Structure of blocks with normal defect and abelian $p'$ inertial quotient

TL;DR

This paper characterizes the structure of blocks with normal defect and abelian inertial quotient in modular representation theory, showing they are matrix algebras over quantized group algebras and providing explicit examples and generators.

Contribution

It introduces a new structural description of such blocks as matrix algebras over quantized group algebras and computes explicit examples with generators and relations.

Findings

Blocks are matrix algebras over quantized semidirect product group algebras.

Explicit generators and relations are provided for specific cases.

Associated graded algebras are analyzed using Jennings--Quillen style theorems.

Abstract

Let $k$ be an algebraically closed field of prime characteristic $p$. Let $kGe$ be a block of a group algebra of a finite group $G$, with normal defect group $P$ and abelian $p'$ inertial quotient $L$. Then we show that $kGe$ is a matrix algebra over a quantised version of the group algebra of a semidirect product of $P$ with a certain subgroup of $L$. To do this, we first examine the associated graded algebra, using a Jennings--Quillen style theorem. As an example, we calculate the associated graded of the basic algebra of the non-principal block in the case of a semidirect product of an extraspecial $p$-group $P$ of exponent $p$ and order $p^3$ with a quaternion group of order eight with the centre acting trivially. In the case $p=3$ we give explicit generators and relations for the basic algebra as a quantised version of $kP$. As a second example, we give explicit generators and relations in the case of a group of shape $2^{1+4}:3^{1+2}$ in characteristic two.