On crystal operators in Lusztig's parametrizations and string cone\n defining inequalities

2012/03/16 by Shmuel Zelikson, Zelikson, Shmuel
Mathematics · #05E10 #16G70 #17B37 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1203.3700

openalex publication_date 2012/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \w₀ be a reduced expression for the longest element of the\nWeyl group, adapted to a quiver of type A_n. We compare Lusztig's and\nKashiwara's (string) parametrizations of the canonical basis associated with\n\w₀. Crystal operators act in a finite number of patterns in\nLusztig's parametrization, which may be seen as vectors. We show this set gives\nthe system of defining inequalities of the string cone constructed by Gleizer\nand Postnikov. We use combinatorics of Auslander-Reiten quivers, and as a\nby-product we get an alternative enumeration of a set of inequalities defining\nthe string cone, based on hammocks.\n

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