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Cluster Categorification of Rank 2 Webs

Ian Le, Emine Yıldırım

Abstract

The homogeneous coordinate ring of the Grassmannian $\rm{Gr}(k,n)$ has a well-known cluster structure. There is a categorification of this cluster structure via a category of modules for a ring $A_{k,n}$ due to Jensen-King-Su, building on work of Geiss-Leclerc-Schröer, in which cluster variables correspond to indecomposable rigid modules. We give a combinatorial description of modules that correspond to rank $2$ cluster variables by using webs. This conjecturally gives the categorification of all rank $2$ cluster variables.

Cluster Categorification of Rank 2 Webs

Abstract

The homogeneous coordinate ring of the Grassmannian has a well-known cluster structure. There is a categorification of this cluster structure via a category of modules for a ring due to Jensen-King-Su, building on work of Geiss-Leclerc-Schröer, in which cluster variables correspond to indecomposable rigid modules. We give a combinatorial description of modules that correspond to rank cluster variables by using webs. This conjecturally gives the categorification of all rank cluster variables.
Paper Structure (7 sections, 17 theorems, 49 equations, 21 figures)

This paper contains 7 sections, 17 theorems, 49 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Webs of Rank $2$
  3. JKS Modules
  4. Webs and Modules
  5. Stretching Functions
  6. Rank 2 webs as cluster variables
  7. Cactus sequence and Main Result

Key Result

Lemma 3.2

Let $\frac{I}{J}$ be a profile consisting of three boxes with $(\frac{U}{U})$ and $(\frac{D}{D})$ interspersed. Then there is a unique indecomposable module of rank $2$ with this profile. All the modules in $\mathcal{M}$ are of this type.

Figures (21)

  • Figure 1: An illustration of rank $1$ and $2$ webs.
  • Figure 2: A rank $2$ web in an exchange relation.
  • Figure 3: The quiver $Q.$
  • Figure 4: The quiver Q and profile of a rank $1$ module $I=\{1,2,5\}$. The contour of I is shaded in gray.
  • Figure 5: Examples of rank $2$ modules with three boxes.
  • ...and 16 more figures

Theorems & Definitions (38)

  • Example 2.1
  • Conjecture 2.2
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Example 4.1
  • Example 4.2
  • Theorem 4.3
  • Definition 5.1
  • ...and 28 more