Dihedral blocks with two simple modules

Details Dihedral blocks with two simple modules Dihedral blocks with two simple modules

2009-12-05

arxiv.org/abs/0912.0987v2

Proc. Amer. Math. Soc. 138 (2010), 3467-3479

Mathematics

Group Theory

11 pages, 5 figures. The arguments work also for non-principal blocks. The paper has been changed accordingly; in particular,

Scientific paper

10.1090/S0002-9939-10-10402-X

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

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