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
Preface.-An Insertion Scheme for Cn Crystals.-On the Combinatorics of Forrester--Baxter Models.-Combinatorial R Matrices for a Family of Crystals: Cn(1) and A 2n-1(2) Cases.-Theta Functions Associated with Affine Root Systems and the Elliptic Ruijsenaars Operators.-A Generalization of the q-Saalschutz Sum and the Burge Transform.-The Bethe Equation at q=0, the Mobius Inversion Formula, and Weight Multiplicities I: The sl(2) Case.-Hidden E-Type Structures in Dilute A Models.-Canonical Basses of High-Level q-Deformed Fock Spaces and Kazhdan--Lusztig Polynomials.-Finite-Gap Difference Operators with Elliptic Coefficients and Their Spectral Curves.