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

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

2012-03-16

arxiv.org/abs/1203.3700v1

Mathematics

Quantum Algebra

To appear in Glasgow Mathematical Journal. Revised version of LMNO preprint 2011-23 UMR CNRS 6139, Caen University, France

Scientific paper

Let $\mathbf{w}_0$ be a reduced expression for the longest element of the Weyl group, adapted to a quiver of type $A_n$. We compare Lusztig's and Kashiwara's (string) parametrizations of the canonical basis associated with $\mathbf{w}_0$. Crystal operators act in a finite number of patterns in Lusztig's parametrization, which may be seen as vectors. We show this set gives the system of defining inequalities of the string cone constructed by Gleizer and Postnikov. We use combinatorics of 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.

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