Representations of constant socle rank for the Kronecker algebra

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 $r \geq 2$ arrows and $1 \leq d \leq r-1$. 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 $\mathfrak{F} \colon \mathrm{mod} \ \mathcal{K}_r \to \mathrm{mod} \ kE_r$ to transfer our results to $\mathrm{mod} \ kE_r$.