Indecomposable Jordan types of Loewy length $2$

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 $q_{Γ_r}(d,d+c) \leq 1$ and $c \geq r-1$, where $q_{Γ_r}(x,y) := x^2 + y^2-rxy$ denotes the Tits form of the generalized Kronecker quiver $Γ_r$. 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.