Representation Theory of Finite Groups on the Unified Field

We present a formally verified framework for the representation theory of finite groups on the unified field U. The formalization establishes 44 theorems and 32 hypotheses covering: (1) the fundamental correspondence between irreducible representations and conjugacy classes; (2) Maschke's theorem on complete reducibility; (3) Schur's lemma in both forms (intertwiner vanishing and scalar endomorphisms); (4) character orthogonality relations (row and column); (5) the dimension formula and degree divisibility; (6) Burnside's p^a q^b theorem and counting lemma; (7) Frobenius reciprocity and induced representation degree; and (8) Artin and Brauer induction theorems. A key contribution is the demonstration that the unified field parameter ρ provides uniform bounds on representation complexity: max irreducible degree, number of irreducibles, and character norms are all controlled by ρ. Evolution on U preserves both group order and ρ. The entire development is machine-verified in the Platonic kernel (107 verified declarations, 0 errors), exportable to Lean 4.

Keywords: Representation theory, finite groups, character theory, Schur's lemma, Burnside theorem, Frobenius reciprocity, formal verification, Lean 4

MSC 2020: 20C15, 20C30, 20D10, 68V15