2008/09/08 by Thorsten Holm, Holm, Thorsten, Guodong Zhou +1
Mathematics · #16G10 #18E30 #20C05 #20C20 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #math.GR #math.RT #msc:16G10 #msc:18E30 #msc:20C05 #msc:20C20
paper · pdf · doi:10.48550/arxiv.0809.1363
23 pages
arxiv created 2008/09/08 · openalex publication_date 2008/09/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In by now classical work, K. Erdmann classified blocks of finite groups with dihedral defect groups (and more generally algebras of dihedral type) up to Morita equivalence. In the explicit description by quivers and relations of such algebras with two simple modules, several subtle problems about scalars occurring in relations remained unresolved. In particular, for the dihedral case it is a longstanding open question whether blocks of finite groups can occur for both possible scalars 0 and 1. In this article, using Kuelshammer ideals (a.k.a. generalized Reynolds ideals), we provide the first examples of blocks where the scalar is 1, thus answering the above question to the affirmative. Our examples are the principal blocks of PGL2(Fq), the projective general linear group of 2x2-matrices with entries in the finite field Fq, where q=pn≡ ± 1 mod 8, with p an odd prime number.