The self-dual indecomposable modules in blocks with cyclic defect groups

TL;DR

This paper classifies self-dual indecomposable modules in blocks with cyclic defect groups for finite groups, identifying their symplectic or orthogonal nature, advancing understanding of module structures in modular representation theory.

Contribution

It provides a complete description of self-dual indecomposable modules in blocks with cyclic defect groups and determines their symplectic or orthogonal types.

Findings

Classification of self-dual indecomposable modules

Determination of symplectic or orthogonal nature

Enhanced understanding of module structures in cyclic defect blocks

Abstract

Let $p$ be an odd prime and let $\mathbf{B}$ be a $p$-block of a finite group, such that $\mathbf{B}$ has cyclic defect groups. We describe the self-dual indecomposable $\mathbf{B}$-modules and for each such module determine whether it is symplectic or orthogonal.