\subsection{Off-wall cells and their canonical lattices}\label{fin:cellssection}

Let \(b/a\) lie strictly between the integers \(r\) and \(r+1\).
The rank-\(r\) domino correspondence sends a signed permutation
\(w\) to \((P_r(w),Q_r(w))\), its insertion and recording tableaux.
These are standard domino tableaux of the same partition shape,
with fixed staircase \(2\)-core
\(\delta_r=(r,r-1,\ldots,1)\). The core is filled by zeroes, and
the remaining boxes are tiled by dominoes labelled \(1,\ldots,n\).
The labels weakly increase along rows and columns, and the two
boxes with a given positive label form one domino. We use the
insertion and inverse insertion conventions of
\cite[\S2]{TaskinDomino}. In particular inversion exchanges the
two tableaux, and replacing every letter by its negative
transposes both tableaux. For a shape \(\lambda=(\lambda_i)_{i\geq1}\)
with core \(\delta_r\), padded by zero row lengths,
we use the fixed-core two-quotient as its ordinary bipartition
label. One way to fix the runner convention is to take
\(\ell\) at least the number of rows with
\(\ell\equiv r+1\pmod 2\), form beta numbers
\(\lambda_i+\ell-i\), \(1\leq i\leq\ell\), and read the even
runner first. On each runner, divide the even numbers by two
and subtract one from the odd numbers before dividing by two.
Subtract the runner staircase from the decreasing list to
obtain its partition. Changing \(\ell\) by two preserves
the label. In these ordinary labels a shape \((\alpha,\beta)\)
corresponds to positive cell character
\((\beta^{\mathsf t},\alpha^{\mathsf t})\).
Removing a last domino removes one box of the two-quotient,
and the dimension of the labelled irreducible is the number
of standard tableaux of that shape. These facts follow from
the fixed-core correspondence with standard bitableaux, or
from the two-runner abacus: each domino addition moves a bead
up one step on one runner, hence adds one box to that runner's
partition. A labelled standard tableau is exactly a chain of
such additions. In the asymptotic range \(n\leq r+1\) the
usual type \(B\) insertion description and its character labels
are classical \cite{BonnafeIancu}.

\paragraph{Leading matrix units and parabolic transport.}
The unconditional balanced leading algebra for the present weights
is a sum of full matrix algebras with original basis elements
\(t_w\) represented by signed matrix units. A column lies in an
actual left cell and a row lies in an actual right cell.
These statements are \cite[Theorem~2.1 and Proposition~2.3]{BGIL},
in the convention at infinity from Section~\ref{fin:section}.
The leading coefficient of every finite Schur element is one
off the walls: in the cancellation-free formula
\cite[Theorem~3.2]{ChlouverakiJacon}, quantum hooks have leading
coefficient one and a cross factor can tie only if \(b/a\)
is integral. The same normalization holds for type \(A\).
The leading algebra here is defined by the leading
representation matrices before its identification with the
\(\gamma\)-product. We call its columns and rows leading
columns and leading rows. Its involution is \(t_w^*=t_{w^{-1}}\).

We give the parabolic calculation controlling the original indices.
For a finite group and for each parabolic define the central
element \(c=\sum_E v^{-A(E)}e_E\), where \(e_E\) is the primitive
central idempotent of the generic block \(E\).
Use subscripts \(W\) and \(I\) to distinguish the ambient and
parabolic elements, and equip each algebra with the rescaled
product \(h\star k=chk\). The map
\[
 h\longmapsto c_W^{-1}c_Ih
\]
is an injective unital homomorphism for the rescaled products,
compatible with the involutions. In a balanced model the
rescaled regular product on canonical coordinates is bounded
as \(v\to+\infty\), and its limit is the leading algebra.
Indeed the map from original coordinates to the tuple of
scaled representation matrices has an invertible limit by
the signed matrix-unit property. In these matrix coordinates
the rescaled product is ordinary matrix multiplication.

