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Imaginary vectors in the dual canonical basis of $U_q(n)$

Bernard Leclerc
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Abstract

Let $n$ be the maximal nilpotent subalgebra of a simple complex Lie algebra $g$. We introduce the notion of imaginary vector in the dual canonical basis of $U_q(n)$, and we give examples of such vectors for types $A_n (n\ge 5)$, $B_n (n\ge 3)$, $C_n (n\ge 3)$, $D_n (n\ge 4)$, and all exceptional types. This disproves a conjecture of Berenstein and Zelevinsky about $q$-commuting products of vectors of the dual canonical basis. It also shows the existence of finite-dimensional irreducible representations of quantum affine algebras whose tensor square is not irreducible.

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Bernard Leclerc. 2002-09-11. Imaginary vectors in the dual canonical basis of $U_q(n)$. https://arxiv.org/abs/math/0202148

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