Categories for Grassmannian Cluster Algebras of Infinite Rank

Abstract

We construct Grassmannian categories of infinite rank, providing an infinite analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. Each Grassmannian category of infinite rank is given as the category of graded maximal Cohen–Macaulay modules over a certain hypersurface singularity. We show that generically free modules of rank $1$ in a Grassmannian category of infinite rank are in bijection with the Plücker coordinates in an appropriate Grassmannian cluster algebra of infinite rank. Moreover, this bijection is structure preserving, as it relates rigidity in the category to compatibility of Plücker coordinates. Along the way, we develop a combinatorial formula to compute the dimension of the $\textrm {Ext}^{1}$-spaces between any two generically free modules of rank $1$ in the Grassmannian category of infinite rank.

Cluster structures for the A∞$A_\infty$ singularity

“…Proof This is immediate from the equivalence between $$\underline{\mathcal {C}}_2$$ and $$\overline{\mathcal {C}}_M$$ and [19, Proposition 3.14]. Note that the computation for finite arcs also follows from [1, Theorem C], and direct calculations can be done using the matrix factorisations in Proposition 2.1.$$\Box$$…”

“…Continuing to follow [1], we will be interested in a particular subcategory of $$\mathcal {C}_2$$ for which the indecomposable objects are in bijection with the Plücker coordinates of the corresponding Grassmannian cluster algebra. To define this subcategory, we need the following.…”

“…We call the subcategory of $$\mathcal {C}_2$$ consisting of the generically free modules $$\mathcal {C}_2^\mathrm{f}$$, and note that this is an extension closed subcategory which is stably 2‐Calabi–Yau by [1, Proposition 3.12]. Moreover, we know by [1, Theorem 3.7] that the generically free modules in $$\mathcal {C}_2$$ are precisely the shifted ideals $$(x,y^k)(j)$$ for $$k \geqslant 0$$, $$j \in \mathbb {Z}$$, and these correspond to Plücker coordinates in the homogeneous coordinate ring of an infinite version of the Grassmannian of planes.…”

“…From a combinatorial perspective, it is natural to extend this to an ∞-gon: Take a discrete set of points on the unit circle 𝑆 1 with one two-sided accumulation point. We may think of the points as being indexed by the integers.…”

“…Specifically, we consider the category $$\mathcal {C}_2$$ of $$\mathbb {Z}$$‐graded maximal Cohen‐Macaulay modules over $$\mathbb {C}[x,y]/(x^2)$$ with $$x$$ in degree 1 and $$y$$ in degree $$-1$$. This is the Grassmannian category of infinite rank associated to an infinite version of the Grassmannian of planes, as introduced by the authors in [1]. We focus on this specific Grassmannian category because it is discrete, in the sense that it has countably many isomorphism classes of indecomposables, allowing us to explicitly describe and classify cluster tilting and related subcategories, as well as their mutations.…”