# Indecomposable Jordan types of Loewy length $2$

## TL;DR

This paper classifies indecomposable modules of Loewy length 2 with specific Jordan types over elementary abelian p-groups, using the Tits form of the Kronecker quiver to determine existence conditions.

## Contribution

It provides a complete characterization of when indecomposable modules of certain Jordan types and Loewy length 2 exist, based on algebraic and quiver-theoretic criteria.

## Key findings

- Existence of indecomposable modules characterized by Tits form inequality.
- Full classification of Jordan types for p > 2.
- Conditions depend on rank r and parameters c, d.

## Abstract

Let $k$ be an algebraically closed field, $\mathop{char}(k) = p \geq 2$ and $E_r$ be a $p$-elementary abelian group of rank $r \geq 2$. Let $(c,d) \in \mathbb{N}^2$. We show that there exists an indecomposable module of constant Jordan type $[1]^c [2]^d$ and Loewy length $2$ if and only if $q_{\Gamma_r}(d,d+c) \leq 1$ and $c \geq r-1$, where $q_{\Gamma_r}(x,y) := x^2 + y^2-rxy$ denotes the Tits form of the generalized Kronecker quiver $\Gamma_r$. Since $p > 2$ and constant Jordan type $[1]^c [2]^d$ imply Loewy length $\leq 2$, we get in this case the full classification of Jordan types $[1]^c [2]^d$ that arise from indecomposable modules.