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\renewcommand{\ker}[1]{\text{Ker}(#1)}
\newcommand{\im}[1]{\text{Im}(#1)}
\renewcommand{\fk}[1]{\mathfrak{#1}}

\usepackage{lipsum}

%% define your title in the usual way
\title[Kra\'skiewicz--Hecke insertion]{Type C \texorpdfstring{$K$}{K}-Stanley symmetric functions and Kra\'skiewicz--Hecke insertion}

%% define your authors in the usual way
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\author[J.~Arroyo, Z.~Hamaker, G.~Hawkes, and J.~Pan]{\texorpdfstring{Joshua Arroyo\thanks{\href{mailto:joshuaarroyo@ufl.edu}{joshuaarroyo@ufl.edu}. Partially supported by NSF Grant DMS-2054423.}\addressmark{1}, Zachary Hamaker\thanks{\href{mailto:zhamaker@ufl.edu}{zhamaker@ufl.edu} Partially supported by NSF Grant DMS-2054423.}\addressmark{1}, Graham Hawkes\thanks{\href{mailto:ghawkes1217@gmail.com}{ghawkes1217@gmail.com}}, \and Jianping Pan\thanks{\href{mailto:jianping.pan@asu.edu}{jianping.pan@asu.edu}}\addressmark{2}}{Joshua Arroyo, Zachary Hamaker, Graham Hawkes, and Jianping Pan}}

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\address{\addressmark{1}Department of Mathematics, University of Florida, Gainesville, FL USA \\ \addressmark{2}School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ USA}

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\received{\today}

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%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{We study Type C $K$-Stanley symmetric functions, which are $K$-theoretic extensions of the Type C Stanley symmetric functions.
Our main contribution is Kra\'skiewicz--Hecke insertion ($KH$), a $K$-theoretic analogue of Kra\'skiewicz insertion.
Much like Kra\'skiewicz insertion enumerates reduced words for signed permutations, $KH$ enumerates their 0-Hecke expressions.
The former enumeration witnesses the Type C Stanley expansion into Schur-$Q$ functions.
We conjecture that $KH$ extends this to give the Type C $K$-Stanley expansion into $GQ$ functions, which are $K$-theory representatives for the Lagrangian Grassmannian introduced by Ikeda and Naruse.
We also show Type C $K$-Stanleys of top fully commutative signed permutations are skew $GQ$'s.
This allows us to prove a conjecture of Lewis and Marberg and to give the first conjectural formulas for the expansion of a skew $GQ$ into $GQ$'s.
The latter specializes to a rule for multiplying two $GQ$ functions where one has trapezoid shape.
This would extend Buch and Ravikumar's Pieri rule, the only known product rule for $GQ$'s.}


%% put your keywords here, or comment this out if you don't have them yet
\keywords{Lagrangian Grassmannian, set-valued tableaux, insertion algorithms, symmetric functions, Schubert calculus, $K$-theory}

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\begin{document}

\maketitle
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\section{Introduction}

Although this paper is combinatorial in methods and results, our underlying objective is to understand geometric properties of the Lagrangian Grassmannian.
A major line of active research is to understand combinatorially more exotic cohomology theories such as $K$-theory, which encodes finer data about the boundaries of intersections of Schubert varieties.
In the Grassmannian, the $K$-theory of such intersections was first computed combinatorially by Buch~\cite{buch2002littlewood}, later extended to the orthogonal Grassmannian~\cite{clifford2014k,buch2016k}.
The $K$-theory of the Lagrangian Grassmannian has proved far more difficult to understand, with the only progress being Buch and Ravikumar's Pieri rule~\cite{buch2012pieri}.
We offer a pathway towards understanding such products based on Type C $K$-Stanley symmetric functions.


The Stanley symmetric functions $F_w$ of the permutation $w$ is a symmetric function that enumerates reduced words for $w$~\cite{stanley1984number} and expands into the Schur basis with non-negative integral coefficients enumerated by Edelman--Greene insertion tableaux~\cite{edelman1987balanced}.
By choosing the permutation $w$ appropriately, this expansion recovers the Littlewood--Richardson rule for products of Schur functions.
Our work continues this line of investigation.
We study the \defn{Type C $K$-Stanley symmetric functions} $G^C_w$, introduced in~\cite{kirillov2017construction}, which is indexed by a signed permutation $w$.
Our insertion, which we call \defn{Kra\'skiewicz--Hecke insertion}, generalizes Kra\'skiewicz insertion, the Type C analogue of Edelman--Greene insertion~\cite{kraskiewcz1989reduced}.
Our insertion enumerates $0$-Hecke words for $w$ and gives a conjectural description of $G^C_w$ into $GQ$ functions, which represent the $K$-theory of the Lagrangian Grassmannian~\cite{ikeda2013k}.

