depending rationally on two parameters q and t. These symmetric func- tions include as particular cases many of those encountered earlier in the book: for example, when q = 0 they are the Hall-Littlewood functions of Chapter III, and when q = t they are the Schur functions of Chapter I. They also include, as a limiting case, Jack's symmetric functions depending on a parameter a. Many of the properties of the Schur functions general- ize to these two-parameter symmetric functions, but the proofs (at present) are usually more elaborate. Finally, Chapter VII (which was originally intended as an appendix to Chapter VI, but outgrew that format) is devoted to a study of the zonal polynomials, long familiar to statisticians. From one point of view, they are a special case of Jack's symmetric functions (the parameter a being equal to 2), but their combinatorial and group-theoretic connections make them worthy of study in their own right. This chapter can be read independently of Chapter VI. London, 1995 I. G. M. Preface to the first edition This monograph is the belated fulfilment of an undertaking made some years ago to publish a self-contained account of Hall polynomials and related topics. These polynomials were defined by Philip Hall in the 1950s, originally as follows. If M is a finite abelian p-group, itis a direct sum of cyclic subgroups, of orders p Al, p A2, ... , pAr say, where we may suppose that 'A1 > A2 >' ... % ArThe sequence of exponents A = A,) is a parti- tion, called the type of M, which describesMup to isomorphism. If now p, and v are partitions, let gµ,(p) denote the number of subgroups N of M such that N has type µ and M/N has type v. Hall showed that gµ (p) is a polynomial function of p, with integer coefficients, and was able to determine its degree and leading coefficient. These polynomials are the Hall polynomials.
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