Crystal Structure on Rigged Configurations

There are (at least) two main approaches to solvable lattice models and their associated quantum spin chains: the Bethe ansatz [6] and the corner transfer matrix method [5]. In his paper [6], Bethe solved the Heisenberg spin chain based on the string hypothesis which asserts that the eigenvalues of the Hamiltonian form certain strings in the complex plane as the size of the system tends to infinity. The Bethe ansatz has been applied to manymodels to prove completeness of the Bethe vectors. The eigenvalues and eigenvectors of the Hamiltonian are indexed by rigged configurations. However, numerical studies indicate that the string hypothesis is not always true [1]. The corner transfer matrix (CTM) method, introduced by Baxter [5], labels the eigenvectors by one-dimensional lattice paths. These lattice paths have a natural interpretation in terms of Kashiwara’s crystal base theory [20, 21], namely as highest-weight crystal elements in a tensor product of finite-dimensional crystals. Even though neither the Bethe ansatz nor the corner transfer matrix method is mathematically rigorous, they suggest the existence of a bijection between the two index sets, namely rigged configurations on the one hand and highest-weight crystal paths on the other (see Figure 1.1). For the special case when the spin chain is defined on V(μ1) ⊗ V(μ2) ⊗ · · · ⊗V(μk),where V(μi) is the irreducible GL(n) representation indexed by the partition (μi) for μi ∈ N, a bijection between rigged configurations and semistandard Young tableaux was given by Kerov, Kirillov, and Reshetikhin [24], Kirillov and Reshetikhin [26].