Level-one global-basis transition polynomials in the symmetric-group convention

Transport of the lower basis and the coefficient identity. By [F1] the QG lower basis satisfies $G_{\mathrm{Q}\mathrm{G}}({\lambda }^{\mathrm{\prime }})=\mathrm{\mid }{\lambda }^{\mathrm{\prime }}⟩+{\sum }_{\nu }{D}_{\mathrm{Q}\mathrm{G}}(\nu ,{\lambda }^{\mathrm{\prime }})\mathrm{\mid }\nu ⟩$ with $D_{\mathrm{Q}\mathrm{G}}(\nu ,{\lambda }^{\mathrm{\prime }})$ the coefficient of $\mathrm{\mid }\nu ⟩$, and $D_{\mathrm{Q}\mathrm{G}}(\nu ,{\lambda }^{\mathrm{\prime }})=d_{\nu {\lambda }^{\mathrm{\prime }}}(q)$; applying $J^{-1}$ and using $J^{-1}\mathrm{\mid }\nu ⟩=M_{{\nu }^{\mathrm{\prime }}}$ gives $G_{\mathrm{s}\mathrm{y}\mathrm{m}}(\lambda )=M_{\lambda }+{\sum }_{\nu }{d}_{\nu {\lambda }^{\mathrm{\prime }}}(q)M_{{\nu }^{\mathrm{\prime }}}={\sum }_{\mu }{d}_{(\lambda ,\mu )}(q)M_{\mu }$ with $d_{(\lambda ,\mu )}(q)=D_{\mathrm{Q}\mathrm{G}}({\mu }^{\mathrm{\prime }},{\lambda }^{\mathrm{\prime }})$, the sum being weightwise finite by [F5], and the congruence $G_{\mathrm{s}\mathrm{y}\mathrm{m}}(\lambda )\equiv M_{\lambda }<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>q(F_0{)}_{\mathrm{s}\mathrm{y}\mathrm{m}}$ follows from $G_{\mathrm{Q}\mathrm{G}}({\lambda }^{\mathrm{\prime }})\equiv \mathrm{\mid }{\lambda }^{\mathrm{\prime }}⟩<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>q{F}_0$.

Finite weight spaces and the ambient form. By [F5] each $V_w$ is finite-dimensional, and $B$ restricts to a nondegenerate symmetric bilinear form on it with the standard basis orthonormal. The restricted labels $\lambda$ of weight $w$ are finitely many by [F5], so $D_{\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}}$ has finitely many columns, and the corresponding $G_{\mathrm{s}\mathrm{y}\mathrm{m}}(\lambda )$ are linearly independent because they are distinct elements of the lower global basis of $M_{\mathrm{s}\mathrm{y}\mathrm{m}}$ from [F2] (respectively the defining family of [F1] transported).

Support of the transition polynomials. By [F1], $d_{(\lambda ,\lambda )}=1$, all off-diagonal coefficients lie in $q\mathbb{Z}[q]$, and a nonzero off-diagonal coefficient requires ${\mu }^{\mathrm{\prime }}$ to be strictly dominated by ${\lambda }^{\mathrm{\prime }}$ with the same $e$-core. By [F6] this dominance condition is equivalent to $\mu$ strictly dominating $\lambda$. Transposition preserves side adjacency, size and $2\times 2$ blocks, and carries the partition complement of a border strip to its transposed complement. It therefore bijects maximal sequences of $e$-strip removals, so uniqueness in [F7] gives $C_e({\nu }^{\mathrm{\prime }})=C_e(\nu {)}^{\mathrm{\prime }}$ for every partition $\nu$. Applying this to $\lambda ,\mu$ transports the same-core condition. For weight support, the $U$/bar compatibility of [F1] gives $\psi K_h=K_{-h}\psi$; since $\psi$ sends $q$ to $q^{-1}$, a weight vector $v$ of weight $w$ satisfies $K_h\psi v=q^{⟨w,h⟩}\psi v$. Thus the bar preserves each weight space, and its triangular correction recursion does too. Each $\mathrm{\mid }{\mu }^{\mathrm{\prime }}⟩$ occurring in $G_{\mathrm{Q}\mathrm{G}}({\lambda }^{\mathrm{\prime }})$ consequently has the same full affine weight as $\mathrm{\mid }{\lambda }^{\mathrm{\prime }}⟩$. Independence of the simple roots gives $m_j({\mu }^{\mathrm{\prime }})=m_j({\lambda }^{\mathrm{\prime }})$ for every $j$; the reflected weight of $\mu$ equals that of $\lambda$.

The ambient upper coordinates. Let $D:=D_{\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}}$, with $D_{\mu \lambda }=d_{(\lambda ,\mu )}$. The matrix $H=D^{\mathsf{t}}D$ is the Gram matrix of the basic lower vectors under the ambient orthonormal form. Since $D(0)$ has distinct coordinate columns, $H(0)=I$ by [F5], proving $\det H\mathrm{\ne }0$. The columns of $D{H}^{-1}$ lie in the column space of $D$, namely $(M_{\mathrm{s}\mathrm{y}\mathrm{m}}{)}_w$, and $(D{H}^{-1}{)}^{\mathsf{t}}D=H^{-1}H=I$, so they pair as the dual family. Nondegeneracy of $H$ makes this family unique inside the basic subspace. In the whole ambient space the pairing equations determine a coset modulo the orthogonal complement of the basic subspace. The coefficients $e_{(\lambda ,\mu )}=(D{H}^{-1}{)}_{\mu \lambda }$ are in $\mathbb{Q}(q)$.

