The Law of Forgetting in Moments of Sato-Tate Measures: Invariant Tensors, Maximal Tori, and the Rank Law
Source: hexagonmath.org/abs/2610.00687v1
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: _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…
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