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Abstract: Given a quantized enveloping algebra $U q\mathfrak g$ and a pair ofdominant weights $\lambda$, $\mu$, we extend a conjecture raised by Lusztigin \cite{Lusztig:1992}to a more general form and then prove this extendedLusztig-s conjecture. Namely we prove that for any symmetrizable Kac-Moodyalgebra $\mathfrak g$, there is a composition series of the $U q\mathfrakg$-module $V\lambda\otimes V\mu$ compatible with the canonical basis. As abyproduct, the celebrated Littlewood-Richardson rule is derived and we alsoconstruct, in the same manner, a composition series of $V\lambda\otimesV-\mu$ compatible with the canonical basis when $\mathfrak g$ is of affinetype and the level of $\lambda-\mu$ is nonzero.

Autor: Bin Li, Hechun Zhang


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