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

TL;DR

This paper uses Kuelshammer ideals to construct the first known examples of blocks with dihedral defect groups where the scalar in relations is 1, resolving a longstanding open problem in modular representation theory.

Contribution

It introduces the first examples of blocks with dihedral defect groups having scalar 1, using Kuelshammer ideals to address a key open question.

Findings

Provided explicit examples of blocks with scalar 1

Used Kuelshammer ideals to distinguish scalar cases

Resolved a longstanding open problem in the field

Abstract

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 PGL_2(F_q), the projective general linear group of 2x2-matrices with entries in the finite field F_q, where q=p^n\equiv \pm 1 mod 8, with p an odd prime number.