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ON CRYSTAL OPERATORS IN LUSZTIG'S PARAMETRIZATIONS AND STRING CONE DEFINING INEQUALITIES

Published online by Cambridge University Press:  02 August 2012

SHMUEL ZELIKSON
SHMUEL ZELIKSON*
Affiliation:
LMNO, UMR 6139 du CNRS, Département de Mathématiques, Université de Caen, B.P. 5186, 14032 Caen Cedex, France e-mail: Shmuel.Zelikson@unicaen.fr
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Abstract

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Let w0 be a reduced expression for the longest element of the Weyl group, adapted to a quiver of type An. We compare Lusztig's and Kashiwara's (string) parametrizations on canonical basis associated with w0. Crystal operators act in a finite number of patterns in Lusztig's parametrization, which may be seen as vectors. We show that this set gives the system of defining inequalities of the string cone constructed by Gleizer and Postnikov (O. Gleizer and A Postnikov, Littlewood–Richardson coefficients via Yang–Baxter equation, IMRN14 (2000) 741–774). We use combinatorics of the Auslander–Reiten quivers, and as a by-product we get an alternative enumeration of a set of inequalities defining the string cone based on hammocks.

Keywords

17B3705E1016G70
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 1 , January 2013 , pp. 177 - 200
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

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