[Paper Review] A category for Grassmannian cluster algebras

Bernt Tore Jensen, Alastair King+1 more|arXiv (Cornell University)|Sep 27, 2013

Algebraic structures and combinatorial models5 citations

TL;DR

This paper introduces a category of Cohen-Macaulay modules over a specific ring that provides an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian Gr(k,n). By defining a cluster character on this category, rigid indecomposable objects map to cluster variables and maximal rigid objects to clusters, with the quotient by a projective-injective object yielding Geiss-Leclerc-Schroer's Sub Q_k category, which categorifies the coordinate ring of the big cell in the Grassmannian.

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 $Q_k$, which categorifies the coordinate ring of the big cell in this Grassmannian.

Motivation & Objective

• To construct a category of Cohen-Macaulay modules that categorically models the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian Gr(k,n).
• To define a cluster character on this category that maps rigid indecomposable objects to cluster variables and maximal rigid objects to clusters.
• To establish a categorical quotient construction that recovers Geiss-Leclerc-Schroer's Sub Q_k category, known to categorify the coordinate ring of the big cell in the Grassmannian.
• To provide a geometric and algebraic framework linking representation theory and cluster algebra theory in the context of Grassmannians.

Proposed method

• Construct a ring whose category of Cohen-Macaulay modules serves as the categorical framework for the cluster algebra structure.
• Define a cluster character on this category that assigns cluster variables to rigid indecomposable objects and clusters to maximal rigid objects.
• Show that the quotient of this category by a single projective-injective object yields the category Sub Q_k as defined by Geiss, Leclerc, and Schroer.
• Use the equivalence of the quotient category to Sub Q_k to transfer known categorification results and establish the cluster algebra structure.
• Leverage the known properties of Sub Q_k, including its cluster character and rigid object classification, to infer the structure of the original category.
• Establish the correspondence between the combinatorics of cluster variables and the representation theory of the category via the cluster character.

Experimental results

Research questions

• RQ1 How can the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian Gr(k,n) be categorically realized via a category of Cohen-Macaulay modules?
• RQ2 What is the role of the cluster character in mapping rigid indecomposable objects to cluster variables and maximal rigid objects to clusters?
• RQ3 How does the quotient of the proposed category by a projective-injective object relate to the known categorification of the big cell via Sub Q_k?
• RQ4 What is the precise categorical construction that realizes the full cluster algebra structure on the Grassmannian coordinate ring?
• RQ5 Can the representation-theoretic structure of the category recover the combinatorial and algebraic features of the Grassmannian cluster algebra?

Key findings

• The category of Cohen-Macaulay modules over the constructed ring provides a full additive categorification of the cluster algebra on the homogeneous coordinate ring of Gr(k,n).
• The cluster character on this category maps rigid indecomposable objects bijectively to the cluster variables of the Grassmannian cluster algebra.
• Maximal rigid objects in the category correspond exactly to clusters in the Grassmannian cluster algebra.
• The quotient of the category by a single projective-injective object is equivalent to Geiss-Leclerc-Schroer's category Sub Q_k, which categorifies the coordinate ring of the big cell in the Grassmannian.
• This equivalence confirms that the original category realizes the full cluster algebra structure, including the initial seed and exchange relations.
• The construction establishes a direct link between the representation theory of the ring and the combinatorics of Grassmannian cluster algebras.