Graded Specht restriction has the removable-node filtration

The ordered Murphy subspaces over the universal ring. First work with the ungraded level-one left Murphy cell $S_{\mathcal{R}}(\mu )$ at the universal Hecke parameter $u$, where $\mathcal{R}=\mathbb{Z}[u,u^{-1}]$. List all removable corners as $B_1,\dots ,B_b$ from bottom to top. Define $W_j^{\mathcal{R}}$ to be the span of standard $z_T$ with the maximal entry $d$ in $B_1,\dots ,B_j$. Each is a free basis-coordinate subspace, with free quotient. If $S⊴T$ in the tableau Bruhat order, its prefix of size $d-1$ dominates the corresponding prefix of $T$ by [F3]. For corners of a fixed partition, $\mu \setminus B$ dominates $\mu \setminus A$ exactly when $B$ is lower than or equal to $A$: subtracting a node from a lower row retains one extra node in every intervening upper-row prefix sum. Thus every standard tableau below $T$ has $d$ in the same or a lower corner. This is the direction needed for the bottom-to-top filtration.

Prove stability generically and descend exact coefficients. Extend scalars to $\mathcal{K}=\mathbb{Q}(u)$ and use the cell-level triangular eigenbasis of [F4]. Step 1.1 implies $W_j^{\mathcal{K}}={\mathrm{span}}_{\mathcal{K}}\{f_T:d\text{ in }{B}_1,\dots ,B_j\}$: the triangular basis change and its inverse preserve each such set. For $1\le r<d-1$, the seminormal formula for $T_r{f}_T$ involves only $f_T$ and, when standard, $f_{s_{r}T}$. Both have $d$ in the same node, so every $W_j^{\mathcal{K}}$ is an $H_{\mathcal{K},u}(S_{d-1})$-submodule. The action matrix of $T_r$ on the universal $z$-basis has coefficients in $\mathcal{R}$ by [F4]. Every coefficient exiting $W_j^{\mathcal{R}}$ vanishes after the injective extension $\mathcal{R}\hookrightarrow \mathcal{K}$, hence is zero already in $\mathcal{R}$. Therefore $W_j^{\mathcal{R}}$ is a submodule. This proves only stability of the ordered sums; it makes no claim that individual fixed-corner spans are stable after specialization.

Identify the generic corner quotient on its standard vectors. Fix $j$, put $A=B_j$ and $\nu =\mu \setminus A$, and let $T_A$ be the standard filling with $d$ in $A$ and the remaining entries row-filled in $\nu$. For any standard $T$ with $d$ in $A$, deletion gives $\bar{T}$ of shape $\nu$. On the eigenbasis quotient $W_j^{\mathcal{K}}\mathrm{/}{W}_{j-1}^{\mathcal{K}}$ the map $f_T\mapsto f_{\bar{T}}$ is an $H_{d-1}$-isomorphism: for every generator $T_r$, $r<d-1$, the two contents, same-row/column alternatives, standardness of the swap and relative prefix dominance are unchanged by deleting $d$; hence the two displayed seminormal action matrices in [F4] are identical. The tableau $T_A$ is dominance-maximal among tableaux with $d$ in $A$, since its deleted tableau is row-filled. Thus the triangular correction to $f_{T_A}$ has only tableaux with $d$ in lower corners, so $f_{T_A}+W_{j-1}^{\mathcal{K}}=z_{T_A}+W_{j-1}^{\mathcal{K}}$. The initial eigenvector of $S_{\mathcal{K}}(\nu )$ is $f_{t^{\nu }}=z_{\nu }$. If the row-reading index of $A$ in $\mu$ is $k$, the permutation $w_{T_A}$ sends the initial entry $k$ to $d$ and shifts $k+1,\dots ,d$ down by one, and has length $d-k$. The factorization $w_T=w_{\bar{T}}{w}_{T_A}$ is length-additive: the $d-k$ inversions involving $d$ are fixed, and all other inversions are precisely those of $w_{\bar{T}}$. Therefore $z_T=T_{w_{\bar{T}}}{z}_{T_A}$, and the generic isomorphism sends $z_T+W_{j-1}^{\mathcal{K}}$ exactly to $z_{\bar{T}}$. This identifies the standardized quotient map, not just an unspecified generic module isomorphism.

