Representations of constant socle rank for the Kronecker algebra

Abstract

Inspired by recent work of Carlson, Friedlander and Pevtsova concerning modules for $p$-elementary abelian groups $E_r$ of rank $r$ over a field of characteristic $p>0$, we introduce the notions of modules with constant $d$-radical rank and modules with constant $d$-socle rank for the generalized Kronecker algebra ${\mathcal{K}}_r=k{\Gamma }_r$ with $r\ge 2$ arrows and $1\le d\le r-1$. We study subcategories given by modules with the equal $d$-radical property and the equal $d$-socle property. Utilizing the Simplification method due to Ringel, we prove that these subcategories in $\mathrm{mod}{\mathcal{K}}_r$ are of wild type. Then we use a natural functor $\mathfrak{F}:\mathrm{mod}{\mathcal{K}}_r\to \mathrm{mod}k{E}_r$ to transfer our results to $\mathrm{mod}k{E}_r$.