Nilpotent extensions of blocks

TL;DR

This paper investigates the structure of certain non-nilpotent blocks in modular representation theory, introduces inertial blocks as a new class, and characterizes their source algebra structures.

Contribution

It identifies and characterizes inertial blocks, a new class that includes nilpotent blocks and is closed under taking normal sub-blocks, and determines their source algebra structures.

Findings

Identification of non-nilpotent blocks as extensions of non-nilpotent blocks

Introduction of inertial blocks as a new class including nilpotent blocks

Determination of source algebra structures for these blocks

Abstract

There are normal sub-blocks of nilpotent blocks which are NOT nilpotent or, equivalently, nilpotent extensions of non-nilpotent blocks. In this paper we determine the source algebra structure of the non-nilpotent blocks involved in these situations. Actually, we introduce a new type of blocks - called the inertial blocks - which include the nilpotent blocks and is closed by taking normal sub-blocks.