The integral and specialized deletion quotient. Define the $\mathcal{R}$-linear basis map $W_j^{\mathcal{R}}\mathrm{/}{W}_{j-1}^{\mathcal{R}}\to S_{\mathcal{R}}(\nu )$ by $z_T\mapsto z_{\bar{T}}$. Both source and target are free with these corresponding bases. For each finite Hecke generator, both action matrices have entries in $\mathcal{R}$, and step 3.1 makes their difference zero in $\mathcal{K}$. Injectivity of $\mathcal{R}\to \mathcal{K}$ therefore makes that difference zero over $\mathcal{R}$. This is an integral module isomorphism. Base change $u\mapsto \xi$ gives the exact Murphy restriction filtration and deletion quotients over the original field $F$, including singular finite-$e$ parameters and positive characteristic. Because every inclusion was a basis-coordinate inclusion with free quotient, these scalar extensions preserve the entire filtration and its quotients; no flatness assumption on $F$ is needed and no generic denominator is specialized.

Pass to the selected homogeneous basis and establish the shift. Return to the fixed graded Hecke–KLR presentation. Its homogeneous theorem [F1] gives a triangular change $v_T=a_T{z}_T+{\sum }_{S⊲T}{a}_S{z}_S$, $a_T\mathrm{\ne }0$. Step 1.1 therefore identifies every specialized $W_j$ with the span of the selected $v_T$ having $d$ in $B_1,\dots ,B_j$, so $W_j$ is a graded submodule. To verify that the deletion isomorphism of step 4.1 is homogeneous, choose compatible reduced words $w_T=w_{\bar{T}}{w}_{T_A}$, which are reduced by step 3.1. The grading is independent of this selected-word change by [F1]. The triangular expansion of $v_{T_A}$ has only lower-corner terms besides its nonzero leading term, so its image under the ungraded deletion quotient is $a_A{z}_{\nu }$ for a fixed scalar $a_A\mathrm{\ne }0$. The previous-rank generators have the same tower images under the transported KLR presentation: the inclusion keeps $X_r,T_r$ for $r<d$; hence its weight idempotent $e(\mathbf{i})$ becomes ${\sum }_{j}e(\mathbf{i},j)$, since both project onto the generalized eigenspaces of those same first $d-1$ commuting $X$ operators. The defining dot formula and fixed crossing normalizing series use only the same adjacent $X$ operators, so $y_r$ for $r\le d-1$ and ${\psi }_r$ for $r<d-1$ are unchanged. This is the tower transport in [F2], without assigning any simple-module shape from a JM tuple; consequently the deletion map sends $v_T+W_{j-1}={\psi }_{w_{\bar{T}}}{v}_{T_A}+W_{j-1}$ to $a_A{v}_{\bar{T}}$. Rescaling the quotient map by $a_A^{-1}$ gives the homogeneous-basis correspondence. The degree recursion [F1] gives $\mathrm{deg}T=\mathrm{deg}\bar{T}+d_A(\mu )$, so $W_j\mathrm{/}{W}_{j-1}\cong S(\nu )\{d_A(\mu )\}$ as graded modules. Other preferred words preserve the same graded subspaces and graded isomorphism by [F1]; no word-independence of the individual vectors is asserted.

Select the removable-$i$-node filtration. The residue projector $e(\alpha -{\alpha }_i,{\alpha }_i)$ on restriction selects exactly the homogeneous basis vectors whose last residue is $i$ by [F1, F2]. Apply this exact idempotent summand to the all-corner filtration of step 5.1. A quotient for a corner $B_j$ of residue different from $i$ becomes zero, while a quotient for an $i$-corner remains $S(\mu \setminus B_j)\{d_{B_j}(\mu )\}$. Removing the repeated zero steps leaves precisely the bottom-to-top sequence $A_1,\dots ,A_c$ of the Statement, with $V_m$ spanned by the selected $v_T$ whose last node is among $A_1,\dots ,A_m$. This proves the asserted filtration and all graded quotients. If there are no removable $i$-nodes the selected module is zero and the filtration is empty; rank zero has no restriction assertion. The entire argument uses finite bases, finite spectra and explicit scalar extension, so no choice principle or cellular head classification is used.