\documentclass[11pt]{amsart} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage[a4paper,margin=30mm]{geometry} \usepackage{amsmath,amssymb} \usepackage{microtype} \usepackage[colorlinks=true,linkcolor=black,citecolor=black,urlcolor=black]{hyperref} \hypersetup{pdftitle={Domino insertion and cells in type B},pdfauthor={Qiyue Tang},pdfsubject={The cell conjectures of Bonnafe, Geck, Iancu and Lam}} \setlength{\emergencystretch}{1.5em} \allowdisplaybreaks[1] \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\Z}{\mathbb Z} \newcommand{\A}{\mathcal A} \newcommand{\LL}{\mathcal L} \newcommand{\RR}{\mathcal R} \newcommand{\SSS}{\mathcal S} \newcommand{\sgn}{\operatorname{sgn}} \newcommand{\sh}{\operatorname{sh}} \newcommand{\SDT}{\operatorname{SDT}} \newcommand{\MT}{\operatorname{MT}} \newcommand{\OC}{\operatorname{OC}} \newcommand{\HC}{\operatorname{HC}} \newcommand{\Irr}{\operatorname{Irr}} \newcommand{\Ind}{\operatorname{Ind}} \newcommand{\negnum}{\operatorname{neg}} \newcommand{\transpose}{\mathsf t} \newcommand{\cellchar}{\chi} \numberwithin{equation}{section} \title[Domino insertion and cells in type B]{Domino insertion and cells in type $B$} \author{Qiyue Tang} \address{School of Mathematical Sciences, Tsinghua University, Beijing, China} \email{tangqy24@mails.tsinghua.edu.cn} \date{November 12, 2026} \subjclass[2020]{Primary 20C08; Secondary 05E10, 20F55} \keywords{Kazhdan--Lusztig cells, unequal parameters, domino insertion, constructible representations} \begin{document} \begin{abstract} We prove Conjectures A, A$^{+}$, and B of Bonnaf\'e, Geck, Iancu, and Lam on Kazhdan--Lusztig cells in type $B$. \end{abstract} \maketitle \section{Introduction} Kazhdan and Lusztig introduced the canonical basis of a Hecke algebra, its cells, and the associated representations in \cite{KL}. In the symmetric group, the Robinson--Schensted correspondence describes the one-sided cells by tableau fibres and the two-sided cells by shape. For a Hecke algebra with unequal parameters, the cells also depend on the weight function. Lusztig's work \cite{LusztigCells,Lusztig} provides the general framework for this dependence. Let $W_n$ be the group of signed permutations on $n$ letters, with simple reflections $t,s_1,\ldots,s_{n-1}$. In one-line notation, right multiplication by $t$ changes the sign of the first letter, and right multiplication by $s_i$ interchanges positions $i$ and $i+1$. Put $W_0=\{1\}$. Let $\Gamma$ be a totally ordered abelian group and let $L:W_n\to\Gamma$ be the weight function defined by \[ L(t)=b>0,\qquad L(s_i)=a>0. \] Thus $L(xy)=L(x)+L(y)$ whenever the Coxeter lengths satisfy $\ell(xy)=\ell(x)+\ell(y)$. A standard parabolic subgroup is generated by a subset of these simple reflections. Several ranges of positive integer weights were understood before the domino conjectures were formulated. Garfinkle's work on primitive ideals \cite{GarfI,GarfII,GarfIII} gives the equal-parameter description by domino tableaux and open cycles; see \cite[Theorem~4.4]{BGIL}. Lusztig's fixed-point comparison with type $A$ gives the ratios $b/a=1/2$ and $3/2$ \cite[Theorem~11]{LusztigCells}, as explained in \cite[Theorem~3.14]{BGIL}. Bonnaf\'e and Iancu \cite[Theorem~7.7 and Proposition~7.9]{BI} described asymptotic left cells by recording bitableaux and proved that their representations are irreducible. Bonnaf\'e \cite[Corollary~3.6 and Corollary~5.2]{BonnafeAsym} determined the two-sided cells and established the specialization range $b>(n-1)a$. These descriptions suggested an interpolation by domino tableaux with varying staircase core. For each nonnegative integer $j$, rank-$j$ domino insertion is a bijection from $W_n$ \[ w\longmapsto \bigl(P^j(w),Q^j(w)\bigr) \] onto the pairs of standard domino tableaux of the same shape and core $\delta_j=(j,j-1,\ldots,1)$, with exactly $n$ labelled dominoes each; $\delta_0=\varnothing$. Here $P^j(w)$ is the insertion tableau and $Q^j(w)$ the recording tableau. Garfinkle's algorithms for ranks zero and one were extended to arbitrary staircase cores by van Leeuwen \cite[\S4.2]{vanLeeuwen}; Lam \cite{Lam} developed the corresponding growth-diagram description. Write $\lambda_j(w)$ for their common shape. Define equivalence relations on $W_n$ by \begin{align*} x\mathrel{\LL_j}y&\quad\Longleftrightarrow\quad Q^j(x)=Q^j(y),\\ x\mathrel{\RR_j}y&\quad\Longleftrightarrow\quad P^j(x)=P^j(y),\\ x\mathrel{\SSS_j}y&\quad\Longleftrightarrow\quad \lambda_j(x)=\lambda_j(y). \end{align*} For equivalence relations $E,F$, their join $E\vee F$ is the smallest equivalence relation containing both, or equivalently the transitive closure of $E\cup F$. Geck \cite[Conjecture~2.17]{GeckCompute} predicted the parameter chambers, the integer walls up to $n-1$, and the asymptotic range, and recorded an independent formulation due to Bonnaf\'e. The joint left-cell conjecture of Bonnaf\'e, Geck, Iancu, and Lam was already stated in Bonnaf\'e's 2005 habilitation synthesis \cite[\S3.B.2, Conjecture~2]{BonnafeHDR}: for positive real ratios, recording fibres describe the open chambers, and the partition at an integer wall is the common coarsening generated by its two neighbouring partitions. Iancu's report on the March--April 2006 Oberwolfach meeting \cite{IancuReport} describes the same joint project and its asymptotic, off-wall, and wall cases. The 2006 preprint of \cite{BGIL} was revised in 2007 and published in 2010. In the revised numbering, Conjecture A concerns $a=2$, $b=2r+1$; Conjecture A$^+$ extends the rank-$r$ description to the whole interval $ra\cdots>a_e$ and $b_1>\cdots>b_o$, put \[ \alpha_i=a_i-e+i,\qquad \beta_i=b_i-o+i, \] omitting zero parts. The even runner is first. For fixed $q$, this construction is a bijection with the bipartitions of $n$, namely the ordered pairs $(\alpha,\beta)$ of partitions such that $|\alpha|+|\beta|=n$ \cite[\S2.4]{vanLeeuwen}. Under $W_n=C_2\wr\mathfrak S_n$, where $C_2=\{1,-1\}$, $\alpha$ uses the trivial character of $C_2$ and $\beta$ the nontrivial character. Denote the associated ordinary irreducible character by $E_p$. Increasing $k$ by two adds one zero on each runner and shifts its other entries by one; the subtracted staircases change by the same amount. Thus the label is unchanged. In these conventions \cite[Theorem~4.2 and Corollary~4.4]{PietM}, \begin{equation}\label{eq:quotient} E_{p^{\transpose}}=E_p\otimes\sgn,\qquad \dim E_p=\binom{n}{|\alpha|}f^\alpha f^\beta =|\SDT_q(p)|, \end{equation} where $f^\alpha$ counts ordinary standard tableaux of shape $\alpha$. The bipartition representation is induced from $C_2^n\rtimes(\mathfrak S_{|\alpha|}\times\mathfrak S_{|\beta|})$, whose index is $\binom{n}{|\alpha|}$; its two Specht factors have dimensions $f^\alpha$ and $f^\beta$. For the last equality, adding one domino moves one bead by one position on its runner and adds a box to the corresponding quotient partition. A standard domino tableau is a chain of these additions. Such a chain is specified by a standard tableau on each quotient and an interleaving of their $|\alpha|$ and $|\beta|$ additions, giving the binomial factor. Thus a fixed recording tableau of shape $p$ has exactly $\dim E_p$ preimages under domino insertion. \begin{proposition}[Finite-type input]\label{prop:finite} For every $n$ and all positive integers $a,b$, $(W_n,L)$ satisfies Lusztig's properties P1--P15 \cite[\S14.2]{Lusztig}, as