Insertion algorithms map a word $(a_1,\dots,a_p)$ to an insertion tableau $P$ and a recording tableau $Q$.
The recording tableaux for Kra\'skiewicz--Hecke insertion are standard shifted set-valued tableaux from~\cite{ikeda2013k}.
We introduce \defn{strict decomposition tableaux} (see \Cref{d:strict-decomposition}) to play the role of insertion tableaux.
For $\lambda = (\lambda_1>\lambda_2> \dots > \lambda_\ell)$ a strict integer partition, let $\ShSet_n(\lambda)$ be the set of standard shifted set-valued tableaux containing $n$ values of shape $\lambda$ and $\SDT(\lambda)$ be the set of strict decomposition tableaux of shape $\lambda$.
For $P$ a strict decomposition tableau, $\rho(P)$ is the usual reading word of $P$.


\begin{theorem}
\label{t:bijection}
	For all $n \in \mathbb{N}$, the map Kra\'skiewicz--Hecke insertion $KH$ is a bijection:
	\begin{center}$ \displaystyle
KH:	  \mathbb{N}^n \to{\sim} \bigsqcup_{\lambda \vdash m \leq n \ \mathrm{strict}} \SDT(\lambda) \times \ShSet_n(\lambda).
	$\end{center}
	Moreover, for $KH(\bfa) = (P,Q)$, the words $\bfa$ and $\rho(P)$ are $0$-Hecke expressions for the same signed permutation.
\end{theorem}

The definition of Kra\'skiewicz--Hecke insertion is given in \Cref{s:insertion} and is extraordinarily technical.
Checking the strict decomposition tableau column condition for two entries requires examining the intermediate segment of the tableau's reading word.
As a consequence, the insertion rules depend on entries in two rows.

For $w$ a signed permutation, let $\mathcal{H}_n(w)$ be the set of $0$-Hecke expressions for $w$ of length $n$.
Also, let $a^C_w(\lambda)$ be the number of strict decomposition tableaux of shape $\lambda$ whose reading word is a 0-Hecke expression for $w$.
By \Cref{t:bijection}, we have:

\begin{corollary}
	\label{c:0-hecke-count}
	For $w$ a signed permutation and $n \in \mathbb{N}$, we have 
	\begin{equation}
	|\mathcal{H}_n(w)| =  \sum_{\lambda \vdash m \leq n\ \mathrm{strict}} a^C_w(\lambda) \cdot |\ShSet_n(\lambda) |.
\label{eq:0-hecke-count}		
	\end{equation}
	\end{corollary}

\Cref{c:0-hecke-count} is precisely analogous to the use of Edelman--Greene and Kra\'skiewicz insertions for reduced word enumeration and Hecke insertion for 0-Hecke expression enumeration in Type A.
All three of these results are enumerative shadows of ($K$-)Stanley symmetric function expansions.
Unfortunately, as discussed in \Cref{rem:no-insertion} an insertion algorithm extending Kra\'skiewicz insertion \textbf{cannot} be used to directly compute the analogous expansion for $G^C_w$.
That said, we believe $KH$ is correctly identifying $GQ$-coefficients.


\begin{conjecture}
	\label{conj:expansion}
	For $w$ a signed permutation,
	\begin{equation}
	G^C_w = \sum_{\lambda \ \mathrm{strict}} (-1)^{|\lambda| - \ell(w)}a^C_w(\lambda) \cdot GQ_\lambda.
\label{eq:GC-conj}		
	\end{equation}
\end{conjecture}

In \Cref{s:GC}, we show $G^C_w \in \Z[GQ_\lambda: \lambda \ \mathrm{strict}]$.
However, establishing the positivity implied by \Cref{conj:expansion} is an open problem, even using geometric methods.
As evidence, the coefficient of $x_1\dots x_n$ on both sides of~\eqref{eq:GC-conj} gives \Cref{c:0-hecke-count}.

In \Cref{s:conjectures}, we present several important corollaries of \Cref{conj:expansion} for $GQ$ expansions.
For $a < b$, let $\tau(a,b) = (b{+}a{-}1,b{+}a{-}3,\dots,b{-}a{+}1)$.
\Cref{conj:expansion} specializes to provide unknown combinatorial $GQ$-expansions for $GQ_\lambda \cdot GQ_{\tau(a,b)}$.
Note $\tau(1,b) =(b)$ is a single row, so this would generalize the Buch--Ravikumar Pieri rule~\cite{buch2012pieri}.
In forthcoming work, the first author reproves their result using strict decomposition tableaux.
To show rules for $GQ_\lambda \cdot GQ_{\tau(a,b)}$ and related results follow from \Cref{conj:expansion}, we show using Stembridge's theory of fully commutative elements~\cite{stembridge97} that $G^C_w$ can express various $GQ$-expansions (see \Cref{c:GQ-GS}).

