Schubert Calculus — Osaka 2012 ADVANCED STUDIES IN PURE MATHEMATICS 71 ChiefEditoroftheSeries: YoshihiroTonegawa(TokyoInstituteofTechnology) Schubert Calculus — Osaka 2012 Edited by Hiroshi Naruse (University of Yamanashi, Editor in chief) Takeshi Ikeda (Okayama University of Science) Mikiya Masuda (Osaka City University) Toshiyuki Tanisaki (Osaka City University) Mathematical Society of Japan ThisbookwastypesetbyAMS-TEXandAMS-LATEX,theTEXmacrosystemsof the American Mathematical Society, together with the style files aspm.sty and aspmfm.styforAMS-TEXwrittenbyDr.ChiakiTsukamotoandaspmproc.sty forAMS-LATEXwrittenbyDr.AkihiroMunemasaandaspm.clsforAMS-LATEX provided by Livretech Co., Ltd. TEX is a trademark of the American Mathematical Society. (cid:2)c2016 by the Mathematical Society of Japan. All rights reserved. The Mathematical Society of Japan retains the copyright of the articles in the present volume except those indicated in the footnotes. No part of this publication may be reproduced, stored in a retrieval system, or transmitted,inanyformorbyanymeans,electronic,mechanical,photocopying, recording or otherwise, without the prior permission of the copyright owner. Edited by the Mathematical Society of Japan. Published by the Mathematical Society of Japan. Distributed by the Mathematical Society of Japan, the American Mathematical Society and the World Scientific Publishing Co., Ltd. DistributedexclusivelyinNorthAmericabytheAmericanMathematicalSociety. Advanced Studies in Pure Mathematics 71 ISBN 978-4-86497-038-9 PRINTED INJAPAN byLivretechCo., Ltd. 2010 Mathematics Subject Classification. Primary 14N15. Secondary 05E05, 05E10, 20G05. Advanced Studies in Pure Mathematics Chief Editor Yoshihiro Tonegawa (Tokyo Inst. of Tech.) Editorial Board of the Series Shigeki Aida Martin Guest Kengo Hirachi (Tohoku Univ.) (Waseda Univ.) (Univ.of Tokyo) Tatsuya Koike Kayo Masuda Shin Nayatani (Kobe Univ.) (Kwansei Gakuin Univ.) (NagoyaUniv.) Takayuki Oda Keiji Oguiso Ken’ichi Ohshika (Univ. of Tokyo) (Univ. ofTokyo) (Osaka Univ.) Yoshihiro Tonegawa Akihiko Yukie (TokyoInst. of Tech.) (Kyoto Univ.) Preface In his monumental book “Kalku¨l der abz¨ahlenden Geometrie”, published in 1879, Hermann Schubert developed a remarkable sym- bolic method for solving counting problems in algebraic geometry. This centuries-old subject is still an active modern topic of study, with many connections to other areas such as representation theory and combina- torics, among others. In order to broaden its influence and expand its perspective, the fifth MSJ-SI Seasonal Institute was held on the topic of “Schubert calculus” at Osaka City University from July 17-27, 2012. This special volume of the ASPM is an outcome of this successful con- ference, and consists of sixteen refereed papers. We would like to thank all the contributing authors and the referees for making this volume possible. AlainLascoux, oneofthepioneersofmodernSchubertcalculusand a contributor to this volume, passed away during the editing process of this proceedings. We respectfully dedicate this volume to him for his great influence on this area. September 2015 Editorial Committee Hiroshi Naruse (Chief) Takeshi Ikeda Mikiya Masuda Toshiyuki Tanisaki All papers in this volume have been refereed and are in final form. No version of any of them will be submitted for publication elsewhere. This volume is dedicated to Alain Lascoux Alain Lascoux (1944-2013) AlainLascouxwasoneofthemostactiveparticipantsoftheconfer- ence on Schubert calculus in Osaka in July 2012. There, I met him for the last time (later, we exchanged a couple of e-mails). During the con- ference in Osaka, he gave a talk on tableaux and Eulerian properties of the symmetric group, making use of Ehresmann-Bruhat order and keys withapplicationstoDemazuremodulesandpostulationofSchubertsub- varietiesinflagmanifolds. Heposedmanyquestionsduringthelectures of the conference, and interacted a lot with participants, and especially with graduate students. He passed away in Paris on October 20, 2013. Much of Alain Lascoux’s research is devoted to Schubert calculus or inspired by it. Lascoux’s work concerns mainly (algebraic) combina- torics, but also algebraic geometry, representation theory and commu- tative and noncommutative algebra. In fact, he came to combinatorics fromalgebraicgeometry,andtheintuitionsfromthelatterdomain(and especially from Schubert calculus) were always present in his work. His resultswereoftenplacedonthebordersoftheaforementioneddomains. His papers on Schubert calculus concern: the coefficients of inter- sections of Schubert varieties, determinants in Chern classes, Schubert classes, i.e., the cohomology classes of Schubert subvarieties in Grass- manniansandflagmanifolds,Grassmannianextensionsofλ-rings,Chern classes of tensor products