Frenkel–Reshetikhin conjecture on q-character substitutions for transfer-matrix eigenvalues

Let $\mathcal{U}_q(\hat{\mathfrak{g}})$ be a quantum affine algebra, let $V$ be a finite-dimensional representation with $q$-character $\chi_q(V)$, and let $W$ be a tensor product of finite-dimensional simple representations. For $1\leq i\leq n$, write $q_i=q^{r_i}$, and let $\mathcal{T}_V(z)$ be the transfer matrix. The polynomials $Q_{i,j}(z)$ and the functions $F_i(z)$ are defined so that the eigenvalues $\lambda_j$ of $\mathcal{T}_V(z)$ are obtained by substituting for every variable $Y_{i,a}$ in $\chi_q(V)$ the expression

$$F_i(az)q^{\operatorname{deg}(Q_{i,j})}\frac{Q_{i,j}(zaq_i^{-1})}{Q_{i,j}(zaq_i)}.$$

Frenkel–Reshetikhin conjecture. For every eigenvalue $\lambda_j$ of $\mathcal{T}_V(z)$ on $W$, the preceding substitution in $\chi_q(V)$ yields $\lambda_j$; here each $Q_{i,j}(z)$ is a polynomial and $F_i(z)$ does not depend on $\lambda_j$. This conjectural prescription generalizes Baxter's formulas for transfer-matrix eigenvalues through the $q$-character of $V$. It was proposed by Frenkel and Reshetikhin and is presented here as an approach to generalizing Baxter formulas; the supplied text does not establish its resolution.