Journal article
Dihedral blocks with two simple modules
Proceedings of the American Mathematical Society, Vol.138(10), pp.3467-3479
12/05/2009
DOI: 10.1090/S0002-9939-10-10402-X
Abstract
Proc. Amer. Math. Soc. 138 (2010), 3467-3479 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 $k \mathrm{PGL}_2(\mathbb{F}_q)$.
Details
- Title: Subtitle
- Dihedral blocks with two simple modules
- Creators
- Frauke M Bleher
- Resource Type
- Journal article
- Publication Details
- Proceedings of the American Mathematical Society, Vol.138(10), pp.3467-3479
- DOI
- 10.1090/S0002-9939-10-10402-X
- ISSN
- 0002-9939
- eISSN
- 1088-6826
- Language
- English
- Date published
- 12/05/2009
- Academic Unit
- Mathematics
- Record Identifier
- 9983986090702771
Metrics
11 Record Views