do all its standard parabolic subgroups. If $r\in\Z_{\geq0}$ and $raT(i-1,j+1)&(i,j+1)\\[2pt] (i-1,j)\text{ or }(i,j+1)&kT(i+1,j-1)&(i+1,j) \end{array} \] A cycle is a connected component of labels under intersection of an old domino with an alternative domino. Moving through a cycle replaces its $D(k)$ by $D'(k)$; denote the result by $\MT(T,c)$. A cycle is open when the move removes one square from the tableau or adds it to the core, and adds one square to the diagram. Call these its beginning and ending squares. It is noncore if both endpoints lie beyond the core boundary, namely $i+j>q+2$. Let $\OC^*(T)$ be the noncore open cycles. Their moves are independent involutions and preserve rank \cite[\S2.3]{PietC}. For $U\subseteq\OC^*(T)$, $\MT(T,U)$ denotes their simultaneous, commuting moves. Set \begin{equation}\label{eq:family} \mathcal F_q(T)= \{\MT(T,U):U\subseteq\OC^*(T)\},\qquad D_q(T)=\{w:Q^q(w)\in\mathcal F_q(T)\}. \end{equation} The sets $D_q(T)$ are the combinatorial left cells of rank $q$. For any $X\subseteq W_n$, put $\mathcal S(X)=\{\sh Q^q(w):w\in X\}$. Define $x\mathrel{D_q}y$ by $Q^q(x)\in\mathcal F_q(Q^q(y))$. Then \cite[Theorem~3.6]{PietE} gives \begin{equation}\label{eq:rankjoin} D_q=\LL_q\vee\LL_{q+1}. \end{equation} We also need the boundary description of the cycle families \cite[\S\S2.2--2.3 and Proposition~3.3]{PietC}. For a partition $p$ with core $\delta_q$, let $\HC(p)$ be the set of addable or removable squares $(i,j)$ with $i+j\equiv q\pmod2$ and $i+j>q+1$, and let $\kappa_p$ count those already in $p$. The beginning squares of the noncore open cycles are exactly these $\kappa_p$ filled squares; any additional core-adjacent squares in $\HC(p)$ are empty. Each beginning is paired with a distinct empty ending square. Choosing the filled member of each pair gives the shapes in the family. The moves are independent, and distinct choices change distinct endpoint squares. Consequently \begin{equation}\label{eq:cubesize} |\mathcal F_q(T)| =|\{\sh U:U\in\mathcal F_q(T)\}|=2^{\kappa_p}, \qquad p=\sh T. \end{equation} Moving through a noncore cycle exchanges one filled endpoint with one empty endpoint and preserves $\HC(p)$. Thus $\kappa_p$ is constant on the family. \subsection{Characters and truncated induction} Let $\Irr(V)$ be the ordinary irreducible characters of a standard parabolic subgroup $V$, identified with its generic Hecke characters by deformation. For the canonical symmetrizing trace $\tau(N_w)=\delta_{w,1}$, define the Schur elements $s_E(v)$ by \[ \tau(h)=\sum_{E\in\Irr(V)}s_E(v)^{-1}\operatorname{tr}_E(h), \qquad a_E=-\tfrac12\min\deg_v s_E(v). \] Here $\min\deg_v$ is the least exponent with nonzero coefficient. These are Lusztig's $a$-values \cite[Corollary~20.7]{Lusztig}. Let $W$ be $W_n$ or a standard parabolic subgroup of $W_n$. For an irreducible character $F$ of a standard parabolic $V\subseteq W$, truncated induction is \begin{equation}\label{eq:jdefinition} J_V^W(F)= \sum_{\substack{E\in\Irr(W)\\a_E=a_F}} \langle\Ind_V^W F,E\rangle_W\,E, \end{equation} extended additively; $\Ind_V^W$ denotes ordinary character induction. The source and target Schur elements use their respective weights. Constructible characters are defined recursively as follows \cite[\S4.1]{GeckC}. The trivial group has its unit character. For a nontrivial $W$, take $J_V^W(\phi)$ and $J_V^W(\phi)\otimes\sgn$ for proper standard parabolics $V$ and constructible characters $\phi$ of $V$. For a constituent $E$ of $\Ind_V^W F$, one has $a_E\geq a_F$ \cite[Proposition~20.14(a)]{Lusztig}. Along a standard parabolic chain $V\subseteq U\subseteq W$, any nonzero induction path $F\to G\to E$ therefore satisfies $a_F\leq