A multiplicative property of quantum flag minors

Caldero, Ph.

doi:10.1090/S1088-4165-03-00156-0

A multiplicative property of quantum flag minors
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by Ph. Caldero

Represent. Theory 7 (2003), 164-176

DOI: https://doi.org/10.1090/S1088-4165-03-00156-0

Published electronically: April 17, 2003

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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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