Kuelshammer理想与带二面体亏群块的标量问题
Kuelshammer ideals and the scalar problem for blocks with dihedral defect groups
摘要：
在目前的经典作品中，K。Erdmann分类块的有限群与二面角亏损群（和更普遍的代数二面角型）森田等价。在用箭图和关系对这类具有两个单模的代数进行的显式描述中，一些关于关系中标量的微妙问题仍然没有解决。特别是，对于二面角的情况，有限群的块是否可以同时出现在可能的标量0和1的情况下，这是一个长期存在的开放问题。在本文中，使用Kuelshammer理想（a.k.a.广义Reynolds理想），我们提供了标量为1的块的第一个例子，从而肯定地回答了上述问题。我们的例子是PGL_2（F_q）的主块，PGL_2（F_q）是有限域F_q中的2 × 2-矩阵的一般射影线性群，其中q=p^n \equiv \pm 1 mod 8，p是奇素数。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
