Combinatorics, Bethe Ansatz, and representations of the symmetric group. (Russian. English summary) Zbl 0617.20024

Kerov, S. V.; Kirillov, A. N.; Reshetikhin, N. Yu.

Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 155, 50-64 (1986).

Some combinatorial problems relating to the representation theory of Lie groups (Young diagrams, rigged configurations, special bases for representations of symmetric groups) are discussed by making use of a technique originally developed in frames of the quantum inverse scattering method. A new combinatorial (bijective) correspondence is found between the standard Young tableaux belonging to the Young diagram \(\lambda\) and the rigged configurations (sets of diagrams together with admissible labels corresponding to nonzero rows of each diagram) introduced by the authors. In this way the completeness of the multiplet system generated by Bethe’s vectors is proved for the SU(p)-invariant Hamiltonian of the integrable quantum system (Heisenberg’s magnetic).

Reviewer: A.Bogush

MSC:

20G45	Applications of linear algebraic groups to the sciences
22E70	Applications of Lie groups to the sciences; explicit representations
05A17	Combinatorial aspects of partitions of integers
20C30	Representations of finite symmetric groups
05A05	Permutations, words, matrices

Keywords:

Young diagrams; representations of symmetric groups; quantum inverse scattering; standard Young tableaux; rigged configurations; integrable quantum system