Combinatorics of Young tableaux and configurations (Q2764193)

The reviewed paper presents the majority of the results on the combinatorial Bethe Ansatz (CBA) obtained by the author in 1982-1994. By the combinatorial Bethe Ansatz the author means a class of problems concerning the study and combinatorial interpretation of the number of solutions and \(q\)-analogs of the solutions, as well as the study of asymptotic behavior of solutions to Bethe-type equations described by integrable models such as the generalized Heisenberg magnet, \(X X Z\)-models, and \(RSOS\)-models with highest spin. The author's main goal is to present those aspects of combinatorics and representation theory to which CBA can be applied and, using interesting examples, demonstrate numerous applications of the rigged configuration method and fermionic formulas for generalized Kostka polynomials \(K_{\lambda,R}(q)\) to various problems of the combinatorics of Young tableaux and the representation theory of the general linear and symmetric groups. The paper contains the following basic results: fermionic formula and spectral decomposition for the Kostka-Foulkes polynomials; generalization of the Gale-Ryser theorem; new (combinatorial) proof and generalizations of the Lascoux-Schützenberger theorem about inequalities for the Kostka-Foulkes polynomials; fermionic formula for atomic polynomials; new proof of the Gupta conjecture on the growth of the Kostka-Foulkes polynomials; the stable behavior and interesting limits of the Kostka-Foulkes polynomials; computation of the generalized exponents of the algebras \(\mathfrak {sl}(3)\) and \({\mathfrak{sl}}(4)\); new properties of the Robinson-Schensted correspondence; fermionic formulas for the constant partition function and \(q\)-analog of this function; combinatorial proof of the unimodality and a fermionic formula for generalized \(q\)-binomial coefficients; rigged configurations and a constructive proof of the unimodality of \(q\)-binomial coefficients; combinatorial proof of the unimodality of principal specialization of the internal product of the Schur functions.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00033].