General type-A decomposition numbers remain unsolved

The two preceding constraints on the decomposition matrix of $H_{F,q}(S_n)$ are real but limited, and neither solves the general problem.

(i) Dominance unitriangularity (The type-A Hecke decomposition matrix is dominance unitriangular) determines only the diagonal entries $d_{\mu \mu }=1$ and the vanishing region $d_{\lambda \mu }\mathrm{\ne }0\Rightarrow \mu ⊵\lambda$; it gives no value for an off-diagonal entry.

(ii) The q-Schaper/Jantzen sum formula (The q-Schaper virtual radical-layer formula) is an identity for the sum of the positive Jantzen filtration submodules. Its individual signed hook terms are not multiplicity bounds. After the whole virtual sum is expanded in simple classes and all cancellations are performed, its nonnegative simple coefficients bound the corresponding radical composition multiplicities from above; it neither determines a general multiplicity nor produces a positive combinatorial rule.

Special coefficient regimes and low-rank calculations determine additional entries (canonically based, weight-one, and small-$n$ cases), and the classification of irreducible Specht modules for symmetric groups in prime characteristic is known (Fayers, Introduction, printed p. 2 of the cited 2012 paper). That classification is distinct from computing all composition multiplicities. No general closed formula for the decomposition numbers of $S_n$ in a fixed positive characteristic is claimed anywhere on this page. This remark records the boundary; it carries no proof obligation and no choice principle.

The two cited results are the constraints proved on this page: unitriangularity is a vanishing statement about the dominance order, and the q-Schaper formula expresses the sum of the positive Jantzen filtration submodules as a virtual combination of Weyl symbols whose coefficients are $\pi$-adic valuations of hook q-integers. Neither statement computes an individual off-diagonal decomposition number in general, and the remark is not used as a supplier to prove any replacement claim.