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The Symmetric Group. Let G Be a Group Written Multiplicatively with Identity E Graduate Texts in Mathematics 203 Editorial Board S. Axler F.W. Gehring K.A. Ribet Springer Science+Business Media, LLC Graduate Texts in Mathematics T AKEUTI/ZARING. Introduction to 35 ALEXANDERIWERMER. Several Complex Axiomatic Set Theory. 2nd ed. Variables and Banach Algebras. 3rd ed. 2 OXTOBY. Measure and Category. 2nd ed. 36 KELLEYINAMIOKA et al. Linear 3 SCHAEFER. Topological Vector Spaces. Topological Spaces. 2nd ed. 37 MoNK. Mathematical Logic. 4 HILTON/STAMMBACH. A Course in 38 GRAUERT/FRITZSCHE. Several Complex Homological Algebra. 2nd ed. Variables. 5 MAC LANE. Categories for the Working 39 ARVESON. An Invitation to C*-Algebras. Mathematician. 2nd ed. 40 KEMENY/SNELL/KNAPP. Denumerable 6 HUGHES/PIPER. Projective Planes. Markov Chains. 2nd ed. 7 SERRE. A Course in Arithmetic. 41 APOSTOL. Modular Functions and Dirichlet 8 TAKEUTIIZARING. Axiomatic Set Theory. Series in Number Theory. 9 HUMPHREYS. Introduction to Lie Algebras 2nd ed. and Representation Theory. 42 SERRE. Linear Representations of Finite 10 CoHEN. A Course in Simple Homotopy Groups. Theory. 43 GILLMANIJERISON. Rings of Continuous II CONWAY. Functions of One Complex Functions. Variable I. 2nd ed. 44 KENDIG. Elementary Algebraic Geometry. 12 BEALS. Advanced Mathematical Analysis. 45 LOEVE. Probability Theory !. 4th ed. 13 ANDERSON/FULLER. Rings and Categories 46 LOEVE. Probability Theory II. 4th ed. of Modules. 2nd ed. 47 MOISE. Geometric Topology in 14 GoLUBITSKY/GuiLLEMIN. Stable Mappings Dimensions 2 and 3. and Their Singularities. 48 SACHS/Wu. General Relativity for 15 BERBERIAN. Lectures in Functional Mathematicians. Analysis and Operator Theory. 49 GRUENBERG/WEIR. Linear Geometry. 16 WINTER. The Structure of Fields. 2nd ed. 17 RosENBLATT. Random Processes. 2nd ed. 50 EDWARDS. Fermat's Last Theorem. 18 HALMOS. Measure Theory. 51 KLINGENBERG. A Course in Differential 19 HALMOS. A Hilbert Space Problem Book. Geometry. 2nd ed. 52 HARTSHORNE. Algebraic Geometry. 20 HUSEMOLLER. Fibre Bundles. 3rd ed. 53 MANIN. A Course in Mathematical Logic. 21 HUMPHREYS. Linear Algebraic Groups. 54 ORA VERIW ATKINS. Combinatorics with 22 BARNES/MACK. An Algebraic Introduction Emphasis on the Theory of Graphs. to Mathematical Logic. 55 BROWN/PEARCY. Introduction to Operator 23 GREUB. Linear Algebra. 4th ed. Theory I: Elements of Functional 24 HOLMES. Geometric Functional Analysis Analysis. and Its Applications. 56 MASSEY. Algebraic Topology: An 25 HEWITT/STROMBERG. Real and Abstract Introduction. Analysis. 57 CROWELL/Fox. Introduction to Knot 26 MANES. Algebraic Theories. Theory. 27 KELLEY. General Topology. 58 KoBLITZ. p-adic Numbers, p-adic Analysis, 28 ZARISKIISAMUEL. Commutative Algebra. and Zeta-Functions. 2nd ed. Vol.!. 59 LANG. Cyclotomic Fields. 29 ZARISKIISAMUEL. Commutative Algebra. 60 ARNOLD. Mathematical Methods in Vol.!!. Classical Mechanics. 2nd ed. 30 JACOBSON. Lectures in Abstract Algebra I. 61 WHITEHEAD. Elements of Homotopy Basic Concepts. Theory. 31 JACOBSON. Lectures in Abstract Algebra II. 62 KARGAPOLOV/MERLZJAKOV. Fundamentals Linear Algebra. of the Theory of Groups. 32 JACOBSON. Lectures in Abstract Algebra 63 BOLLOBAS. Graph Theory. Ill. Theory of Fields and Galois Theory. 64 EDWARDS. Fourier Series. Vol. I. 2nd ed. 33 HIRSCH. Differential Topology. 65 WELLS. Differential Analysis on Complex 34 SPITZER. Principles of Random Walk. Manifolds. 2nd ed. 2nd ed. (continued after index) Bruce E. Sagan The Symmetric Group Representations, Combinatorial Algorithms, and Symmetric Functions Second Edition With 31 Figures Springer Bruce E. Sagan Department of Mathematics Michigan State University East Lansing, MI 48824-1027 USA Editorial Board S. Axler F.W. Gehring K.A. Ribet Mathematics Department Mathematics Department Mathematics Department San Francisco State East Hall University of California University University of Michigan at Berkeley San Francisco, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 94720-3840 USA USA USA Mathematics Subject Classification (2000): 20Cxx, 20C30, 20C32, 05C85 Library of Congress Cataloging-in-Publication Data Sagan, Bruce Eli. The symmetric group : representations, combinatorial algorithms, and symmetric functions I Bruce E. Sagan. p. em. - (Graduate texts in mathematics ; 203) Includes bibliographical references and index. ISBN 978-1-4419-2869-6 ISBN 978-1-4757-6804-6 (eBook) DOI 10.1007/978-1-4757-6804-6 I. Representations of groups. 2. Symmetric functions. I. Title. II. Series. QA171 .S24 2000 512 .. 