For \(h\) in the source, left rescaled multiplication by its
image is ordinary multiplication by \(c_Ih\). Let \(Y\) be
the minimal representatives of \(W_I\backslash W\) and
\(X=Y^{-1}\) those of \(W/W_I\). In the standard basis the
operator is the direct sum of copies of the source operator
on the cosets \(W_I y\) with \(y\in Y\).
The canonical and standard basis transitions tend to the
identity in both algebras. Therefore the limiting action has
exactly the intrinsic coset blocks in the original canonical
coordinates. The image of an element has a limit as well:
apply its operator to \(c_W^{-1}\), whose limit is the signed
sum of diagonal coordinates, since its scaled matrices are
identities. We obtain a unital integral map of leading
algebras, with left action on \(t_{u y}\), \(u\in W_I\),
given by the source action on \(t_u\) with \(y\) unchanged.
Its right counterpart is given by the cosets \(xW_I\) and
acts on \(t_{x u}\) with \(x\) unchanged. The maps are
transitive for nested parabolics, by the exact generic
formula. Ordinary branching multiplicities survive: the
image of \(v^{A(F)}e_F\) has regular action by \(e_F\),
whose orthogonal isotypic projector in an ambient balanced
block has constant trace
\(\dim(F)[\operatorname{Res}_I E:F]\). Taking the limit
preserves this trace and the idempotent equation.

In particular a move in a proper intrinsic column or row
preserves the ambient leading column or row, respectively.
For the parabolics \(B_j\), \(j<n\), the left source
subword is obtained by keeping the letters of absolute
value at most \(j\), and the right source subword by
standardizing the first \(j\) positions. In the first case
the insertion tableau is the part of \(P_r(w)\) on labels
\(\leq j\), by largest-label deletion
\cite[Lemma~2.18]{TaskinDomino}. The second case records the
part of \(Q_r(w)\) on these labels. Here standardization
replaces absolute values by their ranks while retaining
signs. Hence the source shape labels at \(B_{n-1}\) give
the two predecessor bipartitions of a coordinate.
If these predecessors differ, the ambient leading label
is their unique common upper cover. Indeed its restriction
contains both labels, and in the product of two Young
lattices their componentwise union is the only possible
common upper cover.

The complete tableau labels at all proper ranks are useful
as well. A source diagonal unit acts by a coordinate
projection, and the source unit labels a row or column of
the ambient block. The labels along the nested \(B_j\)
determine the full proper tableau by its successive
shapes. Thus a map along an ambient leading row preserves
the part of the insertion tableau on labels at most
\(n-1\), by commutation with all left source projections.
A map along a column preserves the corresponding recording
part by commutation with the right source projections.

\paragraph{A coordinate quotient criterion.}
The original trace form is Euclidean on the standard
basis. Its canonical Gram matrix tends to the identity at
infinity, so the induced form on a coordinate ideal
subquotient has the same limit, by the Schur complement.
In an original coordinate left-ideal subquotient, the
leading coordinates form a union of whole leading columns.
This follows by taking the limit of the bounded rescaled
ambient action, which preserves the original generic
subquotient. The number of columns of label \(E\) equals
its generic multiplicity. In fact the central projector
\(e_E\) is bounded for the original trace form, and its
limiting matrix is the coordinate projector onto that
block's basis indices: its \((x,y)\) entry in the standard
basis is the limit of
\(s_E^{-1}\operatorname{tr}_E(N_xN_{y^{-1}})\), the
orthonormal matrix pairing of the two leading units.
The limiting trace in the subquotient counts these
indices. Each column has \(\dim E\) indices.

Suppose an irreducible target \(E\) has an integral,
bar-fixed basis and a positive invariant Gram matrix
tending to the identity. Let an irreducible proper cell
module \(V_F\) of character \(F\) embed by signed coordinate
vectors, and suppose
\(\deg_v(s_E/s_F)=0\) with leading coefficient one.
Then the induction counit
\[
 H\otimes_{H_I}V_F\longrightarrow E,\qquad
 h\otimes f\longmapsto h\,i(f)
\]
sends each relative canonical vector to zero or one
signed target coordinate, with every target coordinate
occurring exactly once. Here \(i\) is the embedding.
To prove this, let \(P=ii^*\) for the restricted form,
and let \(x\) run over the minimal representatives of
\(W/W_I\). Schur orthogonality, factoring \(w=xu\) with
\(u\in W_I\), gives
\[
 \sum_x \rho_E(N_x)P\rho_E(N_x)^*=(s_E/s_F)1_E .
\]
Indeed the full Schur sum with \(P\) is
\(s_E\dim(F)1_E\), and the inner parabolic sum is
\(s_F\dim(F)P\). In the chosen bases the sum is
\(\sum_x M_xG_F^{-1}M_x^{\mathsf t}
=(s_E/s_F)G_E^{-1}\), where \(M_x\) is the matrix of
\(\rho_E(N_x)i\) and \(G_F,G_E\) are the two Gram
matrices. Positivity makes every entry of every \(M_x\)
bounded at infinity. Relative canonical correction
preserves this bound and gives bar-invariant integral
Laurent coordinates, hence constant integers. Their
column matrix \(M\) satisfies \(MM^{\mathsf t}=1\) by
taking the limit. Each row thus has one entry \(\pm1\),
in distinct columns. This proves the assertion.
By relative induction \cite[Theorem~3.6]{GeckRelative},
the induced canonical module is an original coordinate
left-ideal subquotient. The kernel is coordinate and the
irreducible quotient is one actual left cell: a union
of more than one strongly connected component of its
coefficient graph would have a proper nonempty initial
union, giving a nonzero proper submodule.

