The cluster basis of \(\mathbb Z[x_{1,1},\dots, x_{3,3}]\)
type: Item
instance of: scholarly article
title: The cluster basis of \(\mathbb Z[x_{1,1},\dots, x_{3,3}]\)
zbMATH Open document ID: 1187.13019
author: Mark Skandera
published in: The Electronic Journal of Combinatorics
publication date: 2009-04-07T00:00:00Z
full work available at URL: v14i1r76.html
review text: Summary: We show that the set of cluster monomials for the cluster algebra of type \(D_4\) contains a basis of the \({\mathbb Z}\)-module \({\mathbb Z}[x_{1,1},\dots,x_{3,3}]\). We also show that the transition matrices relating this cluster basis to the natural and the dual canonical bases are unitriangular and nonnegative. These results support a conjecture of Fomin and Zelevinsky on the equality of the cluster and dual canonical bases. In the event that this conjectured equality is true, our results also imply an explicit factorization of each dual canonical basis element as a product of cluster variables.
Mathematics Subject Classification ID: 13F60
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 13F55
zbMATH DE Number: 5540854
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5540854