Peter Orlik Volkmar Welker Algebraic Combinatorics Lectures at a Summer School in Nordfjordeid, Norway, June 2003 ABC
Authors Peter Orlik Department of Mathematics University of Wisconsin-Madison 480 Lincoln Drive Madison, Wisconsin 53706 USA E-mail: orlik@math.wisc.edu Volkmar Welker Philipps-Universität Marburg Fachbereich Mathematik und Informatik Hans Meerwein Strasse, Lahnberge 35032 Marburg, Germany E-mail: welker@mathematik.uni-marburg.de Editor Gunnar Fløystad Department of Mathematics University of Bergen Johs.Brunsgate 12 5008 Bergen, Norway E-mail: Gunnar.Floystad@mi.uib.no Mathematics Subject Classification (2000): 13D02, 32S22, 52C35; 05E99, 55N25, 55U15 Library of Congress Control Number: 2006938280 ISBN-10 ISBN-13 3-540-68375-5 Springer Berlin Heidelberg New York 978-3-540-68375-9 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, 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. Typesetting: by the authors and techbooks using a Springer TEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: 11931218 46/techbooks 543210
Preface Each year since 1996 the universities of Bergen, Oslo and Trondheim have organized summer schools in Nordfjordeid in various topics in algebra and related fields. Nordfjordeid is the birthplace of Sophus Lie, and is a village on the western coast of Norway situated among fjords and mountains, with spectacular scenery wherever you go. As such it is a welcome place for both Norwegian and international participants and lecturers. The theme for the summer school in 2003 was Algebraic Combinatorics. The organizing committee consisted of Gunnar Fløystad and Stein Arild Strømme (Bergen), Geir Ellingsrud and Kristian Ranestad (Oslo), and Alexej Rudakov and Sverre Smalø (Trondheim). The summer school was partly financed by NorFa-Nordisk Forskerutdanningsakademi. With combinatorics reaching into and playing an important part of ever more areas in mathematics, in particular algebra, algebraic combinatorics was a timely theme. The fist lecture series Hyperplane arrangements was given by Peter Orlik. He came as a refugee to Norway, eighteen years old, after the insurrection in Hungary in 1956. Despite now having lived more than four decades in the United States, he impressed us by speaking fluent Norwegian without a trace of accent. The second lecture series Discrete Morse theory and free resolutions was given by Volkmar Welker. These two topics originate back in the second half of the nineteenth century with simple problems on arrangements of lines in the plane and Hilberts syzygy theorem. Although both are classical themes around which mathematics has centered since, there has in recent years been an influx of completely new insights and ideas, and interest in these fields has surged. An attractive feature of both topics is that they relate heavily both to combinatorics, algebra, and topology and, in the case of arrangements, even to analysis, thus giving a rich taste of mathematics. The third lecture series was Cluster algebras by Sergei Fomin. This is a recent topic, of this millennium. It has quickly attracted attention due to it giving new insights into classical mathematics as well as giving us fascinating new algebraic structures to study, relating to combinatorics and discrete
VI Preface geometry. This lecture series is however published elsewhere and so is not included here. But we are pleased to present the first two lecture series in this volume, the topics of which are so natural, classical and inexhaustible that mathematicians will certainly center around them for years to come. November 2006, Gunnar Fløystad
Contents Part I Lectures on Arrangements Introduction................................................. 5 1 Algebraic Combinatorics... 13 1.1 ChamberCounting... 13 1.2 Ranking Patterns........................................ 17 1.3 RandomWalks... 19 1.4 The Orlik-Solomon Algebra............................... 23 1.5 The NBC Complex...................................... 27 1.6 The Aomoto Complex.................................... 33 1.7 Combinatorial Types..................................... 40 1.8 Formal Connections...................................... 41 1.9 Multiplicities........................................... 43 1.10 Ideal Invariance......................................... 47 1.11 Examples............................................... 52 1.12 Exercises............................................... 58 2 Applications... 61 2.1 Topology............................................... 61 2.2 Local System Cohomology................................ 64 2.3 Resonance... 67 2.4 ModuliSpaces... 68 2.5 Gauss-Manin Connections................................ 70 2.6 Exercises... 76 References... 77
VIII Contents Part II Discrete Morse Theory and Free Resolutions 1 Introduction... 83 1.1 Overview... 83 1.2 Enumerative and Algebraic Invariants of Simplicial Complexes 85 1.3 Cohen-Macaulay Simplicial Complexes..................... 92 1.4 Some Open Problems in the Field......................... 97 2 Basic Definitions and Examples...107 2.1 Multigraded Free Resolutions............................. 107 2.2 Basics of CW-Complexes................................. 113 2.3 Basics of Cellular Homology.............................. 117 2.4 Cellular Chain Complexes and Cellular Resolutions.......... 120 2.5 Co-Artinian Monomial Modules........................... 123 3 Cellular Resolution...125 3.1 When Does a CW-Complex Support a Cellular Resolution?.. 125 3.2 ReadingoffTheBettiNumbers...128 3.3 Examples of Cellular Resolutions.......................... 131 4 Discrete Morse Theory...145 4.1 Forman sdiscretemorsetheory...146 4.2 Discrete Morse Theory for Graded CW-Complexes........... 148 4.3 Minimizing Cellular Resolutions using Discrete Morse Theory. 158 4.4 The Morse Differential................................... 159 References...169 Index...173