Cyclotomic Schur Jantzen sum formula from rim-hook valuations

Filtration and radical. For $n=0$ there is one empty multipartition, $S_{\mathcal{O}}=\mathcal{O}$, ${\mathrm{\Delta }}_{\mathcal{O}}=\mathcal{O}$ with form $1$, every positive filtration term is $0$, and every $J$ coefficient is $0$, so the assertions follow directly. For $n\ge 1$, [F1] gives free standard lattices with nondegenerate generic forms. The Smith reduction of [F4] shows that the defined filtration is by $S_F$-submodules, begins at ${\mathrm{\Delta }}_F(\lambda )$, has first term equal to the form radical, and is zero for all sufficiently large $s$. The rank-one cell actions of the nonzero cellular form force every proper submodule into that radical, so the first term is the module radical.

The auxiliary character identity. Introduce ordered variables for the type-row labels; the auxiliary generating series of flagged semistandard tableaux is the product over physical components $a$ of $s_{{\lambda }^{(a)}}(Y_a)$, where $Y_a$ is the alphabet of type components at least $a$, and its coefficient at an actual multipartition $\nu$ is $K_{\lambda \nu }$. Successive ${\mathrm{\Gamma }}_a$ matrices add the horizontal strips carrying each next label and therefore enumerate these tableaux once; the path/Jacobi–Trudi cancellation proves this character identity locally. No actual Schur weight or module for another composition is needed.

The weight-character reduction. Fix an actual multipartition weight $\nu$. The selfadjoint orthogonal decomposition by $e_{\nu }$ makes the standard weight space orthogonal to every other weight space, so [F4] gives ${\sum }_{s>0}{\dim }_F{e}_{\nu }{\mathrm{\Delta }}_F(\lambda {)}^s=V(\det G_{\lambda \nu })$. By [F2] and additivity of the valuation this is the sum over type-$\nu$ semistandard $T$ of the valuations of their explicit monic norms; empty weight spaces use determinant $1$ and contribute $0$.

The beta form of the hook side. Pad the beta lists to $N>n$. Removing a hook of length $h$ lowers one beta bead by $h$ with sorting sign $(-1{)}^{\mathrm{leg}}$, and adding it raises a bead by $h$ with the corresponding sign; the foot content is the lowered bead position minus $N$ plus $1$, up to the unit foot-residue shift. Thus the full signed equal-length hook side is precisely the finite shifted-beta Jacobi–Trudi polynomial: same-component inward ranges, crosscomponent earlier-to-later ranges, occupied-position zero terms and the unit shift are all covered by [F3]. Its coefficient at $\nu$ is ${\sum }_{\mu }{J}_{\lambda \mu }{K}_{\mu \nu }$.

Telescoping the marked-strip commutator. Apply the complete local identity [F3] to the product of successive type-row $\mathrm{\Gamma }$ matrices and telescope $J\mathrm{\Gamma }-\mathrm{\Gamma }J$, starting from the empty shape where $J_0=0$. Each commutator inserts exactly the valuation of the strip factor at that label position, while all other $\mathrm{\Gamma }$ factors retain the same tableau chain; summing the inserted positions gives the total valuation of the product of the factors over the chain, hence of the monic norm by [F2] and [F3]. The left endpoint of the telescope is exactly the auxiliary polynomial of step 2.2, so at the actual multipartition coefficient $\nu$, ${\nu }_{\pi }(\det G_{\lambda \nu })={\sum }_{\mu }{J}_{\lambda \mu }{K}_{\mu \nu }$. Every auxiliary rank keeps the final-rank-$n$ parameters $U_a=Q_a{t}^{na}$, with no rank-dependent deformation substituted during the telescope.

Exactness of the valuations. For a nonzero difference $A{t}^u-B{t}^v$, unequal constants give valuation $0$, and equal constants with $u-v\mathrm{\ne }0$ give valuation $1$ in characteristic zero or $p^{v_p(\mathrm{\mid }u-v\mathrm{\mid })}$ in characteristic $p$ by [F5]. Artificial padded symbols in the crosscomponent minor proofs cancel formally before evaluation, and the retained terms are actual valid hook or node pairs whose exponent bound makes their differences nonzero in $K$; hence step 3.1 is valid over every $F$ and retains the higher positive-characteristic multiplicities.

The integer Grothendieck class identity. Combining steps 2.1 and 3.1 shows that the actual multipartition-weight character of ${\sum }_{s>0}[{\mathrm{\Delta }}_F(\lambda {)}^s]$ equals that of ${\sum }_{\mu }{J}_{\lambda \mu }[{\mathrm{\Delta }}_F(\mu )]$. Weight projection is exact, so these are additive characters on the integer $K_0$ group [F4], and the standard classes form its $\mathbb{Z}$-basis by [F1]; their multipartition-weight matrix is unitriangular, hence invertible over $\mathbb{Z}$. A zero difference of weight characters therefore has all standard coefficients zero, which proves the asserted integer $K_0$ equality, with no rationalization and no Schur-functor injectivity claim.