English

A categorification of Grassmannian cluster algebras

Representation Theory 2017-05-17 v3

Abstract

We describe a ring whose category of Cohen-Macaulay modules provides an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-space. More precisely, there is a cluster character defined on the category which maps the rigid indecomposable objects to the cluster variables and the maximal rigid objects to clusters. This is proved by showing that the quotient of this category by a single projective-injective object is Geiss-Leclerc-Schroer's category Sub QkQ_k, which categorifies the coordinate ring of the big cell in this Grassmannian.

Keywords

Cite

@article{arxiv.1309.7301,
  title  = {A categorification of Grassmannian cluster algebras},
  author = {Bernt Tore Jensen and Alastair King and Xiuping Su},
  journal= {arXiv preprint arXiv:1309.7301},
  year   = {2017}
}

Comments

v2: minor change in title, new Sec 9 on categorification, small changes in exposition and some new figures; v3: Sec 2 rearranged and shortened, small changes in exposition, to appear in Proc. Lond. Math. Soc