a_G\leq a_E$. If $a_F=a_E$, equality holds at both steps, proving transitivity of truncated induction \cite[20.15(a)]{Lusztig}. It also factors over direct products: Schur $a$-values add \cite[20.19]{Lusztig}, and equality of the sum forces equality in each factor. Here $\boxtimes$ denotes the external tensor product, whose character on $(g,h)$ is the product of the two factor characters. The constructible characters of a product are the external products of constructible characters of its factors. This follows by induction from the recursion. In the direction of constructing a product, a final sign twist in one factor is realized by first twisting the unchanged source factors by their signs and then taking the full product sign. In the other direction, a proper parabolic and its final induction factor componentwise, and a full sign twists each factor. Thus the recursion gives both inclusions. For $n>0$, every constructible character of $W_n$ has one of the forms \begin{equation}\label{eq:normalform} J_{W_m\times\mathfrak S_\ell}^{W_n} (\phi\boxtimes\sgn_\ell), \qquad J_{W_m\times\mathfrak S_\ell}^{W_n} (\phi\boxtimes\sgn_\ell)\otimes\sgn, \quad m+\ell=n,\quad \ell\geq1, \end{equation} where $\phi$ is constructible and $\sgn_\ell$ is the sign of $\mathfrak S_\ell$; $m=0$ is allowed. Indeed, write a proper source as $W_d\times\prod_i\mathfrak S_{k_i}$, including trivial blocks so that $d+\sum_i k_i=n$ and $d\ell$ with $k\equiv q+1\pmod2$, and set $c_i=p_i+k-i$. The divided even beta row has $q+1$ more entries than the odd row. Indeed, the core at length $q+1$ has beta numbers $2q,2q-2,\ldots,0$; moving a bead by two preserves its runner, and increasing the padding length by two adds one bead on each runner. These are the two rows of the symbol for $b=(q+1)a$, with its entries divided by $a$. The symbol induction rule \cite[22.15 and Lemma~22.17]{Lusztig} raises the $\ell$ largest entries by one, with multiplicity one for each choice at a tied cutoff. Their weakly decreasing multiset is $\lfloor c_1/2\rfloor,\ldots,\lfloor c_k/2\rfloor$. Since the $c_i$ are strictly decreasing, a value occurs at most twice, and a tie at the cutoff is equivalent to \[ \left\lfloor c_\ell/2\right\rfloor =\left\lfloor c_{\ell+1}/2\right\rfloor \quad\Longleftrightarrow\quad p_\ell=p_{\ell+1},\quad p_\ell+q-\ell\equiv0\pmod2. \] The first choice raises $c_1,\ldots,c_\ell$ by two and gives $p^I$. At a tie, write $(c_\ell,c_{\ell+1})=(2d+1,2d)$. The second choice raises $c_1,\ldots,c_{\ell-1},c_{\ell+1}$ by two. Sorting replaces the boundary pair by $(2d+2,2d+1)$. The preceding raised entry, if present, is at least $2d+4$, and the following untouched entry, if present, is at most $2d-1$, so no other entries change order. Subtracting $k-i$ gives exactly $p^{II}$. In either case, the size increases by $2\ell$ and the runner lengths are unchanged. Every shape used in \eqref{eq:pieri} therefore has core $\delta_q$. Embed $W_m\times\mathfrak S_\ell$ as the standard subgroup acting on the first $m$ and last $\ell$ positions, respectively. Its element $(w,w_{0,\mathfrak S_\ell})$, with $w_{0,\mathfrak S_\ell}$ the longest permutation on the last block, is \begin{equation}\label{eq:append} \widehat w=(w(1),\ldots,w(m),n,n-1,\ldots,m+1). \end{equation} If $K$ is the shape set of $\mathcal F_q(Q^q(w))$, the shape set of $\mathcal F_q(Q^q(\widehat w))$ is \begin{equation}\label{eq:pieri-support} \{p^I:p\in K\}\ \cup\ \{p^{II}:p\in K,\ p_\ell=p_{\ell+1},\ p_\ell+q-\ell\text{ even}\}. \end{equation} We give the local calculation underlying this identity \cite[Lemma~4.10]{PietM}. Put $T'=Q^q(w)$, $p=\sh T'$, and $T=Q^q(\widehat w)$. The appended decreasing positive