Dihedral blocks with two simple modules
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- by Frauke M. Bleher PDF
- Proc. Amer. Math. Soc. 138 (2010), 3467-3479
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 $k \mathrm {PGL}_2(\mathbb {F}_q)$.References
- Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR 1314422, DOI 10.1017/CBO9780511623608
- Frauke M. Bleher and Ted Chinburg, Universal deformation rings and cyclic blocks, Math. Ann. 318 (2000), no. 4, 805–836. MR 1802512, DOI 10.1007/s002080000148
- F. M. Bleher, J. B. Froelich and G. Llosent, Universal deformation rings and dihedral blocks with two simple modules. Preprint, 2009.
- Richard Brauer, On $2$-blocks with dihedral defect groups, Symposia Mathematica, Vol. XIII (Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972) Academic Press, London, 1974, pp. 367–393. MR 0354838
- Richard Brauer and Michio Suzuki, On finite groups of even order whose $2$-Sylow group is a quaternion group, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 1757–1759. MR 109846, DOI 10.1073/pnas.45.12.1757
- M. C. R. Butler and Claus Michael Ringel, Auslander-Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra 15 (1987), no. 1-2, 145–179. MR 876976, DOI 10.1080/00927878708823416
- Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
- Karin Erdmann, Blocks of tame representation type and related algebras, Lecture Notes in Mathematics, vol. 1428, Springer-Verlag, Berlin, 1990. MR 1064107, DOI 10.1007/BFb0084003
- Karin Erdmann, On $2$-modular representations of $\textrm {GU}_2(q),\;q\equiv 3\bmod 4$, Comm. Algebra 20 (1992), no. 12, 3479–3502. MR 1191964, DOI 10.1080/00927879208824526
- B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703, DOI 10.1007/978-3-642-64981-3
Additional Information
- Frauke M. Bleher
- Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242-1419
- Email: fbleher@math.uiowa.edu
- Received by editor(s): July 18, 2009
- Received by editor(s) in revised form: August 21, 2009, and January 6, 2010
- Published electronically: April 27, 2010
- Additional Notes: The author was supported in part by NSF Grant DMS06-51332.
- Communicated by: Ted Chinburg
- © Copyright 2010 Frauke M. Bleher
- Journal: Proc. Amer. Math. Soc. 138 (2010), 3467-3479
- MSC (2010): Primary 20C05; Secondary 16G20
- DOI: https://doi.org/10.1090/S0002-9939-10-10402-X
- MathSciNet review: 2661547