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This work is concerned with combinatorial aspects arising in the theory of exactly solvable models and representation theory. 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 of Hamiltonians. Taking into account the various criss-crossing among mathematical subjects, this text presents results and ideas from three viewpoints: representation theory; integrable models; and combinatorics.

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