Graded Specht classes realize the standard Fock vectors

Evaluate on projective words. For a residue word $\mathbf{i}=(i_1,\dots ,i_d)$ of content $\alpha$, put $P(\mathbf{i})=R_{\alpha }^{\mathrm{\Lambda }}e(\mathbf{i})$ and transport it to the Hecke block. Appending the strands successively realizes it as $F_{i_d}^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}\cdots F_{i_1}^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}$ of the vacuum projective, so [F1] sends its class to $F_{i_d}\cdots F_{i_1}{v}_{{\mathrm{\Lambda }}_0}$. The generator-fixing anti-involution identifies $P(\mathbf{i}{)}^{\psi }$ with the right ideal $e(\mathbf{i})R_{\alpha }^{\mathrm{\Lambda }}$, and multiplication gives $e(\mathbf{i})R_{\alpha }^{\mathrm{\Lambda }}{\otimes }_{R_{\alpha }^{\mathrm{\Lambda }}}S(\mu )\cong e(\mathbf{i})S(\mu )$. Thus the defining bilinear tensor pairing, without any Hom-pairing bar on a shifted first argument, gives ${\mathrm{\Psi }}_{\mathrm{c}\mathrm{a}\mathrm{n}}^{\mathrm{H}\mathrm{e}\mathrm{c}\mathrm{k}\mathrm{e}}([S(\mu )])(F_{i_d}\cdots F_{i_1}{v}_{{\mathrm{\Lambda }}_0})={\sum }_{T:i_T=\mathbf{i}}{q}^{\mathrm{deg}T}$.

The identical standard-coordinate evaluation. If $A$ is a removable $i$-node of $\mu$ and $\alpha =\mathrm{cont}(\mu )$, the Fock formula [F2] gives its removal coefficient $q^{-d^A(\mu \setminus A)}$. The operator $K_i$ on the resulting weight ${\mathrm{\Lambda }}_0-\alpha +{\alpha }_i$ multiplies by $q^{h_i(\alpha )+2}$. Contravariance of [F3] therefore gives the coefficient of $M_{\mu }$ in $F_i{M}_{\mu \setminus A}$ as $q^{h_i(\alpha )+1-d^A(\mu \setminus A)}=q^{d_A(\mu )}$, by [F4]. Iterating additions from the vacuum enumerates precisely the standard tableaux of shape $\mu$ with successive residues $\mathbf{i}$: at each stage the occupied nodes form a partition, and conversely its increasing filling records the addition path. The product of the node factors is $q^{{\sum }_A{d}_A}=q^{\mathrm{deg}T}$ by the degree recursion. Hence the coordinate functional ${\mathrm{Std}}_{\mu }(x)=B(x,M_{\mu })$ satisfies ${\mathrm{Std}}_{\mu }(F_{i_d}\cdots F_{i_1}{v}_{{\mathrm{\Lambda }}_0})={\sum }_{T:i_T=\mathbf{i}}{q}^{\mathrm{deg}T}$, exactly step 1.1.

Equality in the restricted integral dual. The basic highest-weight module over $\mathbb{Q}(q)$ is generated from the vacuum by the negative generators, so its vectors $F_{i_d}\cdots F_{i_1}{v}_{{\mathrm{\Lambda }}_0}$ span every weight space; no assertion that ordinary words alone span the divided-power lattice over $A$ is needed. Steps 1.1–2.1 make the two functionals equal on this field spanning family. Both lie in the restricted integral weight dual by [F1] and the coordinate-functional construction of [F2]; their field equality therefore is equality in that lattice. Under the nondegenerate contravariant form, ${\mathrm{Std}}_{\mu }$ is represented by $S_{\mu }=\pi (M_{\mu })$, giving the stated equation. It holds in both field-characteristic branches because the homogeneous basis and integral categorical pairing do, and uses no cellular head classification or characteristic-zero based theorem. Choice is precisely the inherited [C1].