Mirković-Vilonen polytopes and a quiver construction of crystal basis in type A
type: Item
instance of: scholarly article
zbMATH Open document ID: 1326.17015
author: Yoshihisa Saito
publication date: 2012-10-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.0086
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 52B12
zbMATH DE Number: 6088718
zbMATH Keywords: quantum universal enveloping algebra
zbMATH Keywords: crystal basis
zbMATH Keywords: MV polytopes
zbMATH Keywords: PBW basis
zbMATH Keywords: Kashiwara operators
MaRDI profile type: Publication
OpenAlex ID: W2964233673
arXiv ID: 1010.0086
DOI: 10.1093/IMRN/RNR173
published in: IMRN. International Mathematics Research Notices
title: Mirković-Vilonen polytopes and a quiver construction of crystal basis in type A
review text: Let \(B(\infty)\) be the crystal basis of the negative half of the quantum universal enveloping algebra associated with \(\mathfrak{sl}_{n+1}(\mathbb{C})\). The aim of the paper is to give explicit isomorphisms between the Lagrangian (or quiver) realization, the realization by using the theory of PBW basis, and the realization in terms of Mirkovi\(\acute{c}\)-Vilonen (MV) polytopes of \(B(\infty)\). A quiver theoretical description of MV polytopes is given in type \(A_n\). As a by-product, a new proof of the Anderson-Mirkovi\(\acute{c}\) conjecture on the explicit forms of the actions of lowering Kashiwara operators on the set of MV polytopes is given.
reviewed by: Sorin Dascalescu
description: scientific article; zbMATH DE number 6088718