The Kirillov–Reshetikhin multiplicity formula for Yangian KR modules

Let $\mathfrak{g}$ be a complex simple Lie algebra of rank $n$. Set $y=(y_a)_{a=1}^n$ with $y_a=e^{-\alpha_a}$ for the simple roots $\alpha_a$ of $\mathfrak{g}$. For the KR module $W_m^{(a)}(u)$ of the Yangian $Y(\mathfrak{g})$, let $\mathcal{Q}_m^{(a)}(y)$ be its normalized $\mathfrak{g}$-character, and set $\mathcal{Q}^{\nu}(y):=\prod_{(a,m)}(\mathcal{Q}_m^{(a)}(y))^{\nu_m^{(a)}}$. Let $A=(A_{ab})$ be the Cartan matrix, let $d_a$ be coprime positive integers such that $(d_aA_{ab})$ is symmetric, and let $\Delta_+$ be the positive roots. Kirillov–Reshetikhin conjecture. The formula

$$\mathcal{Q}^{\nu}(y)\prod_{\alpha\in\Delta_+}(1-e^{-\alpha})=\sum_{N=(N_m^{(a)})}\prod_{(a,m)}\binom{P_m^{(a)}(\nu,N)+N_m^{(a)}}{N_m^{(a)}}(y_a)^{mN_m^{(a)}},$$

where

$$P_m^{(a)}(\nu,N)=\sum_{k=1}^{\infty}\nu_k^{(a)}\min(k,m)-\sum_{(b,k)}N_k^{(b)}d_aA_{ab}\min\left(\frac{m}{d_b},\frac{k}{d_a}\right),$$

should hold, with $\binom{a}{b}=\varGamma(a+1)/(\varGamma(a-b+1)\varGamma(b+1))$. This is the proposed multiplicity formula for tensor products of Kirillov–Reshetikhin modules; the paper reformulates it through canonical solutions of $Q$-systems, while its general validity is the subject of the conjecture.