2-dc21 00-040042 Printed on acid-free paper. First edition© 1991 by Wadsworth, Inc., Belmont, California. © 2001 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 2001 Softcover reprint of the hardcover 2nd edition 2001 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC , except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of gl!neral descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Timothy Taylor; manufacturing supervised by Jeffrey Taub. Photocomposed copy prepared from the author's LaTeX files. 9 8 7 6 5 4 3 2 1 ISBN 978-1-4419-2869-6 SPIN 10770720 Preface to the 2nd Edition I have been very gratified by the response to the first edition, which has resulted in it being sold out. This put some pressure on me to come out with a second edition and now, finally, here it is. The original text has stayed much the same, the major change being in the treatment of the hook formula which is now based on the beautiful Novelli-Pak-Stoyanovskii bijection (NPS 97]. I have also added a chapter on applications of the material from the first edition. This includes Stanley's theory of differential posets (Stn 88, Stn 90] and Fomin's related concept of growths (Fom 86, Fom 94, Fom 95], which extends some of the combinatorics of Sn-representations. Next come a couple of sections showing how groups acting on posets give rise to interesting representations that can be used to prove unimodality results (Stn 82]. Finally, we discuss Stanley's symmetric function analogue of the chromatic polynomial of a graph (Stn 95, Stn ta]. I would like to thank all the people, too numerous to mention, who pointed out typos in the first edition. My computer has been severely reprimanded for making them. Thanks also go to Christian Krattenthaler, Tom Roby, and Richard Stanley, all of whom read portions of the new material and gave me their comments. Finally, I would like to give my heartfelt thanks to my editor at Springer, Ina Lindemann, who has been very supportive and helpful through various difficult times. Ann Arbor, Michigan, 2000 v Preface to the 1st Edition In recent years there has been a resurgence of interest in representations of symmetric groups (as well as other Coxeter groups). This topic can be approached from three directions: by applying results from the general theory of group representations, by employing combinatorial techniques, or by using symmetric functions. The fact that this area is the confluence of several strains of mathematics makes it an exciting one in which to study and work. By the same token, it is more difficult to master. The purpose of this monograph is to bring together, for the first time under one cover, many of the important results in this field. To make the work accessible to the widest possible audience, a minimal amount of prior knowledge is assumed. The only prerequisites are a familiarity with elemen­ tary group theory and linear algebra. All other results about representations, combinatorics, and symmetric functions are developed as they are needed. Hence this book could be read by a graduate student or even a very bright undergraduate. For researchers I have also included topics from recent journal articles and even material that has not yet been published. Chapter 1 is an introduction to group representations, with special empha­ sis on the methods of use when working with the symmetric groups. Because of space limitations, many important topics that are not germane to the rest of the development are not covered. These subjects can be found in any of the standard texts on representation theory. In Chapter 2, the results from the previous chapter are applied to the symmetric group itself, and more highly specialized machinery is developed to handle this case. I have chosen to take the elegant approach afforded by the Specht modules rather than working with idempotents in the group algebra. The third chapter focuses on combinatorics. It starts with the two famous formulae for the dimensions of the Specht modules: the Frame-Robinson­ Thrall hook formula and the Frobenius-Young determinantal formula. The centerpiece is the Robinson-Schensted-Knuth algorithm, which allows us to describe some of the earlier theorems in purely combinatorial terms. A thor­ ough discussion of Schiitzenberger's jeu de taquin and related matters is in­ cluded. Chapter 4 recasts much of the previous work in the language of symmet­ ric functions. Schur functions are introduced, first combinatorially as the generating functions for semistandard tableaux and then in terms of sym­ metric group characters. The chapter concludes with the famous Littlewood­ Richardson and Murnaghan-Nakayama rules. My debt to several other books will be evident. Much of Chapter 1 is based on Ledermann's exquisite text on group characters [Led 77]. Chapter Vll viii PREFACE TO THE 1ST EDITION 2 borrows heavily from the monograph of James [Jam 78], whereas Chapter 4 is inspired by Macdonald's already classic book [Mac 79]. Finally, the third chapter is a synthesis of material from the research literature.
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