This work is concerned with 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 the Bethe Ansatz method in quantum spin chains as labels for the eigenstates for Hamiltonians.

Taking into account the various criss-crossing among mathematical subject, Physical Combinatorics presents new results and exciting ideas from three viewpoints; representation theory, integrable models, and combinatorics.

This volume will be of interest to mathematical physicists and graduate students in the the above-mentioned fields.

Contributors to the volume: T.H. Baker, O. Foda, G. Hatayama, Y. Komori, A. Kuniba, T. Nakanishi, M. Okado, A. Schilling, J. Suzuki, T. Takagi, D. Uglov, O. Warnaar, T.A. Welsh, A. Zabrodin

Developments in mathematical physics during the second half of the 20th century influenced a number of mathematical areas, among the more significant being representation theory, differential equations, combinatorics, and algebraic geometry. In all of them, the dynamic role of integrable models has been central, largely due to two essential properties: the fact that integrable models possess infinite degrees of freedom and infinite dimensional symmetries. This volume focuses on the ongoing importance of integrability in covering the following topics: conformal field theory, massive quantum field theory, solvable lattice models, quantum affine algebras, the Painleve equations and combinatorics.

Contributors: H. Au-Yang, R.J. Baxter, H.E. Boos, E. Date, K. Fabricius, V.A. Fateev, B. Feigin, G.Hatayama, A. Its, M. Jimbo, A. Kapaev, A.N. Kirillov, V.E. Korepin, A.Kuniba, J.M. Maillet, B.M. McCoy, C. Mercat, T. Miwa, A. Nakayashiki, M.Okado, C.H.Otto Chui, P.A. Pearce, J.H.H. Perk, V. Petkova, A. Schilling, F.A. Smirnov, T.Takagi, Y. Takeyama, M. Taneda, C.A. Tracy, Z.Tsuboi, H. Widom, J.-B. Zuber 'MathPhys

"Odyssey 2001" will serve as an excellent reference text for mathematical physicists and graduate students in a number of areas."

Integrable models in statistical mechanics and quantum field theory constitute a rich research field at the crossroads of modern mathematics and theoretical physics. An important issue to understand is the space of local operators in the system and, ultimately, their correlation functions and form factors. This book is the first published monograph on this subject. It treats integrable lattice models, notably the six-vertex model and the XXZ Heisenberg spin chain. A pair of fermions is introduced and used to create a basis of the space of local operators, leading to the result that all correlation functions at finite distances are expressible in terms of two transcendental functions with rational coefficients. Step-by-step explanations are given for all materials necessary for this construction, ranging from algebraic Bethe ansatz, representations of quantum groups, and the Bazhanov-Lukyanov-Zamolodchikov construction in conformal field theory to Riemann surfaces and their Jacobians. Several examples and applications are given along with numerical results. Going through the book, readers will find themselves at the forefront of this rapidly developing research field.

'MathPhys Odyssey 2001' will serve as an excellent reference text for mathematical physicists and graduate students in a number of areas.; Kashiwara/Miwa have a good track record with both SV and Birkhauser.

Taking into account the various criss-crossing among mathematical subject, Physical Combinatorics presents new results and exciting ideas from three viewpoints; representation theory, integrable models, and combinatorics. This work is concerned with 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.

The 1990 Hayashibara Forum, "the International Conference on Special Functions," was held at Fujisaki Institute, Hayashibara Biochemical Laboratories, Inc., Okayama, Japan for five days (August 16-20, 1990). This volume is the proceedings for that meeting. On January 14,1985, Heisuke Hironaka and Ken Hayashibara, the president of Chair man, Board of Trustees, Hayashibara Foundation, met and decided to have an international conference on mathematics in the summer of 1990. This was pushed forward by Kiyosi Ito, who proposed "Special functions" as the theme of the conference. He also asked the present editors to join in the organizing committee of the Hayashibara Forum, 1990. On May 13, 1989 the organizing committee sent letters to major Japanese mathemat ical institutions asking their members to give suggestions about whom it should invite. Receiving the replies, the organizing committee decided the invited speakers, and sent invitation letters to them, in which it was written that "Special functions have been created and explored to describe scientific and mathematical phenomena. Trigonometric functions give the relation of angle to length. Riemann's zeta function was invented in order to describe the prime number distribution. Legendre's spherical functions and Bessel's functions were born in connection with the eigenvalue problems for partial differential equations."