Dihedral blocks with two simple modules

TL;DR

This paper determines the parameter c in the description of dihedral blocks with two simple modules over an algebraically closed field of characteristic 2, linking it to central extensions and quaternion defect groups.

Contribution

It shows that the parameter c=0 for certain blocks, especially when G is PGL_2 over a finite field, using central extensions and quaternion defect groups.

Findings

Parameter c=0 for blocks with dihedral defect groups under specified conditions.

Identifies cases where the block is contained in a central extension with quaternion defect groups.

Provides a criterion linking the parameter c to the existence of certain central extensions.

Abstract

Let $k$ be an algebraically closed field of characteristic 2, and let $G$ be a finite group. Suppose $B$ is a block of $kG$ with dihedral defect groups such that there are precisely two isomorphism classes of simple $B$-modules. The description by Erdmann of the quiver and relations of the basic algebra of $B$ is usually only given up to a certain parameter $c$ which is either 0 or 1. In this article, we show that $c=0$ if there exists a central extension $\hat{G}$ of $G$ by a group of order 2 together with a block $\hat{B}$ of $k\hat{G}$ with generalized quaternion defect groups such that $B$ is contained in the image of $\hat{B}$ under the natural surjection from $k\hat{G}$ onto $kG$. As a special case, we obtain that $c=0$ if $G=\mathrm{PGL}_2(\mathbb{F}_q)$ for some odd prime power $q$ and $B$ is the principal block of $k \mathrm{PGL}_2(\mathbb{F}_q)$.