Representations of constant socle rank for the Kronecker algebra

TL;DR

This paper introduces and studies modules with constant $d$-radical and $d$-socle rank for the Kronecker algebra, extending concepts from elementary abelian groups and showing these subcategories are of wild type.

Contribution

It defines new module properties for the Kronecker algebra and proves these subcategories are of wild type, transferring results from $kE_r$ modules.

Findings

Subcategories with constant $d$-radical and $d$-socle rank are of wild type.

Results are transferred from Kronecker algebra modules to elementary abelian group modules.

New notions generalize recent work on modules over elementary abelian groups.

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 $r \geq 2$ 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 $\mathrm{mod} \ kE_r$.