Progress in Mathematics

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2 Progress in Mathematics Volume 191 Series Editors Hyman Bass Joseph Oesterle Alan Weinstein

3 Physical Combinatorics Masaki Kashiwara Tetsuji Miwa Editors Springer Science+Business Media, LLC

4 Masaki Kashiwara Tetsuji Miwa Research Institute for Research Institute for Mathematical Sciences (RIMS) Mathematical Sciences (RIMS) Kyoto University Kyoto University Kyoto, Japan Kyoto, Japan Library of Congress Cataloging-in-Publication Data Physical combinatorics / Masaki Kashiwara, Tetsuji Miwa. p. cm. - (Progress in mathematics ; v. 191) Includes bibliographica1 references. ISBN ISBN (ebook) DOI / Combinatorial analysis-congresses. 2. Representations of algebras-congresses. 3. Integral equations-congresses. 1. Kashiwara, Masaki, II. Miwa, T. (Tetsuji) III. Progress in mathematics (Boston, Mass.); voi QAI64.P l'.6-dc CIP AMS Subject Classifications: 81Qxx, 05AI0, 05A30, 33D15 Printed on acid-free paper Springer Science+Business Media New York Originally published by Birkhituser Boston in 2000 Softcover reprint of the hardcover Ist edition 2000 AII rights reserved. This work may not be translated or copied in whole or in part without the written permission ofthe. publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly anaiysis. U se in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, tradernarks, etc., in this publication, even if the former are not especiaily identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. SPIN ISBN Reformatted from editors' files by lohn Spiegelman, Philadelphia, PA

5 Dedicated to the memory of our friend and colleague Denis UgIQV (January 12, ctober 4, 1999)

6 Contents Preface Masaki Kashiwara and Tetsuji Miwa ix An Insertion Scheme for ell Crystals T. H. Baker 1 On the Combinatorics of Forrester-Baxter Models Omar Foda and Trevor A. Welsh 49 Combinatorial R Matrices for a Family of Crystals: d 1 ) and A ~ ; Cases ; _ l Goro Hatayama, Atsuo Kuniba, Masato Okado, and Taichiro Takagi 105 Theta Functions Associated with Affine Root Systems and the Elliptic Ruijsenaars Operators Yasushi Komori 141 A Generalization of the q-saalschiitz Sum and the Burge Transform Anne Schilling and S. Ole Warnaar 163 The Bethe Equation at q = 0, the Mobius Inversion Formula, and Weight Multiplicities I: The.51(2) Case Atsuo Kuniba and Tomoki Nakanishi 185 Hidden E -Type Structures in Dilute A Models J.Suzuki 217

7 viii Contents Canonical Bases of Higher-Level q-deforrned Fock Spaces and Kazhdan-Lusztig Polynomials Denis Uglov Finite-Gap Difference Operators with Elliptic Coefficients and Their Spectral Curves A. Zabrodin

8 Preface Masaki Kashiwara and Tetsuji Miwa This is the proceedings of a workshop entitled "Physical Combinatorics" held in 1999 during January at the International Institute for Advanced Studies and February 1-2 at Research Institute for Mathematical Sciences, Kyoto University. This conference was concerned with the combinatorial aspects arising in the theory of exactly solvable models and representation theory. Recent developments in integrable models reveal an unexpected link between representation theory and statistical mechanics through combinatorics. For example, Young tableaux, which describe the basis of irreducible representations, appear in Bethe Ansatz method in quantum spin chains as labels for the eigenstates of Hamiltonians. This connection brought new ideas both in representation theory and integrable models in statistical mechanics and also in combinatorics. For example, the classical Rogers-Ramanujan identities were generalized extensively, and their interpretations are given from both representation theory and integrable models. In this workshop "Physical Combinatorics," the most recent developments were discussed by researchers from three different areas: representation theory, integrable models, and combinatorics. All of the papers in this volume were refereed. We thank Kumiko Matsumura and Natsuko Morino for their help for the organization of the workshop and the preparation of these proceedings. We also thank the International Institute for Advanced Studies for their hospitality during the workshop. Denis Uglov, our friend and one of the contributors to this volume, passed away on October 4,1999. We feel much sorrow at his passing. This volume is dedicated to his memory. Masaki Kashiwara and Tetsuji Miwa Kyoto, Japan January 2000