Kirillov–Reshetikhin character conjecture for quantum affine algebras

Let $X_n$ be a finite-dimensional simple Lie algebra of classical or exceptional type, let $W_m^{(a)}$ denote the Kirillov–Reshetikhin module associated with node $a$ and positive integer $m$, and let $Q_m^{(a)}$ be the corresponding solution of the $Q$-system. Write $\operatorname{ch} W_m^{(a)}$ for its character. For classical $X_n$, the $Q_m^{(a)}$ are given by the character formulas in the displayed identities (a1)–(d1); for arbitrary $X_n$, they are given by the expansion in (qexp1), where $P_k^{(b)}$ and $\mathcal{N}_\lambda$ are as defined there. Kirillov–Reshetikhin conjecture. (i) For any classical $X_n$, $\operatorname{ch} W_m^{(a)}$ equals the corresponding right-hand sides of (a1)–(d1). (ii) For any $X_n$, $\operatorname{ch} W_m^{(a)}$ equals the right-hand side of (qexp1). (iii) For any $X_n$, the characters $\operatorname{ch} W_m^{(a)}$ satisfy relation (Q-I) with each $Q_m^{(a)}$ replaced by $\operatorname{ch} W_m^{(a)}$. The modules were originally claimed to exist without identification of their Drinfeld polynomials. The conjecture had been completely proved in the literature for $A_n$ and $D_n$; for classical types, parts (i) and (ii) are equivalent and (iii) follows from (i).