\paragraph{Pure cells.}
We use the preceding criterion to construct at least one
actual cell for each pure label. In the positive convention
let \(\lambda\) be a partition of \(n\), of height \(d\),
with bottom row of length \(l\). Remove that row to obtain
\(\mu\). Let the sign of the short eigenvalue be fixed,
so the pure module is the type \(A\) Specht module of
shape \(\lambda\), inflated with \(N_{s_0}=v^b\) or
\(-v^{-b}\). In the type \(A\) induction from shape
\(\mu\) and the trivial module of \(\mathfrak S_l\),
Pieri gives \(\lambda\) as the unique dominance-minimal
constituent, with multiplicity one. All other constituents
dominate \(\lambda\): a horizontal strip adds at most one
new row, and the partial row sums above row \(d\)
already dominate those of \(\mu\). The type \(A\)
two-sided preorder in the positive convention places
the more dominant shapes below this constituent.
Thus the quotient onto \(\lambda\) is a coordinate
quotient. The identity coset of the induction generates
it, so that proper irreducible coordinate copy embeds
in the quotient, with the original relative basis.
This gives the proper based embedding also for the
inflated module from \(B_{n-l}\times\mathfrak S_l\).
An actual pure \(B_{n-l}\) cell restricts to one
type \(A\) cell basis: its relative-cell filtration
has just one factor because its restriction is
irreducible. Therefore it can be used as the
proper source after the usual type \(A\) based
identification.

With the negative short eigenvalue the Schur ratio of
this induction is the type \(A\) Schur ratio times
\(\prod_{j=1}^l(1+v^{-2b+2a(j-d)})\).
The type \(A\) ratio has degree zero and leading
coefficient one because half the highest degree of a
partition Schur element is \(a\sum_i\binom{\lambda_i}{2}\).
The ratio therefore has the required leading term when
\(b/a>l-d\). For the positive short eigenvalue the
same calculation uses the product with \(2b\) in place
of \(-2b\) and works when \(b/a<d-l\).
These formulas follow by cancelling the unchanged rows
in the pure Schur product of
\cite[Theorem~3.2]{ChlouverakiJacon}. The cross factor
for a box \((i,j)\) in the pure shape is
\(1+v^{\pm2b+2a(j-i)}\) after reversing the box order
within the row. For a negative pure target, use the
first inequality, or apply the second to its sign
twist of transposed shape and use longest-element
duality. Indeed if the largest row has length \(h\)
and occurs \(t\) times, the latter inequality reads
\(b/a<h-t\). We have \(h-t\geq l-d\), with equality
only for a rectangle. The nonintegral ratio therefore
satisfies one of the two strict inequalities. Both
sources have smaller rank. The coordinate quotient
criterion proves by induction that every pure label
has an actual irreducible cell.

