The KNS conjecture for Kirillov–Reshetikhin modules in type E

Let $\mathfrak{g}$ be a simple Lie algebra of type $E$, let $I$ index the vertices of its Dynkin diagram, and let $W^{(i)}_k$ be the Kirillov–Reshetikhin modules. Fix $\ell\in\mathbb{Z}_{>0}$, set $l=\ell+h$, and let $\zeta=\exp(i\pi/l)$. Write

$$Q^{(i)}_k=d_\zeta\left(W^{(i)}_k\right),\qquad (i\in I,\ 0\leq k\leq\ell).$$

The KNS conjecture. The collection of real numbers $Q^{(i)}_k$ is a solution of the $\ell$-restricted $Q$-system. Moreover, for every $i\in I$:

• $d_\zeta\left(W^{(i)}_k\right)=0$ for $k\in\llbracket \ell+1,l-1\rrbracket$;
• $Q^{(i)}_k=Q^{(i)}_{\ell-k}$ for $k\in\llbracket 0,\ell\rrbracket$;
• $Q^{(i)}_k>0$ for $k\in\llbracket 0,\ell\rrbracket$;
• $Q^{(i)}_k<Q^{(i)}_{k+1}$ for $k\in\llbracket 0,\lfloor\ell/2\rfloor-1\rrbracket$.

The paper proves substantial portions of these assertions in types $E_6$, $E_7$, and $E_8$, while the remaining positivity and monotonicity cases are not proved there.