The Kirillov–Reshetikhin conjecture for tensor-product multiplicities

Let ${\mathfrak g}$ be a simple Lie algebra with Cartan matrix $C$, let ${\mathbf R}=\{a_p\omega_{i_p}:1\leq p\leq N\}$ be a collection of dominant integral weights, and let $\zeta_p$ be distinct nonzero complex numbers. Write $M_{\lambda,{\mathbf R}}$ for the multiplicity of the irreducible ${\mathfrak g}$-module $V_\lambda$ in the tensor product of the corresponding localized Kirillov–Reshetikhin modules. Let $n_a^{(i)}$ count the elements of ${\mathbf R}$ equal to $a\omega_i$, and write $\lambda=\sum_i l^{(i)}\omega_i$. The Kirillov–Reshetikhin conjecture. The multiplicity is

$$M_{\lambda,{\mathbf R}} = \sum_{\{m_a^{(i)}\in {\mathbb Z}_+: P_a^{(i)}\geq 0\}} \prod_{a,i} \binom{P_a^{(i)}+m_a^{(i)}}{m_a^{(i)}}$$

where the sum is over $m_a^{(i)}$ satisfying

$$\sum_a a m_a^{(i)}=\sum_{j,a}C^{-1}_{i,j}a n_a^{(j)}-\sum_j C^{-1}_{i,j}l^{(j)},$$

and

$$P_a^{(i)}=\sum_b\min(a,b)n_b^{(i)}+\sum_{b,j\neq i}\min(|C_{i,j}|b,|C_{j,i}|a)m_b^{(j)}-2\sum_b\min(a,b)m_b^{(i)}.$$

Here $C_{i,j}$ is the Cartan matrix. The formula is the fermionic multiplicity formula for the tensor product of KR-modules; it is known in several special cases, including arbitrary KR-modules for type $A_r$, fundamental weights for type $D_r$, and multiples of $\omega_1$ for nonexceptional types, while the general case remains open.