Composition Series of Tensor Product

Details Composition Series of Tensor Product Composition Series of Tensor Product

2010-03-29

arxiv.org/abs/1003.5416v1

Mathematics

Quantum Algebra

19pages, 1 figure

Scientific paper

Given a quantized enveloping algebra $U_q(\mathfrak g)$ and a pair of dominant weights ($\lambda$, $\mu$), we extend a conjecture raised by Lusztig in \cite{Lusztig:1992}to a more general form and then prove this extended Lusztig's conjecture. Namely we prove that for any symmetrizable Kac-Moody algebra $\mathfrak g$, there is a composition series of the $U_q(\mathfrak g)$-module $V(\lambda)\otimes V(\mu)$ compatible with the canonical basis. As a byproduct, the celebrated Littlewood-Richardson rule is derived and we also construct, in the same manner, a composition series of $V(\lambda)\otimes V(-\mu)$ compatible with the canonical basis when $\mathfrak g$ is of affine type and the level of $\lambda-\mu$ is nonzero.

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