Elementary equivalences for blocks with normal elementary abelian defect group of rank 2

Abstract

We consider the effect of performing an elementary equivalence as defined by Okuyama on a group block of form $F(C_p \times C_p)\rtimes C_r$, for a field $F$ of characteristic $p$. If $I=\{0,1,2,\dots r-1\}$ is the set of residues corresponding to the simple modules of $FC_r$, the elementary equivalence is determined by a proper, non-empty subset $I_0 \subset I$, and the corresponding elementary tilting complex is completely determined by $I_0$. We give a catalog of homogeneous maps between irreducible components of the elementary tilting complex. When the subset $I_0$ is an interval, we prove that the maps in the catalog are sufficient to describe all homogeneous maps between two irreducible components.