HEXAGONbeta

The Law of Forgetting in Moments of Sato-Tate Measures: Invariant Tensors, Maximal Tori, and the Rank Law

Primarily human-written textHuman understanding: all partsmath.RT — Representation Theorymath.NT — Number Theory

Contributed by Zeng Chunping ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

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.

Tools used

Magma
Magma

References

  1. P. Diaconis, M. Shahshahani, On the eigenvalues of random matrices , J. Appl.\ Probab.\ 31A (1994), 49--62.
  2. E. Rains, High powers of random elements of compact Lie groups , Probab.\ Theory Related Fields 107 (1997), 219--241.
  3. M. L. Mehta, Random Matrices , 3rd ed., Elsevier, 2004.
  4. R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond , Israel Math.\ Conf.\ Proc.\ 8 (1995), 1--182.
  5. T. Clausen, Über die Fälle, wenn die Reihe y=1+α⋅β1⋅γx+⋯y=1+\frac{\alpha\cdot\beta}{1\cdot\gamma}x+\cdots ein Quadrat von der Form y=1+α′β′γ′1⋅δ′ϵ′x+⋯y=1+\frac{\alpha'\beta'\gamma'}{1\cdot\delta'\epsilon'}x+\cdots hat , J.\ Reine Angew.\ Math.\ 3 (1828), 89--91.
  6. G. Domb, On the theory of cooperative phenomena in crystals , Adv.\ Phys.\ 9 (1960), 149--361.
  7. G. Césnavicij̄us, and BSD reference literature: J. Silverman, The Arithmetic of Elliptic Curves , 2nd ed., Springer, 2009.
  8. J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves , Springer, 1994 (Sato--Tate conjecture, moments).
  9. H. Kesten, Symmetric random walks on groups , Trans.\ Amer.\ Math.\ Soc.\ 92 (1959), 336--354; B. McKay, The expected eigenvalue distribution of a large regular graph , Linear Algebra Appl.\ 40 (1981), 203--216.

Version history

  1. v1 · 16 pagesSubmitted by Zeng ChunpingInitial depositCurrentOct 30, 2026