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 {E_{r}} of rank r over a field of characteristic p > 0 {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 𝒦 r = k ⁢ Γ r {\mathcal{K}_{r}=k\Gamma_{r}} with r ≥ 2 {r\geq 2} arrows and 1 ≤ d ≤ r - 1 {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 mod ⁡ 𝒦 r {\operatorname{mod}\mathcal{K}_{r}} are of wild type. Then we use a natural functor 𝔉 : mod ⁡ 𝒦 r → mod ⁡ k ⁢ E r {\operatorname{\mathfrak{F}}\colon{\operatorname{mod}\mathcal{K}_{r}}\to% \operatorname{mod}kE_{r}} to transfer our results to mod ⁡ k ⁢ E r {\operatorname{mod}kE_{r}} .