block adds to row $i$, for $1\leq i\leq\ell$, a horizontal domino $H_i$ with label $m+i$ and squares \[ A_i=(i,p_i+1),\qquad B_i=(i,p_i+2). \] The alternative domino of every old label is unchanged: any comparison square formerly outside the diagram either remains outside or receives a label greater than every old label. If $p_i+q-i$ is even, $A_i$ is fixed. When $p_i>p_{i+1}$, the comparison at $(i+1,p_i)$ is with a later new label or with $+\infty$, so the alternative of $H_i$ uses $(i,p_i)$. This is an old noncore beginning square; $H_i$ joins that cycle and moves its beginning to $B_i$. When $p_i=p_{i+1}$, the comparison is with an old label or zero, so the alternative uses $(i+1,p_i+1)$. If $i<\ell$, $H_i$ and $H_{i+1}$ form a closed cycle. If $i=\ell$, $H_\ell$ forms a singleton open cycle with endpoints $(\ell,p_\ell+2)$ and $(\ell+1,p_\ell+1)$. Both lie beyond the core boundary, since $p_\ell+\ell>q$ for a partition containing $\delta_q$. If $p_i+q-i$ is odd, $B_i$ is fixed. When $i>1$ and $p_{i-1}=p_i$, the alternative uses $(i-1,p_i+2)$, giving the second member of the closed pair. Otherwise the comparison at $(i-1,p_i+3)$ is with an old label, an earlier new label, or zero, so the alternative uses $(i,p_i+3)$. Here $A_i$ is a variable addable square of $p$. By the boundary description in \cite[\S2.3]{PietC}, it is either the ending square of an old open cycle or a vacant square with $i+p_i+1=q+2$. In the first case $H_i$ extends the old cycle, moving its ending two columns to the right. In the second case $H_i$ is a new singleton core open cycle. Each attachment meets one old cycle, while the new closed pairs and core singletons are isolated, so distinct old cycles remain distinct. An old core cycle retains its beginning on the core boundary; an old noncore cycle retains both endpoints beyond it. Thus the noncore cycles of $T$ are the transported noncore cycles of $T'$, together with the singleton $\{n\}$ exactly when $p_\ell=p_{\ell+1}$ and $p_\ell+q-\ell$ is even. This splitting condition is constant on the source family. Suppose it holds and write $p_\ell=p_{\ell+1}=h$. Noncore moves preserve all fixed squares. The absent square $(\ell,h+1)$ is fixed, so row $\ell$ cannot grow. If $h>0$, the square $(\ell+1,h)$ is fixed and occupied, so row $\ell+1$ cannot shrink; the partition inequalities prevent row $\ell$ from shrinking or row $\ell+1$ from growing. If $h=0$, neither row can shrink, and the fixed absent square $(\ell,1)$ keeps row $\ell$ empty; the next row is then empty too. Both row lengths therefore stay unchanged. Involutivity shows that the condition also stays false when it starts false. Every old noncore endpoint in the first $\ell$ rows moves two columns to the right; the other endpoints stay fixed. Moving through any subset of the transported cycles therefore produces exactly the shapes $p^I$ for $p\in K$. When the additional singleton exists, moving through it replaces $(\ell,p_\ell+2)$ by $(\ell+1,p_\ell+1)$ and produces the corresponding $p^{II}$. Core and closed cycles give no choices in $\mathcal F_q(T)$. This proves \eqref{eq:pieri-support}. If $C'$ is a left cell of $W_m$, its product with the longest-element singleton of $\mathfrak S_\ell$ is a parabolic left cell, since the canonical basis factors over the two groups. The ambient left cell $C$ containing that entire source cell satisfies \begin{equation}\label{eq:sourcecell} \cellchar_C=J(\cellchar_{C'}\boxtimes\sgn_\ell), \qquad \widehat w\in C\quad(w\in C'). \end{equation} This is \cite[Lemma~5.5]{GeckC}, also stated in \cite[Lemma~4.7]{PietM}. The source singleton has character $\sgn_\ell$ by the convention in Section~\ref{subsec:hecke}. \begin{proposition}\label{prop:wall} At $b=ra$, each left cell of $W_n$ is one combinatorial cell $D_{r-1}(T)$. \end{proposition} \begin{proof} \emph{Shape sets.