\input{finite-anchors}

\begin{lemma}\label{fin:domino}
Under the proper-rank induction assumptions, every rank-\(r\)
recording fibre is one original left cell with its labelled
irreducible character. Insertion fibres are the right cells.
\end{lemma}

\begin{proof}
We first identify the leading fibres. Fix an insertion
tableau. In its fibre, classify words by the corner occupied
by the last recording domino. For a fixed corner, reverse
insertion fixes the last signed letter and the prefix
insertion tableau. Thus the words in that corner class
are related by proper \(B_{n-1}\) moves. The noncritical
cases of the reverse-insertion proof of
\cite[Theorem~3.5]{TaskinDomino} join different corner
classes by these moves and ordinary signed-alphabet
Knuth moves. We specify the use of its case analysis.
When the northeast region of a lower corner and the
southwest region of an upper corner contain an
intermediate removable domino, reverse insertion at
the three corners gives a triple \(c,b,a\), with \(b\)
greater than \(c,a\) by its Lemma~2.28. The appropriate
Knuth switch is \(cba\leftrightarrow bca\) if \(c<a\)
and \(cba\leftrightarrow cab\) if \(a<c\).
The recording corner switches by its Corollary~2.32.
For overlapping corners in a rectangle larger than
\(2\times2\), the two corners exposed by the first
removal give the same triple and switch
(Case~3.1.1). Cases~3.2.1, 3.2.2 before its final
subcase, and 4.1 reduce to these switches through a
third or fourth corner. Case~4.2 has \(n\leq r+1\)
and is asymptotic. The two configurations left by this
reduction are the overlapping pair with no larger
rectangle in Case~3.1.2, and the final three-corner
configuration of Case~3.2.2. In the former the shape
and the rank are those in Lemma~\ref{fin:hookanchor}.
In the latter, tracing the staircase boundary gives
the shape in Lemma~\ref{fin:rectangleanchor}, up to
transposition. The lower and upper horizontal corner
classes are already joined by the intermediate-corner
switch, since their region contains the plateau square.
Every Knuth move just used is a type \(A\) move in the
proper subgroup on the permutation generators. The
parabolic formulas therefore preserve the ambient
leading row on all of these moves.

For a nonexceptional shape with at least two removable
corners, choose a recording tableau whose last corner
differs from that of the fixed insertion tableau.
Their proper predecessors differ, so the unique-cover
argument above identifies the ambient leading label.
The full fixed-insertion fibre is in its one leading
row, and its size is the dimension of that label.
The row and the fibre therefore coincide.

For a three-corner shape,
Lemma~\ref{fin:rectangleanchor} gives a full leading
column, with its recording tableau ending in the
overlapping vertical corner. Indeed its anchor pair
joins the two remaining corner components within that
recording fibre, and the dimension gives equality.
Label the rows by the insertion tableaux on this
column, and the columns by the recording tableaux
on the inverse row. Their common diagonal element
is an involution, since the involution fixes a
signed diagonal unit. The two labels agree there.
At a matrix position with distinct last corners, the
proper tableau labels from parabolic transport give
both predecessors. The known block label is their unique
common upper cover. A predecessor tableau has only one
extension to that shape, so both complete tableaux are forced.
For any row ending at a horizontal corner, take a
recording tableau ending at the other horizontal
corner and one ending at the vertical corner.
The two forced coordinates connect the two corner
components in that row. For a row ending vertically,
use its known coordinate in the anchor column and a
coordinate ending horizontally. The proper moves through
these coordinates put the whole insertion fibre in the row.
Its size equals the row dimension. For a two-corner hook,
Lemma~\ref{fin:hookanchor} supplies two full anchor
columns, one for each last corner. Label using the
first column and its inverse row. The second column
is placed by its intersection with a row having the
opposite last corner, which is already forced by
the two predecessors. Each row has an entry from each
last-corner class, one by its known entry in a full anchor
column and the other by the distinct-corner argument.
Proper moves fill the row and the dimension gives equality.
This proves the required
row and column statements for the exceptional shapes.

After these blocks are exhausted, the remaining
labels and shapes have just one corner. They are
pure rectangles. A predecessor of positive size
has at most one one-corner parent in ranks \(n\geq3\):
deleting the southeast box from a rectangle with
both side lengths greater than one determines the
rectangle from its two row lengths, while a single
row or column determines its extension unless
the predecessor is a single box. The remaining
coordinate therefore has its labelled block by
restriction to \(B_{n-1}\). The fixed-corner move
and dimension give its entire row. Inversion
gives every corresponding column.

