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Representation Theory

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A multiplicative property of quantum flag minors

Author: Ph. Caldero
Journal: Represent. Theory 7 (2003), 164-176
MSC (2000): Primary 17B10
Published electronically: April 17, 2003
MathSciNet review: 1973370
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Abstract: We study the multiplicative properties of the quantum dual canonical basis ${\mathcal B}^*$ associated to a semisimple complex Lie group $G$. We provide a subset $D$ of ${\mathcal B}^*$ such that the following property holds: if two elements $b$, $b’$ in ${\mathcal B}^*$ $q$-commute and if one of these elements is in $D$, then the product $bb’$ is in ${\mathcal B}^*$ up to a power of $q$, where $q$ is the quantum parameter. If $G$ is SL$_n$, then $D$ is the set of so-called quantum flag minors and we obtain a generalization of a result of Leclerc, Nazarov and Thibon.

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Additional Information

Ph. Caldero
Affiliation: Institut Girard Desargues, Université Claude Bernard – Lyon 1, 69622 Villeurbanne Cedex, France

Received by editor(s): January 23, 2002
Received by editor(s) in revised form: November 8, 2002, and January 8, 2003
Published electronically: April 17, 2003
Additional Notes: Supported in part by the EC TMR network “Algebraic Lie Representations", contract no. ERB FMTX-CT97-0100
Article copyright: © Copyright 2003 American Mathematical Society