Small modules with interesting rank varieties

Abstract

This paper focuses on the rank varieties for modules over a group algebra FE where E is an elementary abelian p-group and p is the characteristic of an algebraically closed field F. In the first part, we give a sufficient condition for a Green vertex of an indecomposable module to contain an elementary abelian p-group E in terms of the rank variety of the module restricted to E. In the second part, given a homogeneous algebraic variety V, we explore the problem on finding a small module with rank variety V. In particular, we examine the simple module D(kp−p+1,1p−1) for the symmetric group Skp.