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RICHARD P. STANLEY Patricia Hersh THE MATHEMATICAL LEGACY OF RICHARD P. STANLEY Patricia Hersh . Thomas Lam . Pavlo Pylyavskyy . Victor Reiner . Editors 4321 3421 4312 3412 4231 3241 4132 2431 2341 3142 4213 4123 3214 3124 2413 1432 1342 2314 1423 1324 2143 2134 1243 1234 φ THE MATHEMATICAL LEGACY OF RICHARD P. STANLEY https://doi.org/10.1090//mbk/100 THE MATHEMATICAL LEGACY OF RICHARD P. STANLEY Patricia Hersh . Thomas Lam . Pavlo Pylyavskyy . Victor Reiner . Editors AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 2010 Mathematics Subject Classification. Primary 05A15, 05B35, 05Exx, 06A07, 06A11, 13F55, 52B05, 52B20, 52B22, 52C35. For additional information and updates on this book, visit www.ams.org/bookpages/mbk-100 The photo of Richard P. Stanley is courtesy of Thomas Lam. Library of Congress Cataloging-in-Publication Data Names: Hersh, Patricia, 1973– editor. | Lam, Thomas, 1980– editor. | Pylyavskyy, Pavlo, 1982– editor. | Reiner, Victor, 1965– editor. Title: The mathematical legacy of Richard P. Stanley / Patricia Hersh, Thomas Lam, Pavlo Pylyavskyy, Victor Reiner, editors. Description: Providence, Rhode Island : American Mathematical Society, 2016. Identifiers: LCCN 2016004438 | ISBN 9781470427245 (alk. paper) Subjects: LCSH: Stanley, Richard P., 1944– | Mathematicians–United States–Biography. | Combinatorial analysis. | AMS: Combinatorics – Enumerative combinatorics – Exact enumera- tion problems, generating functions. msc | Combinatorics – Designs and configurations – Matroids, geometric lattices. msc | Combinatorics – Algebraic combinatorics – Algebraic combi- natorics. msc | Order, lattices, ordered algebraic structures – Ordered sets – Combinatorics of partially ordered sets. msc | Order, lattices, ordered algebraic structures – Ordered sets – Alge- braic aspects of posets. msc | Commutative algebra – Arithmetic rings and other special rings – Stanley-Reisner face rings; simplicial complexes. msc | Convex and discrete geometry – Poly- topes and polyhedra – Combinatorial properties (number of faces, shortest paths, etc.). msc | Convex and discrete geometry – Polytopes and polyhedra – Lattice polytopes (including rela- tions with commutative algebra and algebraic geometry). msc | Convex and discrete geom- etry – Polytopes and polyhedra – Shellability. msc | Convex and discrete geometry – Discrete geometry – Arrangements of points, flats, hyperplanes. msc Classification: LCC QA29.S6735 M38 2016 | DDC 511/.6–dc23 LC record available at http://lccn.loc.gov/2016004438 Color graphic policy. Any graphics created in color will be rendered in grayscale for the printed version unless color printing is authorized by the Publisher. In general, color graphics will appear in color in the online version. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2016 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 212019181716 Richard P. Stanley Contents Preface ix Richard Stanley’s Short Curriculum Vitae xi Publications, with commentary by the author Richard P. Stanley 1 A survey of subdivisions and local h-vectors Christos A. Athanasiadis 39 Stanley’s major contributions to Ehrhart theory Matthias Beck 53 “Even more intriguing, if rather less plausible...” Face numbers of convex polytopes Louis J. Billera 65 The contributions of Stanley to the fabric of symmetric and quasisymmetric functions Sara C. Billey and Peter R. W. McNamara 83 “Let Δ be a Cohen-Macaulay complex ...” Anders Bjorner¨ 105 Stanley’s work on unimodality Francesco Brenti 119 Five stories for Richard Persi Diaconis 131 Some new applications of the Stanley-Macdonald Pieri rules Adriano Garsia, Jim Haglund, Guoce Xin, and Mike Zabrocki 141 A historical survey on P -partitions Ira M. Gessel 169 Transitive factorizations of permutations and geometry I. P. Goulden and D. M. Jackson 189 Stanley’s influence on monomial ideals Takayuki Hibi 203 Cohen-Macaulay varieties, geometric complexes, and combinatorics Melvin Hochster 219 vii viii CONTENTS Plane partitions in the work of Richard Stanley and his school C. Krattenthaler 231 Combinatorial representation theory of