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Representation Theory

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On the spanning vectors of Lusztig cones

Author: Robert Bédard
Journal: Represent. Theory 4 (2000), 306-329
MSC (2000): Primary 16G20, 16G70, 17B37
Published electronically: July 31, 2000
MathSciNet review: 1773864
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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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Additional Information

Robert Bédard
Affiliation: Département de Mathématiques, Université du Québec à Montréal, C.P. 8888, Succ. Centre-Ville, Montréal, Québec, H3C 3P8, Canada

Received by editor(s): December 2, 1999
Received by editor(s) in revised form: May 27, 2000
Published electronically: July 31, 2000
Article copyright: © Copyright 2000 American Mathematical Society