It remains to exclude mergers of leading columns into
actual left cells. Relative induction fixes one proper
\(B_{n-1}\) left cell for all indices in an actual
ambient cell, by \cite[Theorem~3.6]{GeckRelative}.
If its known leading column has predecessor \(F\),
its character is a multiplicity-free sum of parents
of \(F\), since \(B_{n-1}\) branching is
multiplicity-free. A proper based isomorphism between
source cell lattices induces an isomorphism preserving
the relative canonical basis up to the same signs.
It sends \(N_x\otimes C_u\) to
\(N_x\otimes(\pm C_{u'})\) and commutes with bar, so
canonical uniqueness gives the claim. Consequently
it transports actual cells and their characters along
proper right moves, using the induction hypothesis
that the isomorphism preserves the insertion label.
This also applies to the proper type \(A\) moves,
by ordinary same-shape based equivalence.

If a proper-move component of recording tableaux
has two distinct predecessors, all the actual
characters there coincide and each is a sum of
common parents. The only common parent is the
given shape. This proves actual irreducibility in
all nonexceptional multiple-corner cases.
The single-corner pure rectangles already have
one actual irreducible cell by the pure-cell
construction. Within a fixed last-corner class,
the proper \(B_{n-1}\) based isomorphisms change
the recording prefix arbitrarily while keeping
the insertion prefix and final letter fixed.
Thus that irreducibility propagates to every
recording tableau in the class.
For a three-corner shape the two horizontal
classes have distinct predecessors and are already
irreducible. The vertical class has a pure
predecessor, with just one mixed parent. Since
all the pure columns are already their own
actual cells, no such column can merge with
the mixed column. Multiplicity one proves
irreducibility there. For a hook,
Lemma~\ref{fin:hookanchor} supplies an actual
irreducible anchor for each last corner. The same
proper transport reaches every filling in either
class. This completes the proof.
\end{proof}

\begin{lemma}\label{fin:based}
Under the same assumptions, all actual left cells of a
given character \(E\) have signed based isomorphic
lattices, preserving the insertion labels. When the
proper source of Lemma~\ref{fin:orientation} has
\(E\) as its unique minimum-degree constituent, its
induced canonical lattice surjects onto the original
lattice of every actual left cell with character \(E\).
\end{lemma}

\begin{proof}
The chosen proper irreducible \(F\) occurs with
multiplicity one in \(\operatorname{Res}_I E\) and
\(A(F)=A(E)\). Restrict an actual left cell of
character \(E\). For a minimal representative \(y\) of a right
coset \(W_Iy\), the \(H_I\)-generator edges within the coset
are the intrinsic edges, by \cite[Lemma~4.3]{GeckRelative}.
Thus if the ambient left cell meets \(D y\) for an intrinsic
left cell \(D\), it contains \(D y\). Filtering the restricted
module by cosets and by intrinsic left cells gives factors
of the characters of these \(D\), by the same coefficient
formula and \cite[Proposition~4.4]{GeckRelative}.
There is exactly one such relative cell of character \(F\).
Any edge out of it for a source generator has intrinsic
two-sided label below that of \(F\), by the same proposition.
By proper P4 its spectral degree is at least
\(A(F)\). On the other hand a restriction
constituent has degree at most \(A(E)\):
the restriction of the trace gives
\(s_F^{-1}=\sum_G [\operatorname{Res}_I G:F]s_G^{-1}\)
with positive leading terms, and hence the
inequality. Proper P11 and the proper two-sided
classification now force the same label \(F\).
Multiplicity one prevents an edge to another
relative cell of that label. Thus the chosen relative
cell spans an actual source coordinate submodule.

The target lattice has Gram tending to the identity
as an original cell quotient. The Schur ratio is
a degree-zero unit at infinity by off-wall
normalization. Apply the coordinate quotient
criterion above, using the proper signed based
identification of the source. For any two target
cells the resulting kernels have the same generic
span, since the target constituent has multiplicity
one in induction. Both kernels are coordinate
submodules, so they are identical, yielding the
signed based isomorphism. If the proper source
is obtained for \(E\otimes\operatorname{sgn}\),
exact longest-element cell duality
\cite[Proposition~11.6 and Corollary~11.7]{Lusztig}
transfers the based isomorphism to \(E\) by the signed
transpose of the cell matrices. Finally a constant
based \(H\)-isomorphism intertwines the action of every
\(v^{-A(E)}C_x\) and hence of every limiting leading matrix.
It takes a leading column to the same-row
coordinates of the other column. By
Lemma~\ref{fin:domino} these are exactly the
corresponding insertion labels.
\end{proof}