Lie algebras. Richard Stanley’s work and the way it was continued Cristian Lenart 263 Lessons I learned from Richard Stanley James Propp 279 Richard Stanley through a crystal lens and from a random angle Anne Schilling 287 From poset topology to q-Eulerian polynomials to Stanley’s chromatic symmetric functions John Shareshian and Michelle L. Wachs 301 Stanley character polynomials Piotr Sniady´ 323 Some problems arising from partition poset homology Sheila Sundaram 335 Preface Richard Stanley has had a profound impact on combinatorics. We hope that this book gives readers an opportunity to learn some of the mathematics that he has touched. Stanley and his PhD advisor, Gian-Carlo Rota, were at the vanguard in trans- forming combinatorics from a disparate collection of tricks into an organized and mature area of modern mathematics. In particular, Stanley’s talent for discovering deep examples captured the attention of the mathematical community at large. This book is a recollection of this journey, putting work of the past half-century within the context of the current mathematical scene. ——————– Stanley graduated with a B.S. from the California Institute of Technology in 1966. He then became a graduate student at Harvard University, where he worked with Gian-Carlo-Rota (who was a professor at MIT). In 1971, Stanley graduated with a Ph.D. in combinatorics, entitled Ordered Structures and Partitions;the survey by Gessel in this volume discusses this work. Stanley has supervised 59 doctoral students (56 at MIT, 3 at Harvard) and mentored countless postdocs and visitors. Following this preface, we include Stanley’s abbreviated curriculum vitae and a list of his doctoral students. A hallmark of Stanley’s work has been importation of ideas born outside com- binatorics to crack combinatorial problems. Examples include: • The introduction of tools from commutative algebra (local cohomology, the Cohen-Macaulay property, canonical modules, Stanley-Reisner rings, affine semigroup rings, and invariant rings) in the enumerative theory of - face numbers of simplicial complexes, polytopes, spheres, - solutions to linear homogeneous Diophantine equations, - partition analysis. • Application of the hard Lefschetz Theorem in algebraic geometry, along with representation theory of sl2(C) and of finite groups, to questions of unimodality and Sperner theory of posets, • Application of the Aleksandrov-Fenchel inequalities from convexity to log- concavity questions. • Application of symmetric function theory to partition identities, permu- tation statistics, and enumeration of reduced decompositions. A wealth of combinatorics viewed through his distinctive lens appears in his books, which are paragons of clarity, organization and elegance: • Combinatorics and Commutative Algebra. ix xPREFACE • Enumerative Combinatorics, Volumes 1 and 2 (fondly referred to1 as “EC1” and “EC2”). • Algebraic Combinatorics. • Catalan Numbers. His books do not tell the full story, however. We hope that the surveys within this volume help to round out this picture. ——————– This book grew, in a sense, from two conferences held in Stanley’s honor. On June 22-26, 2004, a conference titled2 “A children’s party” was held at the Massachusetts Institute of Technology in honor of his 60th birthday. A special Stanley Festschrift Volume of The Electronic Journal of Combinatorics (Vol. 11, Issue 2, edited by Bruce Sagan) published forty papers in his honor. A decade later, on June 23-27, 2014, a conference titled “Stanley@70” was held, also at MIT, in honor of Stanley’s 70th birthday. For example, the articles of Billera and Bj¨orner in this volume were developed in conjunction with the preparation of their historical talks given at this conference. Rather than collecting research monographs, in preparing this volume we so- licited survey papers by researchers with a variety of perspectives on Stanley’s work. He also kindly accepted our invitation to contribute a short reflection on each of his own papers, for which we are extremely grateful. Within it one will find several examples of
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