Geodesic
ON CRYSTAL OPERATORS IN LUSZTIG'S PARAMETRIZATIONS AND STRING CONE DEFINING INEQUALITIES
Glasgow mathematical journal, Tome 55 (2013) no. 1, pp. 177-200

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DOI

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.
DOI : 10.1017/S0017089512000432
Mots-clés : 17B37, 05E10, 16G70 
ZELIKSON, SHMUEL. ON CRYSTAL OPERATORS IN LUSZTIG'S PARAMETRIZATIONS AND STRING CONE DEFINING INEQUALITIES. Glasgow mathematical journal, Tome 55 (2013) no. 1, pp. 177-200. doi: 10.1017/S0017089512000432
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     title = {ON {CRYSTAL} {OPERATORS} {IN} {LUSZTIG'S} {PARAMETRIZATIONS} {AND} {STRING} {CONE} {DEFINING} {INEQUALITIES}},
     journal = {Glasgow mathematical journal},
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     year = {2013},
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