Schofield sequences in the Euclidean case

TL;DR

This paper provides an explicit procedure for determining Schofield submodules and factors of modules in the Euclidean case, enhancing understanding of module filtrations in tame quiver representations.

Contribution

It introduces a concrete method to identify Schofield submodules and factors for modules over Euclidean quivers, filling a gap in the existing theory.

Findings

Explicit procedure for Schofield submodules and factors in Euclidean case

Enhanced understanding of module filtrations in tame quivers

Bridging the gap in determining module components for Euclidean quivers

Abstract

Let $k$ be a field and consider the path algebra $kQ$ of the quiver $Q$. A pair of indecomposable $kQ$-modules $(Y,X)$ is called an orthogonal exceptional pair if the modules are exceptional and $\operatorname{Hom}(X,Y)=\operatorname{Hom}(Y,X)=\operatorname{Ext}^{1}(X,Y)=0$. Denote by $\mathcal{F}(X,Y)$ the full subcategory of objects having filtration with factors $X$ and $Y$. By the theorem of Schofield if $Z$ is exceptional but not simple, then $Z\in\mathcal{F}(X,Y)$ for some orthogonal exceptional pair $(Y,X)$, and $Z$ is not a simple object in $\mathcal{F}(X,Y)$. In fact, there are precisely $s(Z)-1$ such pairs, where $s(Z)$ is the support of $Z$ (i.e the number of nonzero components in ${\underline\dim}Z$). Whereas it is easy to construct $Z$ given $X$ and $Y$, there is no convenient procedure yet to determine the possible modules $X$ (called Schofield submodules of $Z$) and then $Y$ (called Schofield factors of $Z$), when $Z$ is given. We present such an explicit procedure in the tame case, i.e when $Q$ is Euclidean.