Association for Women in Mathematics Series Hélène Barcelo Gizem Karaali Rosa Orellana Editors Recent Trends in Algebraic Combinatorics Association for Women in Mathematics Series Volume 16 Series editor Kristin Lauter Microsoft Research Redmond, WA, USA Association for Women in Mathematics Series Focusing on the groundbreaking work of women in mathematics past, present, and future, Springer’s Association for Women in Mathematics Series presents the latest research and proceedings of conferences worldwide organized by the Association for Women in Mathematics (AWM). All works are peer-reviewed to meet the highest standards of scientific literature, while presenting topics at the cutting edge of pure and applied mathematics. 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More information about this series at http://www.springer.com/series/13764 Hélène Barcelo • Gizem Karaali Rosa Orellana Editors Recent Trends in Algebraic Combinatorics 123 Editors Hélène Barcelo Rosa Orellana Mathematical Sciences Research Institute Department of Mathematics Berkeley, CA, USA Dartmouth College Hanover, NH, USA Gizem Karaali Department of Mathematics Pomona College Claremont, CA, USA ISSN 2364-5733 ISSN 2364-5741 (electronic) Association for Women in Mathematics Series ISBN 978-3-030-05140-2 ISBN 978-3-030-05141-9 (eBook) https://doi.org/10.1007/978-3-030-05141-9 Library of Congress Control Number: 2018963041 Mathematics Subject Classification (2010): 05Exx, 16-xx, 20-xx © The Author(s) and the Association for Women in Mathematics 2019 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, express 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface This book is a collection of survey articles on some of the exciting recent devel- opments in algebraic combinatorics. It also contains a tutorial on Schubert calculus, as we felt that such an article would constitute an important addition to the literature. Algebraic combinatorics is a vast area of research, and attempting to be exhaustive in our coverage would have been a mistake. Rather (after reaching out to many experts in the field for advice), we chose a small number of topics that are currently enjoying broad interest and rapid growth. And after many thoughtful discussions, we sought out the authors of this volume—several writers known for their expository skills as well as a few junior researchers with active research programs who were willing to share their knowledge. Writing a good survey is a deceptively tricky endeavor, as decisions on what to include (or not) and how to best present it lurk at the corner of every sentence! Nonetheless, our contributors did a remarkable job of introducing exciting directions of current research in algebraic combinatorics from its foundational questions to the boundaries of what is known today. More specifically, the book contains four surveys focusing on representation theory, symmetric functions, invariant theory, and the combinatorics of Young tableaux. The other five surveys address subjects at the intersection of algebra, combinatorics, and geometry: the study of polytopes, lattice points, hyperplane arrangements, crystal graphs, and Grassmannians. The surveys are written at an introductory level that emphasizes recent developments and open problems. The tutorial on Schubert calculus is written in an interactive way and is intended as a guide for combinatorialists wishing to understand and appreciate the geometric and topological aspects of Schubert calculus, as well as for geometric-minded researchers seeking to gain familiarity with the relevant combinatorial tools in this area. Each article in this volume was reviewed independently by two referees, and we are simply amazed by, and deeply grateful for, the generosity of those referees. Their care in reviewing the articles and their constructive and judicious suggestions have been invaluable. The authors of this volume wish to warmly thank the referees v vi Preface for their contributions, and we three editors in turn offer our heartfelt thanks to both the authors and the referees. We are very proud to have edited such a fine book. We hope that you will enjoy reading these articles as much as we have, and (as one referee wrote) you will find that the stories flow well and are inspiring! Berkeley, USA Hélène Barcelo Claremont, USA Gizem Karaali Hanover, USA Rosa Orellana Contents Partition Algebras and the Invariant Theory of the Symmetric Group ................................................... 1 Georgia Benkart and Tom Halverson Affine Grassmannians and Hessenberg Schubert Cells .............. 43 Linda Chen and Julianna Tymoczko A Survey of the Shi Arrangement ............................. 75 Susanna Fishel Variations on a Theme of Schubert Calculus ..................... 115 Maria Gillespie Combinatorics of the Diagonal Harmonics ....................... 159 Angela Hicks On Positivity of Ehrhart Polynomials ........................... 189 Fu Liu Recent Trends in Quasisymmetric Functions ..................... 239 Sarah K. Mason On Standard Young Tableaux of Bounded Height ................. 281 M. J. Mishna A Tale of Centrally Symmetric Polytopes and Spheres ............. 305 Isabella Novik Crystal Constructions in Number Theory ....................... 333 Anna Puskás vii Partition Algebras and the Invariant Theory of the Symmetric Group Georgia Benkart and Tom Halverson Abstract The symmetric group Sn and the partition algebra Pk (n) centralize one ⊗k another in their actions on the k-fold tensor power Mn of the n-dimensional permu- tation module Mn of Sn. The duality afforded by the commuting actions determines Φ : ( ) → ( ⊗k ) an algebra homomorphism k,n Pk n EndSn Mn from the partition algebra ( ⊗k ) , ∈ Z to the centralizer algebra EndSn Mn , which is a surjection for all k n ≥1, and an isomorphism when n ≥ 2k. We present results that can be derived from the dual- ity between Sn and Pk (n), for example, (i) expressions for the multiplicities of the ⊗k irreducible Sn-summands of Mn , (ii) formulas for the dimensions of the irreducible ( ⊗k ) modules for the centralizer algebra EndSn Mn , (iii) a bijection between vacillat- ing tableaux and set-partition tableaux, (iv) identities relating Stirling numbers of the second kind and the number of fixed points of permutations, and (v) character values for the partition algebra Pk (n). When 2k > n,themapΦk,n has a nontrivial kernel which is generated as a two-sided ideal by a single idempotent. We describe the kernel and image of Φk,n in terms of the orbit basis of Pk (n) and explain how the surjection Φk,n can also be used to obtain the fundamental theorems of invariant theory for the symmetric group. Keywords Symmetric group · Partition algebra · Schur–Weyl duality Invariant theory Mathematics Subject Classification (2010) MSC 05E10 · MSC 20C30 The second author gratefully acknowledges partial support from Simons Foundation grant 283311. G. Benkart (B) University of Wisconsin - Madison, Madison, WI 53706, USA e-mail: [email protected] T. Halverson Macalester College, Saint Paul, MN 55105, USA e-mail: [email protected] © The Author(s) and the Association for Women in Mathematics 2019 1 H. Barcelo et al. (eds.), Recent Trends in Algebraic Combinatorics, Association for Women in Mathematics Series 16, https://doi.org/10.1007/978-3-030-05141-9_1 2 G. Benkart and T. Halverson 1 Introduction Throughout we assume F is a field of characteristic zero. The symmetric group Sn has a natural action on its n-dimensional permutation module Mn over F by ⊗k permuting the basis elements. The focus of this article is on tensor powers Mn of Mn, which are Sn-modules under the diagonal action (see (1.2)). For n, k ∈ Z≥1, the partition algebra Pk (n) is an associative F-algebra with basis indexed by the set partitions of {1, 2,...,2k} and with multiplication