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$\ell$-restricted $Q$-systems and quantum affine algebras

Abstract : Kuniba, Nakanishi, and Suzuki (1994) have formulated a general conjecture expressing the positive solution of an $\ell$-restricted $Q$-system in terms of quantum dimensions of Kirillov-Reshetikhin modules. After presenting this conjecture, we sketch a proof for the exceptional type $E_6$ following our preprint (2013). In types $E_7$ and $E_8$, we prove positivity for a subset of the nodes of the Dynkin diagram, and we reduce the positivity for the remaining nodes to the conjectural iterated log-concavity of certain explicit sequences of real algebraic numbers.
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Anne-Sophie Gleitz. $\ell$-restricted $Q$-systems and quantum affine algebras. 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. pp.1-10. ⟨hal-01207615⟩

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