Asymptotic dimensionality for identical-index GE-PI spaces

Prove the conjectured asymptotic formula for the dimension of the group-equivariant and permutation-invariant space of functions with respect to the irreducible representation D^L of SU(2) or SO(3) when all one-particle indices are identical. Specifically, let l > 0 and N ≥ 1, and set the index tuple to (l, …, l) of length N. Let L be an allowed equivariance order (integer for SO(3) or half-integer/integer for SU(2) as appropriate) and define Var = N l (N + 2l + 1) / 6. Show that, for L much smaller than sqrt(Var), the dimension of the GE-PI space satisfies dim = C(N+2l, 2l) * (2L+1) / (2 sqrt(2π) Var^{3/2}) + O(1/N^{5/2}).

Background

Within the paper’s framework for constructing group-equivariant and permutation-invariant (GE-PI) bases for rotation groups SU(2)/SO(3), the authors obtain exact and asymptotic dimension formulas for several settings. For GE spaces they prove an asymptotic estimate (Proposition on asymptotic dimensionality), while for GE-PI spaces with identical one-particle indices they provide strong numerical evidence for a sharper asymptotic law.

They formulate a conjecture asserting a precise asymptotic for the GE-PI dimension when all indices equal l, involving a binomial prefactor and a Gaussian-type factor depending on Var = N l (N+2l+1)/6, with a remainder O(1/N^{5/2}) in the regime L ≪ sqrt(Var). They note that existing results in the literature imply related limits under different asymptotic regimes, but the stated bound is stronger and remains unproven.