Liam Riordan | Search arXiv

arXiv2026

Quiver representations and idempotent loops

We study $\mathbb{C}^{\times}$-actions on quiver representations by adding idempotent loops. We construct a category mod $\widehat{\operatorname{W}}(\mathrm{Q})$ and relate it to $\mathbb{C}Q$ modules with $\mathbb{C}^{\times}$-actions. We further describe a subcategory $\widehat{\operatorname{W}}_{T}(\mathrm{Q})$ equivalent to the category of equivariant modules where $\mathbb{C}^{\times}$ acts on $\mathbb{C}Q$ via a character $T \in \mathbb{Z}^{Q_1}.$ This allows us to study $\mathbb{C}^{\times}$-actions on quiver Grassmannians and we can recover known combinatorial descriptions of their Euler characteristics via the category mod $\widehat{\operatorname{W}}(\mathrm{Q}).$ We directly relate morphisms of quivers to full subcategories of mod $\widehat{\operatorname{W}}(\mathrm{Q}).$ Finally we show that the representation theory of quivers with idempotents can be applied in a useful way to preprojective algebras. We show that this gives constructions of Galois covers and cluster characters used by Geiss, Leclerc, and Schröer.

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arXiv2026

Cohen Macaulay modules and positroid varieties

Jensen, King, and Su described a category $\operatorname{CM}(C)$ which categorifies the cluster structure on the homogeneous coordinate ring of a Grassmannian. In this paper we describe subcategories $\operatorname{R}(v,w) \subseteq \operatorname{CM}(C)$ which lift Leclerc's categories $\mathcal{C}_{v,w}$ in the case where $v \in \left(W^{k}\backslash W\right)^{\max}$ and $w \geq v.$ As such, these categories are Frobenius, stably 2-CY, have natural cluster characters, and induce a cluster structure in lifts of open positroid varieties.

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arXiv2026

Grassmannian cluster subcategories and positroid varieties

A class of subcategories GP $B$ of the Grassmannian cluster category CM $C_{k, n}$ was constructed by Jensen--King--Su from certain superorders $B$ of $C_{k, n}$, which they showed are in bijection with Grassmannian positroids of type $(k, n)$. We prove that GP $B$ admits a cluster substructure of CM $C_{k, n}$, giving rise to a cluster algebra $A_{clu}$. This naturally raises questions regarding the relationship of $A_{clu}$ to $C[Gr(k, n)]$ and to the coordinate ring of the positroid variety associated to $B$. Using the cluster substructure, we show that the ice Gabriel quiver $Q^\circ_U$ of a cluster tilting object $U\in$ GP $B$, consisting of rank one modules, is a subquiver of $Q^\circ_T$ with $T$ a cluster tilting object in CM $C_{k, n}$ containing $U$ as a summand. We also deduce that $A_{clu}$ is a subalgebra of $C[Gr(k, n)]$. Moreover, applying a result of Canakci--King--Pressland on the Gabriel quiver $Q_U$ in the case where $B$ is connected (i.e., has no repeated direct summands), we deduce that $Q^\circ_U$, for arbitrary $B$, coincides with the quiver constructed by Muller-Speyer from a plabic graph whose face labels agree with the indices of the indecomposable summands of $U$. Consequently, the localised algebra $(A_{clu})_B$ is isomorphic to the cluster algebra $A_{MS}$ of Muller-Speyer. We then construct bases for certain subalgebras and for an ideal of $C[Gr(k, n)]$, and apply these to prove that $(A_{clu})_B$ is naturally isomorphic to the coordinate ring of the open positroid variety. As a consequence, we obtain a new proof of Galashin--Lam's Theorem, identifying $A_{MS}$ with the coordinate ring of the open positroid variety, which was originally conjectured by Muller-Speyer. In the connected case, we note also that Pressland gave a categorification of the cluster structure following Galashin-Lam.

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