Standard Fock vectors expand in the upper global basis

The restricted coefficient matrix is dominance-unitriangular. Fix a content, and take the square minor $D_{\mathrm{r}\mathrm{e}\mathrm{s}}$ with rows and columns the restricted labels and row-$\mu$, column-$\lambda$ entry $d_{(\lambda ,\mu )}$. By [F2], the diagonal is $1$ and the only off-diagonal support has $\mu$ strictly dominating $\lambda$. Choose a finite linear order refining this dominance direction; then $D_{\mathrm{r}\mathrm{e}\mathrm{s}}=I+N$ is triangular with nilpotent $N$ and coefficients in $\mathbb{Z}[q]$, and $(I+N{)}^{-1}={\sum }_{k\ge 0}(-N{)}^k$ is a finite polynomial matrix. In particular $\det D_{\mathrm{r}\mathrm{e}\mathrm{s}}=1$ and the minor is invertible over $A$.

The restricted form is nondegenerate. Let $D$ be the rectangular matrix with all standard rows and restricted lower columns on the fixed content. The Gram matrix is $H=D^{\mathsf{t}}D$ by [F1]. By [F2] its columns at $q=0$ are distinct coordinate vectors, so $H(0)=I$ and $\det H(0)=1$. Thus $H$ is invertible over $\mathbb{Q}(q)$; the form on the basic subspace is nondegenerate and identifies its field dual with that subspace.

The standard functional and its representing vector. For $x={\sum }_{\lambda }{a}_{\lambda }G(\lambda )\in V$, the definition of the transition coefficients gives ${\mathrm{Std}}_{\mu }(x)=B(x,M_{\mu })={\sum }_{\lambda }{a}_{\lambda }{d}_{(\lambda ,\mu )}$, a finite sum over the labels of the content, with values in $A$ whenever $x$ lies in the lower $A$-lattice [F1, F2]. By step 2.1 the functional ${\mathrm{Std}}_{\mu }$ on $V$ is represented by a unique vector $\pi (M_{\mu })\in V$, characterized by $B(\pi (M_{\mu }),x)={\mathrm{Std}}_{\mu }(x)$ for all $x\in V$; equivalently $B(x,\pi (M_{\mu }))=B(x,M_{\mu })$ for all $x\in V$, so $M_{\mu }-\pi (M_{\mu })$ is $B$-orthogonal to $V$ and $\pi$ is the orthogonal projection along $V^{\perp }$ in the ambient space.

Expansion in the dual upper vectors. Put $v:={\sum }_{\lambda }{d}_{(\lambda ,\mu )}{G}^{\ast }(\lambda )$, a finite sum over the restricted labels of the content [F3]. Using the duality $B(G(\lambda ),G^{\ast }({\lambda }^{\mathrm{\prime }}))={\delta }_{\lambda {\lambda }^{\mathrm{\prime }}}$ from [F3] and step 3.1, for every restricted ${\lambda }^{\mathrm{\prime }}$ $B(v,G({\lambda }^{\mathrm{\prime }}))={\sum }_{\lambda }{d}_{(\lambda ,\mu )}B(G^{\ast }(\lambda ),G({\lambda }^{\mathrm{\prime }}))=d_{({\lambda }^{\mathrm{\prime }},\mu )}={\mathrm{Std}}_{\mu }(G({\lambda }^{\mathrm{\prime }})),$ and both sides are bilinear in their arguments, so $B(v,x)={\mathrm{Std}}_{\mu }(x)$ for all $x\in V$. By uniqueness in step 3.1, $v=\pi (M_{\mu })$, which is the displayed expansion.

Upper dual basis and the failure of lower integrality. The matrix expressing the restricted family $\{{\mathrm{Std}}_{\lambda }\}$ in the restricted dual family $\{G^{\ast }(\lambda )\}$ is the transpose of $D_{\mathrm{r}\mathrm{e}\mathrm{s}}$ of step 1.1, hence unitriangular over $A$ with unit diagonal; therefore the restricted ${\mathrm{Std}}_{\lambda }$ form an $A$-basis of the upper dual lattice and the full ambient family may be redundant [F2, F3]. The coefficients $d_{(\lambda ,\mu )}$ are Laurent polynomials, but $\pi$ need not preserve the lower integral lattice: in $e=2$, at the content of the two-part partition $2$, the recorded transition data give $G_{11}=M_{11}+q{M}_2$, whose dual-form upper vector is $G_{11}^{\ast }=(M_{11}+q{M}_2)\mathrm{/}(1+q^2)$ and hence ${\mathrm{Std}}_{11}=\pi (M_{11})=(M_{11}+q{M}_2)\mathrm{/}(1+q^2)$ has the genuine field denominator $1+q^2$, which is not a unit of $A$; the raw standard vector $M_{11}$ is nevertheless an ambient lattice basis vector.

The canonical recursion, finiteness and Choice. All sums at fixed content are finite. The form identification of [F3] gives $B(F_{i}x,y)=B(x,q^{-1}{K}_i{E}_{i}y)$, and both $V$ and $V^{\perp }$ are stable under the quantum action: for $x\in V^{\perp }$ and $y\in V$, contravariance gives $B(ux,y)=B(x,\sigma (u)y)=0$. Hence the orthogonal projection $\pi$ commutes with the action. In the symmetric linear-$J$ model, conjugating the QG diagram changes lower-content nodes to above nodes and reflects colors; the transported removal formula is therefore $E_i{M}_{\mu }={\sum }_{A\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{o}\mathrm{v}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}}{q}^{-d^A(\mu )}{M}_{\mu \setminus A}$, where $d^A$ is the addable-minus-removable count strictly above $A$ in the smaller diagram. Applying $\pi$ gives the same recursion for ${\mathrm{Std}}_{\mu }$, as used by the later Specht comparison. No new choice is made; [C1] is inherited.