Polynomial quantum Schur first levels: Weyl filtration and the actual Hecke corner

Establish the finite cell interfaces directly. Give the actual right Weyl cell the left action $u\cdot w=w\rho (\tau (u))$. Its canonical vector $h$ is highest: the semistandard weight bounds exclude $\lambda +a{\alpha }_i$, and its toral character is $\lambda$; the canonical cell form has $B_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(h,h)=1$. The cell law gives $h{\phi }_{T^{\lambda }T}=w_T$, and [F1]'s integral surjection makes it cyclic highest. For an actual polynomial Schur module $X=U^{-}x$ cyclic highest of weight $\lambda$, triangular spanning gives weights in $\lambda -Q_{+}$, so every partition projection $1_{\eta }$ with $\eta >\lambda$ annihilates it. The higher cellular ideal is generated by these projections: in decreasing shape order ${\phi }_{S,T^{\eta }}{1}_{\eta }{\phi }_{T^{\eta },T}\equiv {\phi }_{ST}$ modulo higher shapes, with coefficient one. Hence that ideal kills $X$. Thus $S{1}_{\lambda }\to X$, $s{1}_{\lambda }\mapsto sx$, factors through the actual left Weyl cell $S{1}_{\lambda }\mathrm{/}{I}^{>\lambda }{1}_{\lambda }$; its basis is exactly the shape-$\lambda$ columns because semistandard type $\lambda$ requires shape at least $\lambda$. The unit diagonal $C={\sum }_{\mu }{c}_{\mu }{1}_{\mu }$, $c_{\mu }=v^{-{\sum }_j\left(\genfrac{}{}{0px}{}{{\mu }_j}{2}\right)}$, of [F1] transports this left-star cell to the displayed left quantum action, giving its universal highest property. On its actual weight-$\mu$ vectors define $B=(c_{\lambda }\mathrm{/}{c}_{\mu })B_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}$ and different-weight pairings zero. The exact identity $\rho (\tau (u))=C^{-1}\rho (u{)}^{\ast }C$ makes this symmetric integral form $\tau$-contravariant: the right-action weight factor $c_{\nu }\mathrm{/}{c}_{\mu }$ is canceled by $(c_{\lambda }\mathrm{/}{c}_{\nu })(c_{\nu }\mathrm{/}{c}_{\mu })=c_{\lambda }\mathrm{/}{c}_{\mu }$. Its highest value is one. All factors are units, so its radical is precisely the actual cell radical and its simple quotient is self-dual. These are finite cellular and matrix arguments, valid after arbitrary base change and with no use of the free global highest lattice.

Construct the corner over the original $R,q$. Put $M_0={⨁}_{\alpha \in \mathrm{\Lambda }(N-1,d-1)}{m}_{\alpha }{H}_{d-1}$. Length-additive parabolic coset bases identify $M_1={⨁}_{\alpha }{m}_{(\alpha ,1)}{H}_d$ with $M_0{\otimes }_{H_{d-1}}{H}_d$, by $m_{\alpha }h\otimes a\mapsto m_{\alpha }ha$. Hence $f\mapsto f\otimes 1$, extended by zero on the other summands, embeds the lower Schur algebra with identity $p={\sum }_{{\alpha }_N=1}{1}_{\alpha }$. It is injective because the unit parabolic coset supplies a direct summand. The lower regular composition induces the upper regular type ${\omega }_{\ast }=(1^{d-1},0^{N-d},1)$, so $e_{\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}pV=e_{\ast }V$. On that summand the lower Hecke corner acts by left multiplication by exactly $h\in H_{d-1}$ on $H_d$, since $h\otimes a\mapsto ha$.

Use the root convention $F_{i,i+1}=F_i$ and $F_{i,j}=v{F}_{i+1,j}{F}_i-F_i{F}_{i+1,j}$. In the negative algebra, its last-simple-root-degree-one part is spanned by $x{F}_{i,N}$ with $x$ in the negative lower Levi. Here is a direct spanning proof, requiring no specialized PBW independence. Every divided-simple-generator word of this degree has exactly one $F_{N-1}$, exponent one; all other generators belong to the lower Levi. Move each factor to its right past the current root using $F_{i,N}{F}_{i-1}^{(s)}=v^{-s}{F}_{i-1}^{(s)}{F}_{i,N}+v^{-1}{F}_{i-1}^{(s-1)}{F}_{i-1,N},F_{i,N}{F}_i^{(s)}=v^s{F}_i^{(s)}{F}_{i,N}\mathrm{.}$ Other lower simple divided generators commute with $F_{i,N}$. The first identity follows from $v{F}_{i,N}{F}_{i-1}=F_{i-1}{F}_{i,N}+F_{i-1,N}$ and $F_{i-1,N}{F}_{i-1}=v{F}_{i-1}{F}_{i-1,N}$: induction on ordinary powers gives coefficient $v^{-1}[s]$, canceled with $[s]!$ in the generic algebra, leaving the displayed integral identity. The second and remaining commutations follow by inserting the root recurrence into the rank-two Serre identity and then inducting on the interval length; the only noncommuting lower adjacent endpoint is $i-1$, already treated. Each move removes one right factor, so it terminates. Triangular spanning and the highest-vector relations now show $W^1={\sum }_i{U}_{N-1}^{-}{F}_{i,N}{z}_{\lambda }$. If ${\lambda }_i={\lambda }_{i+1}$, the weight $\lambda -{ϵ}_i+{ϵ}_{i+1}$ is absent: its sorted weight strictly dominates $\lambda$, contrary to the tableau support. Thus $F_i{z}_{\lambda }=0$ and the root recurrence gives $F_{i,N}{z}_{\lambda }=-F_i{F}_{i+1,N}{z}_{\lambda }$. Iteration removes all nonremovable rows. Rows beyond the height contribute zero by nonnegative weight support. Therefore the claimed removable-row generators span integrally.

