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

TL;DR

This paper studies elementary equivalences of blocks with normal elementary abelian defect groups of rank 2, providing a detailed catalog of maps and conditions under which these maps describe all relations.

Contribution

It characterizes elementary equivalences for blocks with specific defect groups and provides a complete description of the associated tilting complex maps when certain conditions are met.

Findings

Catalog of homogeneous maps between irreducible components

Elementary equivalences determined by subsets of residues

Sufficiency of maps when subset is an interval

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.