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 Advanced Studies in Pure Mathematics 1996; 453 pp; hardcover Volume: 24 ISBN-10: 4-314-10119-9 ISBN-13: 978-4-314-10119-6 Order Code: ASPM/24 This volume consists of thirteen papers on algebraic combinatorics and related areas written by leading experts around the world. There are four survey papers illustrating the following currently active branches of algebraic combinatorics: vertex operator algebras, spherical designs, Kerdock codes and related combinatorial objects, and geometry of matrices. The remaining nine papers are original research articles covering a wide range of disciplines, from classical topics such as permutation groups and finite geometry, to modern topics such as spin models and invariants of 3-manifolds. Two papers occupy nearly half the volume and present a comprehensive account of new concepts: "Combinatorial Cell Complexes" by M. Aschbacher and "Quantum Matroids" by P. Terwilliger. Terwilliger's theory of quantum matroids unites a part of the theory of finite geometries and a part of the theory of distance-regular graphs--great progess is expected in this field. K. Nomura's paper bridges the classical and the modern by establishing a connection between certain bipartite distance-regular graphs and spin models. All contributors to this volume were invited speakers at the conference "Algebraic Combinatorics" in Fukuoka, Japan (1993) and participated in the Research Institute in the Mathematical Sciences (RIMS) research project on algebraic combinatorics held at Kyoto University in 1994. Volumes in this series are freely available electronically 5 years post-publication. Published for the Mathematical Society of Japan by Kinokuniya, Tokyo, and distributed worldwide, except in Japan, by the AMS. Readership Graduate students, research mathematicians, physicists, and engineers working in coding theory, numerical integration, combinatorics, group theory, finite geometry, and low-dimensional topology. Table of Contents M. Aschbacher -- Combinatorial cell complexes T. Deguchi -- Generalized generalized spin models associated with exactly solvable models C. Dong and G. Mason -- Vertex operator algebras and moonshine: a survey C. D. Godsil -- Covers of complete graphs A. A. Ivanov and J. Saxl -- The character table of $$^2E_6(2)$$ acting on the cosets of $$Fi_{22}$$ F. Jaeger -- Towards a classification of spin models in terms of association schemes W. M. Kantor -- Orthogonal spreads and translation planes T. Kohno and T. Takata -- Level-rank duality of Witten's 3-manifold invariants R. A. Liebler -- Incidence matrix diagonal forms and integral Hecke algebras K. Nomura -- Spin models and almost bipartite 2-homogeneous graphs J. J. Seidel -- Spherical designs and tensors P. Terwilliger -- Quantum matroids Z. Wan -- Geometry of matrices