math.RT — Representation Theory
The Law of Forgetting in Moments of Sato-Tate Measures: Invariant Tensors, Maximal Tori, and the Rank Law
For a compact group G with a finite-dimensional representation V, the moments of the character are, by a one-line argument, the dimensions of the invariant tensor spaces: <chi_V^m>_G = dim(V^{otimes m})^G. We call the passage from averaging over G to averaging over a maximal torus T the "law of forgetting": T-invariants only require weights to sum to zero, while G-invariants impose the additional invariance constraint. For the standard representation of SU(2) the two ladders differ by the exact factor F_k = binom(2k,k)/Cat(k) = k+1 (proved via the Temperley-Lieb non-crossing matching basis). The general forgetting factor admits a complete asymptotic F_k = c_V * k^s * (1 + gamma_V/k + O(1/k^2)), where s is the number of positive roots; both constants are given in closed form: c_V = |W| / (2^s <h_d>), a ratio of the Weyl denominator's lowest homogeneous term at the saddle to its Gaussian average, and gamma_V = (epsilon_T - epsilon_G)/2, whose inputs are the quartic term of -log|chi/dim V| and the next term of the Weyl denominator. The exponent s = (dim G - rank G)/2 is verified across six compact Lie groups (A_1, A_2, B_2, C_2, C_3, G_2) through the theorem F(1) = dim V (proved: weight multiplicities one, Schur's lemma), and the universal U(N) family gives the closed form F(N,k) ~ R_N k^{N choose 2} with R_N = (2/N)^{N choose 2} / G(N+1), G the Barnes function, identified with the reciprocal of a Mehta constant. On the arithmetic side, the master identity is verified on elliptic-curve Frobenius data; the two-copy test E x E versus E_1 x E_2 reads Schur's lemma directly from the second moment (a single number, 4 versus 2, separates isogenous from distinct copies); and the (lambda,lambda) pairing identity of Howe duality is identified with the Dirichlet coefficients of the Rankin-Selberg L-function, the N <-> k symmetry reading as convolution symmetry. A parallel shape fingerprint W = E[Y^2]/E[Y]^2 bypasses the heavy-tail divergence of the Jensen-gap functional J = E[Y]E[1/Y], with closed forms for twelve distribution families. All conclusions carry explicit status labels (proved / numerically anchored / identified), and every error found during the research is recorded with its correction in an appendix.