The Canonical Inductive Character Ring of Modular Representations

This paper introduces and rigorously constructs the concept of a canonical inductive character ring for modular representations of finite groups. While classical representation theory over fields of characteristic zero benefits from the well-established character ring and Brauer's induction theorem, modular representation theory, dealing with fields of positive characteristic $p$ dividing the group order, presents significant challenges due to the non-semisimple nature of group algebras. We formalize a construction that integrates induction and restriction functors, Mackey's decomposition theorem, and the Grothendieck group framework to define a novel ring structure, denoted as $R_k^{ind}(G)$. This ring captures the combinatorial and algebraic properties of modular representations under induction and restriction. We detail its additive and multiplicative structures, identifying a natural basis derived from indecomposable modules. The canonicity is established by demonstrating its universal properties and its relationship to the Green ring and the Burnside ring. The implications for understanding the intricate structure of modular representations, especially in relation to block theory and fusion systems, are discussed, proposing new avenues for research within this complex field.