The worked instance at $e=2$. The weight space contains the standard shapes $(2),(1,1)$, with $(1,1)$ the only restricted label. On the QG side the node formulas give $F_0\mathrm{\mid }\mathrm{\varnothing }⟩=\mathrm{\mid }1⟩$ and $F_1\mathrm{\mid }1⟩=\mathrm{\mid }2⟩+q\mathrm{\mid }11⟩$: the top addable $1$-node has no corner to its right, while the bottom addable $1$-node has the top addable node to its right. This vector is bar-fixed because the vacuum and the lowering generators are bar-fixed by [F1], and it is congruent to $\mathrm{\mid }2⟩$ modulo $q{F}_0$. Uniqueness in [F1] therefore gives $G(2)=\mathrm{\mid }2⟩+q\mathrm{\mid }11⟩$. Applying $J^{-1}$ yields $G_{\mathrm{s}\mathrm{y}\mathrm{m}}(1,1)=M_{1,1}+q{M}_2$. With rows $(2),(1,1)$, $D_{\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}}=\left(\genfrac{}{}{0px}{}{q}{1}\right)$, $H=1+q^2$, and step 2.2 gives $G_{\mathrm{s}\mathrm{y}\mathrm{m}}^{\ast }(1,1)=(q{M}_2^{\ast }+M_{1,1}^{\ast })\mathrm{/}(1+q^2)$. The coefficient $1\mathrm{/}(1+q^2)$ is not Laurent integral: a Laurent polynomial can have a finite pole only at $0$, whereas this rational function has poles at the nonzero roots of $1+q^2$.

The full ambient dual matrix. On the whole Fock space, $D_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}$ is unitriangular by [F1] with off-diagonal entries in $q\mathbb{Z}[q]$; its dual basis with respect to the ambient form $B$ has coefficient matrix the transpose inverse $D_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}^{-\mathsf{t}}$, by the same computation as step 2.2 applied to the square unitriangular matrix (here the standard basis of the whole space is orthonormal and the family $\{G(\mu )\}$ is a basis). We claim $D_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}^{-1}$ has entries in $\mathbb{Z}[q]$: in every finite truncation the strictly triangular part $N$ of $D_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}=1+N$ satisfies $N^m=0$ for large $m$, so $(1+N{)}^{-1}=1-N+N^2-\dots$ is a finite sum with entries in $\mathbb{Z}[q]$; hence $D_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}^{-\mathsf{t}}$ is Laurent integral, in contrast with the non-integral ambient coordinates of step 3.1.

The exact integral upper basis and the form identification. On the QG standard frame the ambient form is orthonormal. For an addition $\mu =\nu +\gamma$ of residue $i$, the node coefficient of $F_i$ is $q^{N_i^r}$, while that of $q^{-1}{K}_i{E}_i$ on the reverse edge is $q^{N_i(\nu )-1-N_i^l}$. The splitting identity $N_i^r+N_i^l=N_i(\nu )-1$ makes them equal (The node order and the signed exponents $N_i^l$ and $N_i^r$, The Fock space node formulas satisfy the quantum affine relations). The diagonal toral operators are self-adjoint, and symmetry supplies the reverse $E_i$ identity; thus the ambient form is contravariant for $\sigma (F_i)=q^{-1}{K}_i{E}_i$ and $\sigma (E_i)=q{F}_i{K}_i^{-1}$. Its restriction to the vacuum basic module is normalized by $B(\mathrm{\varnothing },\mathrm{\varnothing })=1$, and uniqueness of the normalized highest-weight contravariant form identifies it with the QG form (The contravariant form on an integrable highest weight module). The linear relabelling $J$ permutes the orthonormal standard frame and transports the action, so the same identification holds on $M_{\mathrm{s}\mathrm{y}\mathrm{m}}$. Consequently [F3] makes the vectors of step 2.2 the transported upper global basis, free over $A$ in the restricted integral dual lattice and perfectly paired with the lower $A$-span. Their standard-frame coefficients can be rational as in step 3.1; this does not contradict integrality in the upper basis and implies no lower-$A$ selfduality or $A$-unit Gram matrix.

Upper bar and charts. By [F3] the upper bar ${\psi }^{\vee }$ fixes each upper basis element, the upper basis comes with its own dual zero and infinity charts whose balanced $\mathbb{Z}$-intersection is the upper lattice, and the upper operators satisfy the exact mixed adjunction with the lower operators, with reductions at $0$ equal to the lower crystal operators; transporting these statements along $J^{-1}$ gives the corresponding upper bar, charts and operators in the symmetric-group convention, distinct from the ordinary lower bar of [F1].

Distinctions and boundary. The construction of steps 1.1 and 2.1 is a linear relabelling: $J$ fixes $q$ and only transposes labels, whereas the semilinear involution $\mathrm{\mid }\mu ⟩\mapsto \mathrm{\mid }{\mu }^{\mathrm{\prime }}⟩$ is used in the ambient adjoint comparison reported in [F4], $G^{†}(\lambda {)}^{\mathrm{\prime }}=G^{-}({\lambda }^{\mathrm{\prime }})$. That bibliographic comparison is separate from the local construction in steps 1.1–5.1; in particular it does not identify the lower coefficient matrix with its transpose, and by [F1] the equality $d_{\lambda \mu }=d_{{\mu }^{\mathrm{\prime }}{\lambda }^{\mathrm{\prime }}}$ for all lower coefficients is false. Finally, the polynomials $d_{(\lambda ,\mu )}$ are Fock-space transition coefficients between the standard and lower global bases; identifying them with decomposition numbers of Hecke algebras requires the separate graded/Ariki comparison and the correct Hecke parameter dictionary, which is not asserted here, and every application on this page uses them only as the stated transition data.