Composition Series of Tensor Product

TL;DR

This paper proves an extended version of Lusztig's conjecture, establishing a composition series for tensor products of modules over quantized enveloping algebras of symmetrizable Kac-Moody algebras, and derives the Littlewood-Richardson rule.

Contribution

It extends Lusztig's conjecture to a broader class of Kac-Moody algebras and constructs compatible composition series for tensor products of modules.

Findings

Proved the extended Lusztig's conjecture for all symmetrizable Kac-Moody algebras.

Established a composition series compatible with the canonical basis for tensor products.

Derived the Littlewood-Richardson rule as a consequence.

Abstract

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.