Order these rows as $i_1>\cdots >i_t$, so ${\mu }_j=\lambda (i_j)$ decreases in dominance, and let $W_j$ be the negative lower-Levi span of the first $j$ generators, with $W_0=0$. Inductively suppose $W_{j-1}$ is invariant. In $W^1\mathrm{/}{W}_{j-1}$ every weight lies below a remaining generator weight by step 2.1, and ${\mu }_j$ is their maximum. Every positive divided generator kills the image of $F_{i_j,N}{z}_{\lambda }$, since its target weight is strictly above ${\mu }_j$. Triangular spanning makes its negative span invariant. The actual universal highest property gives a surjection $W_{N-1}({\mu }_j)\to W_j\mathrm{/}{W}_{j-1}$. This proves invariance and these integral surjections at each stage, without dividing a quantum integer.

Exactly one entry $N$ in a semistandard $\lambda$-tableau lies at a removable corner, and deleting it gives a semistandard $\lambda (i)$-tableau on $1,\dots ,N-1$. Conversely appending $N$ at that corner preserves semistandardness. Consequently the actual free ranks satisfy $\mathrm{rank}{W}^1={\sum }_j\mathrm{rank}{W}_{N-1}({\mu }_j)$. Over $\mathbb{Q}(v)$ the surjections of step 3.1 and this dimension equality force equality of every quotient dimension. The kernel of each integral surjection lies in the free $A$-module $W_{N-1}({\mu }_j)$ and has generic rank zero; injectivity into its fraction-field extension therefore makes that kernel zero. Each integral quotient is thus the actual free lower Weyl module. Its freeness makes the successive short exact sequences split as $A$-module sequences, so arbitrary base change preserves the filtration injections, quotients and surjections. Generic dimensions were used only to prove kernels of explicit integral surjections vanish, not to replace a specialized construction.

Over a field let $B^{-}$ denote the actual lower Borel image. A $B^{-}$-map $W^1\to F_{\mu }$ is determined by its values at the generators of step 2.1; their weights are distinct, so its dimension is at most one and is zero off ${\mu }_j$. At ${\mu }_j$, quotient by $W_{j-1}$. This quotient has no weight above ${\mu }_j$, its ${\mu }_j$-space is the one-dimensional generator line by step 4.1, and later generator weights lie strictly below it. Taking that generator's coefficient, and zero on every other weight, is a well-defined nonzero $B^{-}$-map, since a nonconstant negative operator cannot reach ${\mu }_j$. This proves dimension exactly one. The universal highest property gives $\mathrm{Hom}(W_{N-1}(\mu ),X)={\mathrm{Hom}}_{B^{+}}(F_{\mu },X)$ for every polynomial $X$: maps are exactly primitive vectors of that weight. Taking contravariant duals identifies this with ${\mathrm{Hom}}_{B^{-}}(X^{\tau },F_{\mu })$. With $X=(W^1{)}^{\tau }$ this proves the first Hom assertion. With $X=L^1$, self-duality and the injection $L^1\hookrightarrow (W^1{)}^{\tau }$ give the upper bound and label restriction. For a primitive vector $x$ of any composition weight, the functional $y\mapsto B(x,y)$ on $L^1$, pulled back to $W^1$, is nonzero by the nondegenerate homogeneous form and kills every nonconstant negative lower operator by contravariance. Thus it is a $B^{-}$-map to the character of that weight. The same generator bound excludes all other primitive weights, without presupposing their dominance. The form's toral weight orthogonality ensures first-level projection commutes with this duality.

Record the precise coefficient test needed by lowering arguments. If $L^1\to {\mathrm{\nabla }}_{N-1}(\lambda (i))$ is nonzero, compose with $W^1\to L^1$ and with the highest-weight coefficient in $\mathrm{\nabla }$. The resulting $B^{-}$-functional is nonzero, hence nonzero at $F_{i,N}{z}_{\lambda }$ by step 5.1. Every nonempty cut product $F_{i,b_1}{F}_{b_1,b_2}\cdots F_{b_s,N}{z}_{\lambda }$ has a nonconstant lower-Levi factor to the left and is killed by this functional. Multiplying factors by the unit toral normalization used for $\widehat{F}$ changes only their nonzero scalar factors on these fixed weight vectors. Thus the same zero/nonzero test applies to the normalized cut products. This argument needs neither an inherited tensor norm nor a claimed full-Hom surjection at singular parameters.

The summands of types ${\omega }_{\ast }=(1^{d-1},0^{N-d},1)$ and $\omega =(1^d,0^{N-d})$ are both the actual regular module $H_d$. Their identity isomorphisms give block matrix units $u,v$ in $S$ with $uv=e_{\ast }$ and $vu=e$. Multiplication by $u$ and $v$ consequently identifies $eV$ with $e_{\ast }V$, transporting every lower Hecke operator to its actual upper corner inclusion. This proves the natural restriction isomorphism, including $d=1$, whose lower algebra and corner are $R$. In the tensor realization of [F1], on words containing exactly one $N$ every lower divided generator leaves that position untouched and applies its subset-sum formula to the remaining ordered positions. These are exactly the induced operators of step 1.2 under the actual tensor/permutation intertwiner. Weight projections agree as well, so [F1]'s lower surjectivity proves equality of the full lower actions. The weight-unit diagonal conjugation of [F2] restricts to this subspace and intertwines the same actions after left/right conversion. This verifies the quantum/cellular comparison at every specialization by explicit generators.