Combinatorial Formulae for Nested Bethe Vectors

Quantum Jacobi-Trudi and Giambelli Formulae for $U_q(B_r^{(1)})$ from Analytic Bethe Ansatz

Analytic Bethe ansatz is executed for a wide class of finite dimensional $U_q(B^{(1)}_r)$ modules. They are labeled by skew-Young diagrams which, in general, contain a fragment corresponding to the spin representation. For the transfer matrix spectra of the relevant vertex models, we establish a number of formulae, which are $U_q(B^{(1)}_r)$ analogues of the classical ones due to Jacobi-Trudi and Giambelli on Schur functions. They yield a full solution to the previously proposed functional relation ($T$-system), which is a Toda equation

量子齐次代数$U_q(\widehat{\mathfrak{gl}}_N)$的多重交换关系，嵌套Bethe向量与Gelfand-Tsetlin基

Multiple commutation relations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$, nested Bethe vector and the Gelfand-Tsetlin basis

本文研究了量子齐次代数$U_q(\widehat{\mathfrak{gl}}_N)$的多重交换关系，推导了向量表示的Gelfand-Tsetlin基构造方法，并探讨了其与Bethe向量的关系。

Off-shell Bethe vectors and Drinfeld currents

In this paper we compare two constructions of weight functions (off-shell Bethe vectors) for the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_N)$. The first construction comes from the algebraic nested Bethe ansatz. The second one is defined in terms of certain projections of products of Drinfeld currents. We show that two constructions give the same result in tensor products of vector representations of $U_q(\hat{\mathfrak{gl}}_N)$.

On Bethe vectors in the XXZ model at roots of unity

We give a construction of creation operators responsible for appearance of Bethe vectors with the same eigenvalues of the transfer-matrix for the inhomogeneous arbitrary spin XXZ model at roots of unity with particular quasiperiodic boundary conditions. This construction generalizes the similar one, recently obtained by K.Fabricius and B.M.McCoy, for the homogeneous six-vertex model with periodic boundary conditions. Even in the last case, the given proof for the main formulae are simpler than the original one.

A method for solve integrable $A_2$ spin chains combining different representations

A non homogeneous spin chain in the representations $ \{3 \}$ and $ \{3^*\}$ of $A_2$ is analyzed. We find that the naive nested Bethe ansatz is not applicable to this case. A method inspired in the nested Bethe ansatz, that can be applied to more general cases, is developed for that chain. The solution for the eigenvalues of the trace of the monodromy matrix is given as two coupled Bethe equations different from that for a homogeneous chain. A conjecture about the form of the solutions for more general chains is presented. PACS: 75.10.Jm, 05.50+q 02.20 Sv

Remarks on Fermionic Formula

Fermionic formulae originate in the Bethe ansatz in solvable lattice models. They are specific expressions of some q-polynomials as sums of products of q-binomial coefficients. We consider the fermionic formulae associated with general non-twisted quantum affine algebra U_q(X^{(1)}_n) and discuss several aspects related to representation theories and combinatorics. They include crystal base theory, one dimensional sums, spinon character formulae, Q-system and combinatorial completeness of the string hypothesis for arbitrary X_n.