Combinatorial Aspect of Integrable Systems (Mathematical Society of Japan Memoirs)

This volume is a collection of six papers based on the expository lectures of the workshop "Combinatorial Aspect of Integrable Systems" held at RIMS during July 26-30, 2004, as a part of the Project Research 2004 "Method of Algebraic Analysis in Integrable Systems". The topics range over crystal bases of quantum groups, its algebra-geometric analogue known as geometric crystal, generalizations of Robinson-Schensted type correspondence, fermionic formula related to Bethe ansatz, applications of crystal bases to soliton celluar automata, Yang-Baxter maps, and integrable discrete dynamics. All the papers are friendly written with many illustrative examples and intimately related to each other. This volume will serve as a good guide for researchers and graduate students who are interested in this fascinating subject. 1, Title and Copyright Pages Combinatorial Aspect of ... 2, Preface Combinatorial Aspect of ... Preface 3, Table of Contents Combinatorial Aspect of ... Table of Contents 4, Part 1 LECTURE NOTES ON GEOMETRIC ... 1. FIRST LECTURE: GEOMETRIC ... 2. SECOND LECTURE: POSITIVE ... REFERENCES 5, Part 2 COMBINATORICS OF CRYSTAL ... 1. INTRODUCTION 2. COMBINATORICS OF CRYSTALS ... 3. BASICS ON CRYSTALS ... 4. TABLEAUX FOR TYPES ... 5. PLACTIC MONOIDS FOR ... 6. BUMPING ALGORITHMS ... 7. ROBINSON-SCHENSTED ... 8. REVERSE BUMPING ALGORITHM ... 9. SLIDING ALGORITHM FOR ... REFERENCES 6, Part 3 $X=M$ CONJECTURE 1. INTRODUCTION 2. CRYSTALS 3. EXAMPLES OF CRYSTALS 4. PATHS AND lD SUMS 5. $X=M$ REFERENCES 7, Part 4 $X=M$ THEOREM: FERMIONIC ... 1. INTRODUCTION 2. BETHE ANSATZ AND RIGGED ... 3. ONE-DIMENSIONAL CONFIGURATION ... 4. $\overline{X}=\overline{M}$ 5. $X=M$ 6. $X^{\ell}=M^{\ell}$ REFERENCES 8, Part 5 SOLITON CELLULAR AUTOMATA 1. INTRODUCTION 2. A BRIEF REVIEW OF COMBINATORIAL ... 3. A BRIEF REVIEW OF COMBINATORIAL ... 4. CELLULAR AUTOMATA: ... Theorem 4.5 ... Theorem 4.6 ... 5. CELLULAR AUTOMATA: ... Theorem 5.16. ... Theorem 5.19. ... APPENDIX A. PROOF OF LEMMA ... REFERENCES 9, Part 6 YANG-BAXTER MAPS: DYNAMICAL ... INTRODUCTION YANG-BAXTER MAPS AND TRANSFER ... MATRIX FACTORISATIONS ... INTERACTION OF SOLlTONS ... POISSON LIE GROUPS AND ... REFERENCES