The q-Schaper virtual radical-layer formula

Work on a fixed weight space of the lifted free Weyl module. Every hook quantum integer is nonzero because its index is at most $n$ and $[n]{!}_{\widehat{q}}\mathrm{\ne }0$ in the domain. The balanced Gram formula [F1] gives its determinant, and the unit power of $\widehat{q}$ has valuation zero. The Jantzen Gram-valuation lemma [F2] thus gives the sum of positive filtration-submodule dimensions as ${\sum }_{(a,b),(a,c),b<c}({\nu }_{\pi }([h_{ab}{]}_{\widehat{q}})-{\nu }_{\pi }([h_{ac}{]}_{\widehat{q}}))d_{\mu }(\beta +h_{ac}{\mathbf{e}}_b-h_{ac}{\mathbf{e}}_c)\mathrm{.}$ The signed number is exactly the weight dimension of the corresponding balanced virtual Weyl symbol, including zero and cutoff cases.

Apply step 1.1 to every weight. The Weyl weight-character matrix is unitriangular, so [F3] detects equality of Grothendieck classes and proves part 1. These are classes of the filtration submodules $W_F^{\lambda }(i)$, not their successive quotients and not simply the radical class.

Establish filtration transport before applying the exact functor. On the lifted Weyl module $W$, the identity $e^{\ast }=e$ makes $We\perp W(1-e)$. For $x\in We$ and every $i\ge 0$, pairing $x$ with all of $W$ is divisible by ${\pi }^i$ exactly when its pairing with $We$ is. Thus $W(i)e=(We)(i)$. Reducing modulo $\pi$ commutes with the direct summand projection, giving $W_F(i)e=(We{)}_F(i)$. The actual corner isomorphism and factor-$1$ form identity [F5] identify the latter with the Murphy Specht filtration. Applying the exact functor to the proved Weyl-class identity therefore transports each submodule and every signed term, proving part 2 even when a reduced cell form is identically zero. For the ordinary convention, apply the same $\theta$ to both modules and conjugate the labels; the independently proved Laurent-unit form factor preserves these divisibility filtrations as well.

Explain the valid multiplicity information. The Weyl form has nonzero simple head, $W_F(1)=\mathrm{rad}{W}_F$, and $W_F(i)\subseteq \mathrm{rad}{W}_F$ for $i>0$. Consequently for $\nu \mathrm{\ne }\lambda$ its actual multiplicity in $W_F$ is its radical multiplicity, bounded above by its multiplicity in the whole positive sum. This whole-sum multiplicity is positive if and only if that factor occurs in the radical, since the first submodule is the radical and later ones lie inside it. No such conclusion is drawn from an individual positive or negative Weyl coefficient before cancellation. All claims follow from steps 1.1-3.1 and this submodule inclusion.