Differential Algebraic Closure Framework for Modular Representation Theory:Explicit Constructions and Combinatorial Structures

This paper establishes a rigorous differential algebraic framework for the explicit construction of modular representations and their Brauer characters. We introduce the modular representation closure KG,p, a differentially closed field extension that systematically contains solutions to a broad class of modular representation problems. Within this closure, we derive explicit analytic expressions for Brauer characters, decomposition numbers, and modular representations of finite groups in characteristic p.We provide complete constructive foundations with rigorous proofs, derive combinatorial expressions with explicit connections to symmetric function theory in characteristic p, present detailed algorithmic implementations with complexity analysis, and situate our results within classical modular representation theory. Numerical experiments across diverse finite groups confirm spectral convergence with maximum residual errors of O(10−15) in exact arithmetic and demonstrate the necessity of combinatorial corrections for higher-dimensional modular representations. This work establishes that explicit analytic expressions exist in KG,p for a significant class of modular representation problems, providing a new algebraic perspective on modular-theoretic solvability while maintaining consistency with classical complexity results. The framework is limited to finite groups satisfying specific technical conditions, including the requirement that p divides |G| and that the Brauer character inner product matrix is non-singular. We further discuss the treatment of singular cases via regularization techniques.