Külshammer ideals and the scalar problem for blocks with dihedral defect groups

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 Külshammer 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 2×2-matrices with entries in the finite field Fq, where q = p n ≡ ±1 mod 8 with p an odd prime number. Mathematics Subject Classification (2000). Primary 20C05; secondary 16G10, 18E30, 20C20 .