Reduction of characteristic-zero simple modules modulo p

The lattice. Apply the rational Garnir annihilator line of [F2] to the left Murphy cell module and its initial homogeneous cyclic generator: the finite integral row-permutation crossing matrices define a rational simultaneous Garnir-annihilator line, the strict crossing filtration proves standard spanning, and the homogeneous standard basis proves generator-preserving scalar-extension equality, factoring through the finite integer cyclotomic algebra by faithful rational scalar extension [F2]. Then the integer cyclic presentation descent of [F2] identifies the radical by the regular trace pairing, proves rational radical base change, and descends the specified simple head together with the initial-generator image; absolute Schur makes every quotient map to the fixed normalized head differ by a scalar, so the generator image lies on the proved rational line [F1, F2]. Hence $L(\lambda )=H_{\alpha }^{\mathrm{\Lambda }}(\mathbb{Z})v_{\lambda }$ is a finitely generated torsion-free $\mathbb{Z}$-module inside the rational simple, invariant under the integral form, and the $e(i)$-projections cut out lattices in the weight spaces [F1, F2].

Base change and the q-character. The integral orbit is finitely generated and torsion-free and spans the rational head by cyclicity, so it is a full lattice of rank ${\dim }_K\tilde{D}(\lambda )$; word idempotents give graded direct summands whose ranks survive tensoring with $F$, so $J(\lambda )=F{\otimes }_{\mathbb{Z}}L(\lambda )$ is a finite-dimensional graded $H_{\alpha }^{\mathrm{\Lambda }}(F)$-module with the same $q$-character as $\tilde{D}(\lambda )$ and is nonzero [F1, F2]. Any other full homogeneous stable lattice has the same word-component ranks, and the extremal-word character lemma proves $G_0$ injectivity, so the class of the reduction is independent of the lattice [F3, F5].

The transported identification. The restricted Specht classes are an $A$-basis of the upper dual and realize the standard Fock vectors in a field-independent way [F4]; expand both module classes in their restricted Specht bases. The corresponding Specht characters agree across fields by the homogeneous tableau formula [F2]. Character injectivity [F3] makes the expansion coefficients equal, since the restricted Specht characters are linearly independent over $A$. Their field-independent images [F4] therefore give equal images under ${\mathrm{\Psi }}_{\mathrm{c}\mathrm{a}\mathrm{n}}^{\mathrm{H}\mathrm{e}\mathrm{c}\mathrm{k}\mathrm{e}}$. Applying this to $J(\lambda )$ and the normalized characteristic-zero simple and invoking the characteristic-zero upper-basis theorem gives ${\mathrm{\Psi }}_{\mathrm{c}\mathrm{a}\mathrm{n}}^{\mathrm{H}\mathrm{e}\mathrm{c}\mathrm{k}\mathrm{e}}([J(\lambda )])=D_{\lambda }$ in the restricted integral weight-dual lattice $V_A({\mathrm{\Lambda }}_0{)}^{\vee }$, equivalently $[J(\lambda )]=({\mathrm{\Psi }}_{\mathrm{c}\mathrm{a}\mathrm{n}}^{\mathrm{H}\mathrm{e}\mathrm{c}\mathrm{k}\mathrm{e}}{)}^{-1}(D_{\lambda })$ [F4].

Choice. The lattice construction, the base change and the character comparison are finite-dimensional linear algebra over $\mathbb{Z}$, $K$ and $F$; the only choice is inherited through the transported global-basis and duality suppliers.