Indecomposable Jordan types of Loewy length 2

Abstract

Let k be an algebraically closed field, char(k)=p≥2 and Er be a elementary abelian p-group of rank r≥2. Let (c,d)∈N2. We show that there exists an indecomposable module of constant Jordan type [1]c[2]d and Loewy length 2 if and only if qΓr(d,d+c)≤1 and c≥r−1, where qΓr(x,y):=x2+y2−rxy denotes the Tits form of the generalized Kronecker quiver Γr.Since p>2 and constant Jordan type [1]c[2]d imply Loewy length ≤2, we get in this case the full classification of Jordan types [1]c[2]d that arise from indecomposable modules.