Indecomposable Jordan types of Loewy length $2$

Abstract

Let $k$ be an algebraically closed field, $\mathrm{char}(k)=p\ge 2$ and $E_r$ be a $p$-elementary abelian group of rank $r\ge 2$. 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 $q_{{\Gamma }_r}(d,d+c)\le 1$ and $c\ge r-1$, 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 $\le 2$, we get in this case the full classification of Jordan types $[1{]}^c[2{]}^d$ that arise from indecomposable modules.