Dihedral blocks with two simple modules

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 $c$ whose value 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 $k \mathrm {PGL}_2(\mathbb {F}_q)$.