Cyclotomic Jantzen equivalence is Schur and Hecke block linkage

Standard classes form a $\mathbb{Z}$-basis. The matrix of standard classes to simple classes is triangular with diagonal one by [F1], hence [F4] makes the classes $[\mathrm{\Delta }(\lambda )]$ a $\mathbb{Z}$-basis of the integer Grothendieck group of the Schur cover, and the classes of the block projections are independent as well.

Nonzero $J$-coefficients link the same block. Suppose $J_{\lambda \mu }\mathrm{\ne }0$ and project the Jantzen sum identity of [F2] to all blocks other than the block of $\mathrm{\Delta }(\lambda )$. Its left side is zero because each ${\mathrm{\Delta }}_F(\lambda {)}^i$ is a submodule of the block-pure standard $\mathrm{\Delta }(\lambda )$; its right side is a $\mathbb{Z}$-linear combination of standard classes lying in those other blocks, and the coefficient of $[\mathrm{\Delta }(\mu )]$ in it is exactly $J_{\lambda \mu }\mathrm{\ne }0$. Basis independence from step 1.1 forces that coefficient to vanish, a contradiction unless $\mathrm{\Delta }(\mu )$ lies in the block of $\mathrm{\Delta }(\lambda )$.

Nonzero decomposition numbers force a $J$-chain. Put $d_{\lambda \mu }=[\mathrm{\Delta }(\lambda ):L(\mu )]$. For $\lambda \mathrm{\ne }\mu$, the radical $\mathrm{\Delta }(\lambda {)}^1=\mathrm{rad}\mathrm{\Delta }(\lambda )$ contains every composition factor $L(\mu )$ of $\mathrm{\Delta }(\lambda )$, and all higher filtration terms lie in that radical, so the coefficient of $[L(\mu )]$ in the left side of the Jantzen identity is nonzero exactly when $d_{\lambda \mu }\mathrm{\ne }0$; on the right it is ${\sum }_{\nu }{J}_{\lambda \nu }{d}_{\nu \mu }$. If $d_{\lambda \mu }\mathrm{\ne }0$, this integer is nonzero, so some $\nu$ satisfies $J_{\lambda \nu }\mathrm{\ne }0$ and $d_{\nu \mu }\mathrm{\ne }0$, with $\lambda ⊳\nu ⊵\mu$. Repeating strictly decreases the index in the finite dominance poset until $\nu =\mu$, so $\lambda {\sim }_J\mu$. Consequently any two standards sharing a simple factor are both $J$-connected to that factor's index, and therefore to each other.

Blocks and Hecke linkage. By [F1], two standards share a Schur block exactly when they are linked by a finite chain of standards whose neighbouring terms share a simple factor. Step 2.2 converts each such shared factor into a $J$-edge to the common index, and step 2.1 shows every $J$-edge lies inside one block; hence sharing a Schur block is equivalent to ${\sim }_J$-connectivity. Applying the established Schur/Specht block correspondence of [F1] transfers this equivalence to the Murphy Specht modules and the Hecke algebra. In rank $0$ there is one empty standard, all positive filtration terms vanish, and the single empty class is its own Jantzen class; the argument never reduces $J$ coefficients modulo the field characteristic.