Canonical bases do not solve arbitrary positive-characteristic decomposition numbers

Assume the Axiom of Choice (The Axiom of Choice), inherited through the cited crystal-label and global-basis results.

The characteristic-zero theorems of this page (Ariki's canonical basis theorem in characteristic zero, Graded decomposition numbers equal canonical-basis coefficients in characteristic zero) compute graded decomposition numbers of graded Specht modules as global-basis coefficients. In characteristic $p>0$ the factorization Positive-characteristic decomposition polynomials factor through adjustment reduces the graded decomposition matrix to a product of the adjustment matrix and the SAME global-basis coefficients, but the adjustment matrix is generally nontrivial: the theorem computes no adjustment entry and asserts no identity adjustment matrix in general. Consequently the characteristic-zero canonical-basis theory does not by itself solve the decomposition-number problem in arbitrary positive characteristic; the general problem retains the boundary recorded in Triangularity does not compute every modular decomposition number and General type-A decomposition numbers remain unsolved. This remark records the boundary; it carries no proof obligation and uses no additional choice.

Source comparison. Nonidentity is a separate literature fact, not a consequence of leaving entries undetermined. Fayers, Weight two blocks of Iwahori–Hecke algebras in characteristic two, Corollary 2.4 (printed p. 9), gives an off-diagonal ordinary adjustment entry $1$ at $e=3$, characteristic $2$, rank $8$, with ordinary row label $(3,2,2,1)$ and column label $(4,4)$. In the source's Section 1.4 notation, the core $(1,1)$ has bottom bead positions $(0,4,5)$, so ${}_0{B}_1=0$, ${}_1{B}_2=1$; the two labels are $\{2,3\}$ and $\{3,5\}$, respectively, exactly the off-diagonal case of the corollary. Conjugating labels and applying the same sign pullback to both sides gives $a_{(2,2,2,2),(4,3,1)}(1)=1$ in this page's restricted convention. The parameter is a primitive third root, so $\xi \mathrm{\ne }1$ and $2\nmid 3$; this witness lies in branch (CP). It supports the boundary observation and is not used as a supplier for the factorization theorem.

The two cited characteristic-zero theorems determine exactly the graded decomposition numbers in branch (C0), where the transition polynomials $d_{\lambda ,\mu }(q)$ are the global-basis coefficients. The positive-characteristic factorization leaves the adjustment coefficients $a_{\lambda ,\nu }(q)$ undetermined beyond bar-invariance, nonnegativity, the diagonal $1$ and the dominance support recorded in The graded adjustment polynomials in positive characteristic; consequently the "same" global-basis coefficients are not by themselves the answer in branch (CP). The source's announced specialization of the adjustment matrix to the matrix of James concerns the degenerate parameter and lies outside this page's nondegenerate regime, as recorded in the batch coverage. No general positive-characteristic formula is claimed, and this remark is not consumed to prove any replacement statement. The boundary it states is the same one recorded in the two unsolved-decomposition-number remarks cited in the statement.