On the spanning vectors of Lusztig cones

Bédard, Robert

doi:10.1090/S1088-4165-00-00090-X

On the spanning vectors of Lusztig cones
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by Robert Bédard

Represent. Theory 4 (2000), 306-329

DOI: https://doi.org/10.1090/S1088-4165-00-00090-X

Published electronically: July 31, 2000

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Abstract:

For each reduced expression ${\mathbf i}$ of the longest element $w_0$ of the Weyl group $W$ of a Dynkin diagram $\Delta$ of type $A$, $D$ or $E$, Lusztig defined a cone ${\mathcal C}_{\mathbf i}$ such that there corresponds a monomial in the quantized enveloping algebra ${\mathbf U}$ of $\Delta$ to each element of ${\mathcal C}_{\mathbf i}$ and he asked under what circumstances these monomials belong to the canonical basis of ${\mathbf U}$. In this paper, we consider the case where ${\mathbf i}$ is a reduced expression adapted to a quiver $\Omega$ whose graph is $\Delta$ and we describe ${\mathcal C}_{\mathbf i}$ as the set of non-negative integral combination of spanning vectors. These spanning vectors are themselves described by using the Auslander-Reiten quiver of $\Omega$ and homological algebra.

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