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
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action affine affine Lie algebra affine Weyl group algorithm analogue Bethe ansatz Bethe equation cell coefficients column insertion combinatorial condition configuration consider corresponding crystal bases crystal basis crystal graph define definition delete the entry denote diagram dilute eigenvalue elliptic empty box energy function expression fermionic Fock space follows formula functional relations Hence identities implies insertion scheme integer interfacial irreducible Ising model isomorphism Kashiwara Kazhdan–Lusztig polynomials Knuth Kostka polynomial Kuniba Lemma letter Lie algebra Math module move Note obtain odd band off-diagonal solutions pair particle partition polynomial pos(a PROOF Proposition q-deformed Fock space QTMs quantum quantum affine algebra representation resp result reverse bump reverse slide root systems scoring vertices Section spectral parameter striking sequence string solutions symmetric Theorem Tin(x type I slide vector vertex wedge wedge product weight wit(h Xi+1