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Bethe Ansatz and combinatorics of the Young tableaux. (Russian. English summary) Zbl 0617.20025
The study of combinatorial aspects of the quantum inverse scattering method and Bethe’s ansatz applied to the \(\mathrm{GL}(N)\) (\(\mathrm{GL}(N/M)\)) invariant integrable quantum systems is continued. A bijective correspondence is built between the standard Young tableaux (bitableaux) and the rigged configurations which is a generalization of the bijection found in the preceding paper for standard tableaux without repetitions [S. V. Kerov, A. N. Kirillov and N. Yu. Reshetikhin [J. Sov. Math. 41, No. 2, 916–924 (1988; Zbl 0639.20028); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 155, 50-64 (1986; Zbl 0617.20024)]. Some important functionals on the standard tableaux are expressed in terms of the numerical parameters of the rigged configurations. In particular, a simple combinatorial representation for Kostka’s polynomial is given.
Reviewer: A.Bogush

MSC:
20G45 Applications of linear algebraic groups to the sciences
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
81R12 Groups and algebras in quantum theory and relations with integrable systems
05E15 Combinatorial aspects of groups and algebras (MSC2010)
05E05 Symmetric functions and generalizations
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