Hatayama et al.'s identification of transfer matrices with Kirillov–Reshetikhin q-characters

Let $X_n$ be a non-exceptional simple Lie algebra with simple roots $\alpha_a$ for $1\leq a\leq n$. Let $T^{(a)}_m(u)$ be the transfer matrices in the $T$-system, with $T^{(a)}_0(u)=1$, and let $W^{(a)}_m(u)$ be the Kirillov--Reshetikhin module whose highest weight monomial is

$$\prod_{j=1}^mY_a\left(u+\frac{(\alpha_a\mid\alpha_a)}{2}(m+1-2j)\right).$$

Transfer-matrix q-character conjecture. The identification

$$T^{(a)}_m(u)=\chi_q\left(W^{(a)}_m(u)\right)$$

solves the $T$-system. In other words, the transfer matrices are the $q$-characters of the corresponding Kirillov--Reshetikhin modules and therefore satisfy the stated functional relations.