Shmuel Zelikson | Searcharxiv

arXiv2012

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

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.

math.QA↗

arXiv2002

Auslander-Reiten quivers and the Coxeter complex

Let Q be a quiver of type ADE. We construct the corresponding Auslander-Reiten quiver as a topological complex inside the Coxeter complex associated with the underlying Dynkin diagram. We use the notion of chamber weights coming from the theory of the canonical basis of quantized envelopping algebras, and show this set has a special linearity property in our setting. Finally, we consider A_n case, and describe how Auslander-Reiten quivers correspond to particular wiring diagrams.

math.QA↗