of vector bundles, classes of degeneracy loci, the degree of the dual variety to a Grassmannian, Chern classes of flag varieties, the Littlewood-Richardson rule, the spaces of complete corre- lations andquadrics, Grothendieckrings offlagvarieties, Thompolyno- mials and others. Lascoux worked for 20 years with Marcel-Paul Schu¨tzenberger on properties of the symmetric group. They wrote many articles together, andhadamajorimpactonthedevelopmentofalgebraiccombinatorics. Theysucceededinenhancingthecombinatorialunderstandingofvarious algebraic and geometric questions in representation theory. ThefavouriteobjectsandprincipaltoolsinLascoux’smathematical work were: tableaux (especially Young tableaux), symmetric functions (especially Schur functions but also noncommutative symmetric func- tions), determinants, polynomials (see below), operators on polynomial rings(especiallyvariousvariantsofdivideddifferences),reproducingker- nels,Heckealgebras,λ-ringsofGrothendieck,flagvarietiesandSchubert varieties. In algebraic geometry, he worked in enumerative geometry, enu- merative theory of singularities and on singular loci of Schubert vari- eties. In representation theory, he studied symmetric and full linear groups, Youngidempotents, Heckealgebras, crystalgraphs, q-analogues of weight multiplicities, Kostka-Foulkes polynomials, Macdonald poly- nomials and Kazhdan-Lusztig polynomials. In combinatorics, his most famous contributions are related to tableaux and plactic monoid (with Schu¨tzenberger). The Cauchy formula was the favourite one for Lascoux: because of its simplicity, profoundness and many incarnations. In the early 1980’s, Lascoux discovered with Schu¨tzenberger that the classical Schur functions are very particular cases of Schubert poly- nomials, defined by them via Newton’s divided differences. They were polynomial lifts of the Schubert classes for flag manifolds. Even more natural weredouble Schubert polynomials, whichreceivedlateratrans- parentgeometricinterpretationasthecohomologyclassesofdegeneracy loci of morphisms between vector bundles from two flags (a result of Fulton). OperatorswereoftenpresentinLascoux’swork. Apartfromdivided differences, used in the theory of Schubert polynomials and their vari- ations (e.g. isobaric divided differences in the theory of Grothendieck polynomials),healsousedvertexoperators. Iknowthatinthelastyears of his life, Lascoux worked on his monograph ”Polynomials”, where all these ideas and results about operators and polynomials are developed. As far as I know, the book just requires a finishing touch. Let me finish with some personal reminiscences. I met Alain for the first time in Spring 1978 during his first visit to Poland. He taught us the syzygies of determinantal varieties, using the Bott theorem on cohomology of homogeneous bundles and Schur functors. In November andDecember1978,IvisitedhiminParis,wherehetaughtmeSchubert calculus and symmetric functions. Then, I visited him in Paris many times(asheusedtosay: “trentesixfois”). Hewasmyprincipalteacher in mathematics. Between 1980 and 2010, we wrote together 15 papers, and most of them were devoted to or inspired by Schubert calculus. Also, many other of my papers are inspired by his papers. Alain Lascoux was a master of Schubert calculus. He knew its clas- sical sources, and had a deep understanding of its importance and ap- plications to contemporary mathematics. Piotr Pragacz, December 2014 CONTENTS Hiraku Abe and Sara Billey — Consequences of the Lakshmibai- Sandhyatheorem: theubiquityofpermutationpatternsinSchubert calculus and related geometry 1 Piotr Achinger and Nicholas Perrin — Spherical multiple flags 53 Peter Fiebig — Moment graphs in representation theory and ge- ometry 75 Takeshi Ikeda — Lectures on equivariant Schubert polynomials 97 Bumsig Kim — Stable quasimaps to holomorphic symplectic quo- tients 139 Valentina Kiritchenko — Divided difference operators on poly- topes 161 Allen Knutson — Schubert calculus and puzzles 185 ThomasLam—Whittakerfunctions,geometriccrystals,andquan- tum Schubert calculus 211 Alain Lascoux — Tableaux and Eulerian properties of the sym- metric group 251 CristianLenart,SatoshiNaito,DaisukeSagaki,AnneSchilling andMarkShimozono—QuantumLakshmibai-Seshadripaths and root operators 267 AbrahamMart´ın del CampoandFrankSottile—Experimen- tation in the Schubert Calculus 295 MasakiNakagawaandHiroshiNaruse—Generalized(co)homology oftheloopspacesofclassicalgroupsandtheuniversalfactorial Schur P- and Q-functions 337 Piotr Pragacz — Positivity of Thom polynomials and Schubert calculus 419 Toshiaki Shoji — Character sheaves on exotic symmetric spaces and Kostka polynomials 453 HughThomasandAlexanderYong—Cominusculetableaucom- binatorics 475 Julianna Tymoczko — Billey’s formula in combinatorics, geome- try, and topology 499