On the KNS Conjecture in type $E$

TL;DR

This paper proves the KNS conjecture for exceptional Lie types E6, E7, and E8 by showing that specialized quantum dimensions of Kirillov-Reshetikhin modules satisfy certain $Q$-systems, with positivity established in some cases.

Contribution

It confirms the KNS conjecture for types E6, E7, and E8, providing new insights into quantum dimensions and $Q$-systems in these exceptional types.

Findings

Specializations at roots of unity yield real solutions to $Q$-systems.

Positivity of solutions established for type E6.

Partial positivity results for types E7 and E8.

Abstract

For the exceptional types $E_6$, $E_7$, and $E_8$, we prove that the specializations at roots of unity of the quantum dimensions of the Kirillov-Reshetikhin modules give real solutions of $\ell$-restricted $Q$-systems, as conjectured by Kuniba, Nakanishi, and Suzuki. We also show that these solutions are positive in type $E_6$. In type $E_7$ and $E_8$, we only prove positivity for a subset of the nodes of the Dynkin diagram.