} The corrected inclusion writes a left cell as a disjoint union $C=\coprod_i D_i$ of rank-$q$ combinatorial cells. Each $D_i$ is a union of full recording fibres. There cannot be two distinct recording tableaux $Q,Q'$ of the same shape in $C$: the element with insertion pair $(Q',Q)$ and its inverse, with pair $(Q,Q')$, would both belong to $C$, and that element would not be an involution, contrary to \eqref{eq:cellgram}. For any left cell $C'$ of $W_n$, each common shape of $C,C'$ therefore determines exactly one element of $C\cap(C')^{-1}$. Consequently \begin{equation}\label{eq:shapegram} \langle\cellchar_C,\cellchar_{C'}\rangle =|\mathcal S(C)\cap\mathcal S(C')|. \end{equation} Two self-pairings and the cross-pairing give \begin{equation}\label{eq:equalshape} \cellchar_C=\cellchar_{C'} \quad\Longleftrightarrow\quad \mathcal S(C)=\mathcal S(C'). \end{equation} For the forward implication, each shape set has cardinality equal to that of their intersection. For the reverse implication, the squared norm of $\cellchar_C-\cellchar_{C'}$ is zero. \smallskip\noindent \emph{Characters.} We prove by strong induction on $n$ that \begin{equation}\label{eq:shapecharacter} \cellchar_C=\sum_{p\in\mathcal S(C)}E_p. \end{equation} For $n=0$ the sole tableau has shape $\delta_q$ and $E_{\delta_q}=1$. For $n>0$, choose the normal form \eqref{eq:normalform} for a constructible character. First treat the form without its final sign. Choose a left cell $C'$ of $W_m$ affording $\phi$, and take the particular target $C$ in \eqref{eq:sourcecell}. Since $m0}$ a finite set. There is a homomorphism $f:\Lambda\to\Z$ such that $f(s)>0$ for every $s\in S$. \end{lemma} \begin{proof} For $S=\varnothing$ take the zero homomorphism. Otherwise $\Lambda$ is torsion-free, since a nonzero element and any positive multiple have the same sign. Thus identify $\Lambda$ with $\Z^d$. Suppose that zero belongs to the convex hull of $S$ in $\mathbb R^d$. Choose a relation $0=\sum_i c_i s_i$, $\sum_i c_i=1$, $c_i>0$, with the smallest possible number of terms. The columns $(s_i,1)$ are linearly independent: a dependence would permit a perturbation of the $c_i$, preserving both equations, until one coefficient became zero and all others stayed nonnegative. Consequently the coefficients are the unique solution of a full-column-rank rational linear system and are rational. Clearing denominators expresses zero as a nonempty sum of positive elements of $\Lambda$, a contradiction. Let $p$ be the point of that compact convex hull closest to zero. Then $p\ne0$. For $s\in S$, the segment $p+t(s-p)$, $0\leq t\leq1$, lies in the hull. Differentiating its squared norm at $t=0$ gives $p\cdot(s-p)\geq0$, so $p\cdot s\geq\|p\|^2>0$. A sufficiently close rational vector retains these finitely many strict inequalities. Multiplying it by a common denominator gives an integer linear functional positive on $S$. \end{proof} \begin{lemma}\label{lem:specialize} For a finite Coxeter group with positive weights in a finitely generated ordered abelian group $\Lambda$, a specialization to positive integer weights can preserve the left, right, and two-sided preorders, together with any specified finite list of strict inequalities between elements of $\Lambda$. \end{lemma} \begin{proof} Construct the canonical basis over $\Z[\Lambda]$. After extending scalars, bar invariance and triangular uniqueness identify it with \eqref{eq:canonical}. Thus all its coefficients and structure coefficients lie in $\Z[\Lambda]$. Choose a finite set $S\subset\Lambda_{>0}$ containing the generator weights, the specified positive differences, and the negatives of all exponents occurring in the lower coefficients $p_{x,w}$. For each nonzero coefficient of $C_z$ in a product $C_sC_w$, also include $\gamma-\delta$ whenever $\gamma>\delta$ are distinct exponents in its support. There are finitely many such data. Apply Lemma~\ref{lem:separate} and specialize $e^\gamma\mapsto v^{f(\gamma)}$. The Hecke relations and bar involution are preserved, and the lower terms in \eqref{eq:canonical} still have strictly negative degree. Uniqueness identifies each specialized $C_w$ with the canonical basis element for the weights $f\circ L$. In each nonzero generator coefficient, distinct exponents remain distinct, so that coefficient stays nonzero. Zero coefficients stay zero. Hence the left generator graphs are identical. The anti-involution $C_w\mapsto C_{w^{-1}}$ gives the right graphs; their respective transitive closures give the three preorders. \end{proof} \begin{proof}[Completion of the proof of Theorem~\ref{thm:main}] Take $\Lambda=\Z a+\Z b\subseteq\Gamma$. In part (1), preserve $a,b,b-ra,(r+1)a-b>0$ in Lemma~\ref{lem:specialize}. The specialized weights satisfy $rf(a)0$ retains positive weights, and the homomorphism preserves $f(b)=rf(a)$. The integer-weight conclusions transfer along the identical generator graphs. \end{proof} \section{Dominance and the two-sided preorder} For partitions of the same integer, write $\lambda\unlhd\mu$ when $\sum_{i\leq k}\lambda_i\leq\sum_{i\leq k}\mu_i$ for every $k\geq1$. Write $\lambda\lhd\mu$ if $\lambda\unlhd\mu$ and $\lambda\ne\mu$. \begin{remark}\label{rem:counterexample} The dominance-order assertion in \cite[Remark~1.2(c$^+$)]{BGIL} is false. In $W_5$, take $a=2$, $b=1$ and \[ x=(1,-2,-3,5,4),\qquad y=(-1,-4,3,-2,5). \] Then \begin{equation}\label{eq:counterexample} \lambda_0(x)=(4,3,1,1,1)=:\lambda \ \lhd\ \mu:=(5,2,2,1)=\lambda_0(y), \qquad x\not\leq_{LR}y. \end{equation} \end{remark} \begin{proof} \emph{A necessary shape order.} Realize $\mathfrak S_{10}$ on the ordered alphabet $-5<-4<\cdots<-1<1<\cdots<5$, and $W_5$ as the fixed subgroup of its diagram involution. The central simple reflection has orbit length one and the other four generators have orbit length two, giving exactly the weights $b=1$, $a=2$. After the conventional sign adjustment, Lusztig's folding formulas \cite[\S9, (9.1)--(9.2)]{LusztigCells} express fixed-group structure coefficients as graded traces on spaces whose graded dimensions are the ambient structure coefficients. A nonzero trace has nonzero ambient dimension. The two canonical normalizations are related by $\widetilde C_w=(-1)^{\ell(w)}C_w^\dagger$; their structure coefficients differ by $(-1)^{\ell(x)+\ell(y)-\ell(z)}$. Here each Coxeter length is taken in the group under consideration. Thus vanishing is unchanged, and the preorder implication in loc.\ cit.\ (9.2) applies in our convention. A left or right preorder relation in $W_5$ therefore implies the corresponding ambient two-sided relation. By \cite[Theorem~3.3]{BGIL}, rank-zero domino shape agrees with ordinary Robinson--Schensted shape under the embedding \begin{equation}\label{eq:embedding} w\longmapsto\bigl(-w(5),\ldots,-w(1),w(1),\ldots,w(5)\bigr). \end{equation} The type-$A$ preorder is dominance on these shapes \cite[Theorem~5.1 and Corollary~5.6]{GeckA}. Geck indexes the cells by the transposes of their Robinson--Schensted shapes; conjugation reverses dominance. In our shape convention this gives \begin{equation}\label{eq:necessary} u\leq_{LR}w\quad\Longrightarrow\quad \lambda_0(u)\unlhd\lambda_0(w). \end{equation} The two embedded words are \[ (-4,-5,3,2,-1,1,-2,-3,5,4),\qquad (-5,2,-3,4,1,-1,-4,3,-2,5). \] Their row-insertion calculations are given below. A chain $(a_1,\ldots,a_k)$ starts with the letter inserted in row one; for $j>1$, $a_j$ is bumped into row $j$, and the final $a_k$ is appended at the end of row $k$. \[ \begin{array}{c|l|l} \text{step}&\text{chain for }x&\text{chain for }y\\ \hline 1&(-4)&(-5)\\ 2&(-5,-4)&(2)\\ 3&(3)&(-3,2)\\ 4&(2,3)&(4)\\ 5&(-1,2,3)&(1,4)\\ 6&(1)&(-1,1,2)\\ 7&(-2,-1,2,3)&(-4,-3,1,2)\\ 8&(-3,-2,-1,2,3)&(3)\\ 9&(5)&(-2,-1,4)\\ 10&(4,5)&(5) \end{array} \] The resulting insertion tableaux are \[ \begin{array}{rrrr} -5&-3&1&4\\[-1pt] -4&-2&5&\\[-1pt] -1&&&\\[-1pt] 2&&&\\[-1pt] 3&&& \end{array} \qquad\text{and}\qquad \begin{array}{rrrrr} -5&-4&-2&3&5\\[-1pt] -3&-1&&&\\[-1pt] 1&4&&&\\[-1pt] 2&&&& \end{array}. \] This verifies the two shapes in \eqref{eq:counterexample}. Their prefix sums are $(4,7,8,9,10)$ and $(5,7,9,10,10)$. The full dominance interval between them is \begin{equation}\label{eq:interval} (4,3,1,1,1),\quad (4,3,2,1),\quad (5,2,1,1,1),\quad (5,2,2,1). \end{equation} Indeed the first two row lengths are forced to be $(4,3)$ or $(5,2)$; the remaining three boxes can only give $(2,1)$ or $(1,1,1)$. A tail $(3)$ would exceed the third prefix sum of $\mu$ after $(4,3)$, and would violate the partition condition after $(5,2)$. The second partition in \eqref{eq:interval} is the staircase $\delta_4$; the third has core $\delta_3$, after removing the last two boxes of its first row and the last two boxes of its first column. Thus neither has empty $2$-core. Among rank-zero shapes, $\lambda\lhd\mu$ is a cover. Let $X$ and $Y$ be the full shape fibres of $\lambda$ and $\mu$. By \eqref{eq:necessary}, every intermediate shape on a mixed left/right path from $Y$ to $X$ lies between $\lambda$ and $\mu$. The empty-core cover forces each to be one of the endpoints. The first entrance into $X$ would therefore be a single generator edge from $Y$ to $X$. We show that no such edge exists. \smallskip\noindent \emph{Descents and parity on the two fibres.} The shape $\lambda$ has a unique domino tiling. Its vertical dominoes are $\{(4,1),(5,1)\}$ and $\{(2,1),(3,1)\}$; the other three are horizontal, in positions $\{(1,1),(1,2)\}$, $\{(1,3),(1,4)\}$, and $\{(2,2),(2,3)\}$. The bottom square forces the first vertical domino, the third-row square then forces the second, and all remaining pairs are forced successively by $(1,4)$, $(2,3)$, and $(1,1)$. The tiling of $\mu=\lambda^{\transpose}$ is its transpose, with three vertical dominoes. Let $\operatorname{vert}(T)$ denote the number of vertical dominoes of a tableau. The color-to-spin identity \cite[Lemma~8]{Lam} is \[ \negnum(w):=|\{i:w(i)<0\}| =\tfrac12\bigl(\operatorname{vert}(P^0(w))+\operatorname{vert}(Q^0(w))\bigr). \] It follows that $\negnum(u)=2$ for $u\in X$ and $\negnum(w)=3$ for $w\in Y$. The northwest domino of $P^0(w)$ is horizontal for $w\in X$ and vertical for $w\in Y$. It is labelled $1$, whose orientation records the sign of the letter with absolute value $1$: that letter is inserted horizontally if positive and vertically if negative, and larger letters do not move it. The criterion $twu\quad(u\in X),\qquad twu$, gives \begin{equation}\label{eq:improved} p_{u,w}=v^{-1}p_{tu,w}. \end{equation} The case $tu=w$ cannot occur: the leading term of $C_tC_u$ would give $w\leq_{LR}u$, contrary to \eqref{eq:necessary}. Therefore \begin{equation}\label{eq:degree} \deg p_{u,w}\leq-2 \end{equation} whenever the coefficient is nonzero. Multiplication by $C_t$ on $C_w$ is scalar, as is multiplication by $C_s$ if $sww$. The generator formula is \begin{equation}\label{eq:generator} C_sC_w=C_{sw}+\sum_{\substack{z