Asymptotic behavior of modular representations over abelian p-groups

Referee: [Embedding and asymptotic calculation] The non-recursiveness claim rests on the embedding of the representation ring into the real algebra of functions (described in the second paragraph of the abstract and the corresponding section). It is necessary to verify explicitly that this map is injective on the relevant elements and that extraction of the core submodule commutes with the functional representation in a way that preserves the distinction between recursive and non-recursive integer sequences; otherwise the derived asymptotic C γ^n n^α could be an artifact that does not rule out eventual recursiveness.

Authors: We agree that explicit verification strengthens the argument. In the revised manuscript we will insert a new lemma (in the section on the embedding) proving that the ring homomorphism φ: R(C_{p}) → A (A the real algebra of functions) is injective on the subring generated by the classes of the syzygies Ω^k(k) and cosyzygies Ω^{-k}(k) of the trivial module. Injectivity follows because φ sends the Z-basis of indecomposable modules to a set of functions that are linearly independent over R, which can be checked by evaluating at sufficiently many points on the unit circle corresponding to the group elements and using the distinct growth rates of their dimensions. The core extraction commutes with φ because the projective summands are precisely the kernel of the augmentation map on the representation ring, and φ maps this ideal to the subspace of functions whose asymptotic growth is strictly larger than that of any non-projective module; thus dim(core(M)) is recovered exactly as the constant term (or appropriate coefficient) in the functional expansion. Consequently the asymptotic C γ^n n^α computed in the function algebra is identical to the integer sequence of core dimensions. Since any eventually recursive sequence (in the sense of Benson–Symonds) must be D-finite and therefore cannot admit an asymptotic of this form with γ transcendental or irrational algebraic (by standard results on singularity analysis), the non-recursiveness conclusion is preserved. We will add the lemma and a short reference to the relevant analytic-combinatorics fact. revision: yes