Induced Characters as a Canonical Basis for Modular Representation Rings

This paper rigorously investigates the role of induced characters in constructing a canonical basis for modular representation rings of finite groups. While induced characters play a fundamental role in classical representation theory, their applicability and properties in the modular setting, particularly as a basis for the Green ring, require careful re-evaluation. We define the modular representation ring as a Z-algebra generated by isomorphism classes of indecomposable modules and then demonstrate that a carefully chosen set of induced characters, specifically those derived from indecomposable modules of p-subgroups, forms a canonical Z-basis. The method involves leveraging the properties of vertices and sources, Green's correspondence, and the Brauer character map to establish both the spanning property and linear independence. The canonicity of this basis is justified by its direct connection to the underlying block structure and defect groups of the group, offering a natural and structurally significant framework for understanding the intricacies of modular representations. This work provides a powerful new tool for simplifying calculations and gaining deeper insights into the algebraic structure of modular representation rings, paving the way for further research into p-block theory and related algebraic invariants.