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Springer Proceedings in Mathematics & Statistics Jianxun Hu Changzheng Li Leonardo C. Mihalcea Editors Schubert Calculus and Its Applications in Combinatorics and Representation Theory Guangzhou, China, November 2017 Springer Proceedings in Mathematics & Statistics Volume 332 Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. More information about this series at http://www.springer.com/series/10533 Jianxun Hu • Changzheng Li • Leonardo C. Mihalcea Editors Schubert Calculus and Its Applications in Combinatorics and Representation Theory Guangzhou, China, November 2017 123 Editors Jianxun Hu Changzheng Li School of Mathematics School of Mathematics Sun Yat-sen University Sun Yat-sen University Guangzhou, Guangdong, China Guangzhou, Guangdong, China Leonardo C. Mihalcea Department of Mathematics Virginia Tech Blacksburg, VA, USA ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-981-15-7450-4 ISBN 978-981-15-7451-1 (eBook) https://doi.org/10.1007/978-981-15-7451-1 Mathematics Subject Classification: 14N15, 14M15, 53D45, 17B45, 37K10 © Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Preface With roots in enumerative geometry and Hilbert’s 15th problem, modern Schubert Calculus studies classical and quantum intersection rings on spaces with symme- tries, such as flag manifolds. The presence of symmetries leads to particularly rich structures, and it connects Schubert Calculus to many branches of mathematics, including algebraic geometry, combinatorics, representation theory, and theoretical physics. For instance, the study of the quantum cohomology ring of a Grassmann manifold combines all these areas in an organic way. The current volume show- cases some of the newest developments in the subject, as presented at the “International Festival in Schubert Calculus”, a conference held at Sun Yat-sen University in Guangzhou, China, during November 6–10, 2017. The event included a 1-day mini-school and a 4-day international conference entitled “Trends in Schubert Calculus”. There were over 80 participants, more than one half of which were international, from countries such as Australia, France, Germany, India, Japan, Korea, Poland, Russia, United Kingdom, and the U.S.A. This event continued the tradition of conferences in Schubert Calculus with large international participation; there were three such conferences in the past decade (Toronto 2010, Osaka 2012 and Bedlewo 2015). The current volume includes 12 papers authored by some of the speakers, covering a large array of topics, including several high-quality surveys. Each of the papers was refereed by two anonymous experts in the field. This volume could not have existed without the combined efforts of the authors and referees, and we are grateful for everyone’s contribution. Problems with roots in classical Schubert Calculus attracted significant attention. The factorial Grothendieck polynomials, which investigate polynomials repre- senting Schubert classes in K theory, were discussed in the paper by Matsumura and Sugimoto. The related problem of finding formulas for cohomology classes of various degeneracy loci is addressed in a paper by Darondeau and Pragacz. Yet another problem with roots in Schubert’s classical work, that of finding formulas for the order of contact between manifolds, is investigated in the paper by Domitrz, Mormul, and Pragacz. Finally, Duan and Zhao address and survey Schubert’s v vi Preface classical problem of characteristics, and find presentations for the integral coho- mology rings of flag manifolds, including those of exceptional Lie types. There are rich connections between Schubert Calculus and the combinatorics of symmetric functions and polynomials. A survey by Pechenik and Searles highlights the properties of some of the most important bases of polynomials relevant for geometry. Expanding on this, a topic of current high interest is to relate and apply Schubert Calculus methods to problems in (combinatorial and geometric) repre- sentation theory. Three papers in the volume address such connections: Anderson and Nigro investigate the geometric Satake correspondence in relation to minuscule Schubert Calculus; McGlade, Ram, and Yang wrote a survey on the combinatorics and geometry of integrable representations of quantum affine algebras with a particular focus on level 0; this is motivated by Schubert Calculus on semi-infinite flag manifolds. Finally, Su and Zhong wrote a survey showcasing applications of Maulik and Okounkov’s theory of stable envelopes on the cotangent bundle of a flag variety to various problems in geometry and representation theory. This recent direction, which one may call “Cotangent Schubert Calculus”, is closely related to the study of characteristic classes of singular varieties; from this viewpoint, it is studied by Fehér, Rimányi, and Weber. Methods and questions from Schubert Calculus can be adapted to varieties related to flag manifolds or to generalizations of the cohomology ring. A survey by Abe and Horiguchi is investigating the properties of the cohomology rings of Hessenberg varieties; these generalize the usual flag varieties, the Peterson variety, and the Springer fibre. In another direction, Hudson, Matsumura, and Perrin address the problem of defining stable Bott-Samelson classes in the algebraic cobordism; this is closely related to the outstanding problem of defining Schubert classes in more general oriented cohomology theories. Finally, a paper by Kim, Oh, Ueda, and Yoshida gives an expository account of quasimap theory, and proves a generalization of toric residue mirror symmetry to Grassmannians. These papers provide a broad overview of current interests in Schubert Calculus and related areas. We would like to thank again all the anonymous referees for their invaluable help, and the Springer editorial staff for the assistance with various technical parts. Finally, we are grateful to Sun Yat-sen University for generously providing funds for this conference and to Springer Nature for providing us the opportunity to publish these papers. All papers in this volume have been refereed and are in a final form. No version of any of them will be submitted for publication elsewhere. Guangzhou, China Jianxun Hu Guangzhou, China Changzheng Li Blacksburg, USA Leonardo C. Mihalcea May 2020 Contents Factorial Flagged Grothendieck Polynomials ..................... 1 Tomoo Matsumura and Shogo Sugimoto Flag Bundles, Segre Polynomials, and Push-Forwards .............. 17 Lionel Darondeau and Piotr Pragacz Order of Tangency Between Manifolds ......................... 27 Wojciech Domitrz, Piotr Mormul, and Piotr Pragacz On Schubert’s Problem of Characteristics ....................... 43 Haibao Duan and Xuezhi Zhao Asymmetric Function Theory ................................. 73 Oliver Pechenik and Dominic Searles Minuscule Schubert Calculus and the Geometric Satake Correspondence ........................................... 113 Dave Anderson and Antonio Nigro Positive Level, Negative Level and Level Zero .................... 153 Finn McGlade, Arun Ram, and Yaping Yang Stable Bases of the Springer Resolution and Representation Theory .................................................. 195 Changjian Su and Changlong Zhong Characteristic Classes of Orbit Stratifications, the Axiomatic Approach .................................... 223 László M. Fehér, Richárd Rimányi, and Andrzej Weber A Survey of Recent Developments on Hessenberg Varieties .......... 251 Hiraku Abe and Tatsuya Horiguchi vii viii
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