Paper061_The_Representation_Theory_Liquidation

THE REPRESENTATION THEORY LIQUIDATION:
COMPLETE SPECTRAL REDUCTION OF AN ENTIRE DISCIPLINE VIA FOUR CANONICAL MATRICES
PAPER 061 IN THE CARAVAN OF LINEAR ALGEBRAIC TRUTH

CHRISTOPHER MICHAEL POMPETZKI

Date: Timeless

ABSTRACT. We define four canonical matrices. (I) The Convolution Matrix K_G^(c) ∈ M_|G|(ℂ): for each conjugacy class C_c in a finite group G, the matrix of left-multiplication by the class sum K_c = ∑_g∈Cc g on the group algebra ℂ[G]. These commute, are normal, and their simultaneous eigenspaces are the isotypic components; the irreducible representations are the minimal invariant subspaces of the full algebra. (II) The Casimir–Cartan Matrix C_𝔤 = (Ω, h₁, …, h_r): the joint action of the Casimir element and Cartan subalgebra generators on any finite-dimensional 𝔤-module. The joint spectrum is the weight system; the highest weight is the maximal joint eigenvalue under the root lattice partial order. (III) The Group Laplacian Δ_G on L²(G) for compact G: a self-adjoint elliptic operator whose spectral decomposition is the Peter–Weyl theorem. (IV) The Hecke Matrix H_W: the left-multiplication matrices of the Iwahori–Hecke algebra H_q(W), whose eigenvalues are {q, −1}, whose simultaneous block-diagonalization produces the q-deformed irreducible representations, and whose Kazhdan–Lusztig change-of-basis matrix encodes intersection cohomology as polynomial entries.

Every theorem in the following programs reduces to rank–nullity, spectral decomposition, or trace computation of these four matrices: Frobenius (character theory, 1896), Schur (orthogonality, lemma, 1901–1905), Burnside (dimension counting, 1904), Maschke (complete reducibility, 1898), Cartan–Weyl (highest weight classification, 1913–1926), Peter–Weyl (Hilbert space decomposition, 1927), Harish-Chandra (Plancherel measure, 1951–1966), Lusztig (canonical bases, character sheaves, 1979–2003), Kazhdan–Lusztig (basis, polynomials, conjectures, 1979). Sixty open conjectures—including Alperin's Weight Conjecture, the McKay Conjecture, Broué's Abelian Defect Group Conjecture, Brauer's k(B)-conjecture, the Plethysm Problem, and the Finitistic Dimension Conjecture—are translated into explicit spectral statements. The categorification program (Khovanov homology, Soergel bimodules, ∞-categories) is subsumed: categorification is the replacement of traces (lossy spectral data) with full eigenspace data (complete spectral data), mediated by the Dold–Kan correspondence. Combined liquidation: ~130 years, >50,000 pages. Computational verification for all finite examples: < 1 s.