This monograph systematically extends the core methodology of Operational Mathematics — the extension of the repetition count of basic operations from natural numbers to integers, rational numbers, real numbers, complex numbers and even infinity — to a wide class of fundamental operations arising in representation theory. These representation operations include linear actions (representation operators ρ(g)) and their inverses ρ(g)−1 = ρ(g−1); tensor product operations V ⊗ − and their virtual inverses; direct sum operations V ⊕ − and their decomposition inverses; duality operations V 7 → V ∗ (involutive); restriction ResGH and induction IndGH (adjoint pair); Hom functors Hom(V, −) and their adjoints − ⊗ V ; and Ext functors Exti(V, −) and their derived analogues. For each family we treat the iteration count as an independent complex parameter t and define a rigorous analytic family F ◦tA satisfying F ◦tA ◦ F ◦sA = F ◦(t+s)A , coinciding with ordinary k-fold iteration for integer t = k, and depending analytically on t (meromorphically when zero eigenvalues occur). The construction uses spectral decomposition, λ-ring structures, Adams operations, Mackey functor algebra, sinc interpolation, weighted parametrisation, and extensions to super representations and categorical duality. A unified axiom system A1–A9 is established. Fractional tensor powers V ⊗t are defined via the character formula χV ⊗t (g) = Pi λi(g)t; we prove that this definition satisfies the semigroup property if and only if dim V = 1; for higher dimensions the multiplication semigroup property is impossible, and we provide an additive semigroup construction instead (Theorem 5.5.1). Fractional direct sums V ⊕t are defined by χV ⊕t = tχV and yield an entire family. Fractional dual powers Dt are defined via the spectral decomposition of the involution D (eigenvalues ±1), giving Dt = P+ + (−1)tP−. Weighted parametrisation introduces a critical weight wc = 1/ρ(FA); for |w| wc exponential blow-up, and at |w| = wc polynomial growth or oscillation. Fractional Hom and Ext are defined via Hom(V ⊗t, W ) and sinc interpolation, respectively; the fractional long exact sequence for Ext is proved in full detail (Theorem 9.2). The representation zeta functions are studied: ζ⊕,V (s) = (dim V )−sζ(s) is directly linked to the Riemann hypothesis, while the local tensor product zeta function ζ⊗,V,g(s) =Piλi(g)−s1−λi(g)−s is meromorphic. Numerical algorithms for finite groups are given with complexity O(|G|3); explicit verifications for S3, A4, Z/3Z and SU (2) are provided. Applications to quantum groups, topological quantum field theory, the Langlands program, and the fractional quantum Hall effect are discussed. All open problems are treated rigorously: solved ones are moved to main theorems, the rest are clearly stated with partial results.
Liu S (Wed,) studied this question.