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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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