\ 

\noindent \textbf{Paper Structure:}
\Cref{s:background} introduces necessary background material on signed permutations, set-valued tableaux and $GQ$ functions.
In \Cref{s:GC}, we give a precise definition for Type C $K$-Stanley symmetric functions and identify the signed permutations whose $G^C$'s are used in our conjectures.
We introduce strict decomposition tableaux and define Kra\'skiewicz--Hecke insertion in \Cref{s:insertion}, giving a sketch of the proof for \Cref{t:bijection}.
We discuss \Cref{conj:expansion} and its applications, most notably \Cref{conj:skew}, in \Cref{s:conjectures}.

\section{Background}\label{s:background}
\smallskip
For $n$ a positive integer, let $\overline{n} = -n$, $[n] = \{1,2,\dots,n\}$ and $[\overline{n}] = \{\overline{1},\overline{2},\dots,\overline{n}\}$.
Define the total order $\prec$ on $\Z - \{0\}$ by $\overline{1} \prec 1 \prec \overline{2} \prec 2 \prec \dots$.



%\label{ss:signed-perms}
A \defn{signed permutation} $w$ is a permutation of $[\overline{n}]\cup[n]$ such that $w(i)=-w(\overline{i})$.
By antisymmetry, $w$ is determined by $w([n])$.
For example $v = \overline{2}31$ is a signed permutation.
Signed permutations with composition form $W_n$, the Coxeter group of Type B/C.
The generators of $W_n$ are $s_0, s_1, \ldots s_{n-1}$ where $s_0 = (\overline{1},1) = \overline{1}2\dots n$ and $s_i = (\overline{i{+}1},\overline{i})(i,i{+}1) = 1\dots i{+}1\,i \dots n$ for $i > 0$.
These generators satisfy the relations 
\begin{center}$
(i)\ s_i^2 = 1,\ (ii)\ s_i s_j = s_j s_i\ |i-j|>1,\ (iii)\ s_i s_{i+1}s_i = s_{i+1}s_is_{i+1}\ i >0,\ (iv)\ s_0s_1s_0s_1 = s_1s_0s_1s_0,
$\end{center}
which are called $(i)$ Self-inverse, $(ii)$ Commutation, $(iii)$ Braid and $(iv)$ Long Braid.

The \defn{0-Hecke monoid} $(W_n,\circ)$ is the monoid on signed permutations obtained by replacing the Self-inverse relation with the Idempotent relation $s_i \circ s_i = s_i$.
A \defn{0-Hecke expression} for $w \in W_n$ is an expression of the form
$
w = s_{a_1} \circ s_{a_2} \circ \ldots \circ s_{a_p}
$
with associated \defn{Hecke word} $(a_1,a_2,\dots,a_p)$.
Let $\HH_p(w)$ be the set of Hecke words for $w$ with $p$ letters and $\HH(w) = \cup_{p \geq 0} \HH_p(w)$.
The \defn{length} $\ell(w)$ of $w$ is minimal so that $\HH_{\ell(w)}(w)$ is non-empty, and $\HH_{\ell(w)}(w)$ is the set of \defn{reduced words} for $w$.

We now introduce several important families of signed permutations.
A signed permutation $w$ is \defn{Grassmannian} if $w(i) < w(i+1)$ for all $i\in [n-1]$, that is $w_1, \dots, w_n$ is an increasing sequence.
Grassmannian signed permutations are in bijection with strict partitions: Grassmannian $w$ maps to $\lambda$ with $\lambda_i = \overline{w(i)}$ for each $i$ such that $w(i) < 0$.
For example, $w = \overline{4}\overline{1}23$ is a Grassmannian signed permutation associated with $\lambda = (4,1)$.
Note the identity corresponds to the empty partition.
A signed permutation $w$ is \defn{vexillary} if it avoids a list of $18$ patterns~\cite[Theorem~7]{billey1998vexillary}.
Each vexillary permutation has an associated shifted shape $\lambda(w)$; see \cite{billey1998vexillary} for an algorithmic construction of $\lambda(w)$.

%\label{ss:tableaux}
The \defn{shifted Young Diagram} of strict partition $\lambda$ is $D_\lambda = \{(i,j) \in \Z^2: 1 \leq i \leq j \leq \lambda_i + i\}$.
A \defn{shifted tableau} is a function $T$ whose domain is $D_{\lambda}$.
A \defn{set-valued tableau} $T$ has entries that are finite, non-empty sets of integers.
The shifted set-valued tableau $T$ is \defn{standard} if its entries partition $[n]$ and for each entry (assuming such cells exist):
\begin{center}$
\max T_{ij} \leq \min T_{i{+}1\,j}, \min T_{i\,j{+}1}.
$\end{center}
Similarly, $T$ is \defn{semistandard} with entries in $\Z - \{0\}$ if $\max(T_{ij}) \preceq \min(T_{i{+}1\,j})$ with equality only for negative values and $\max(T_{ij}) \preceq \min(T_{i\,j{+}1})$ with equality only for positive values.
For $T$ a set-valued tableau of shape $\lambda/\mu$, the \defn{size} of $T$ is $|T| = \sum_{(i,j) \in D_{\lambda/\mu}} |T_{ij}|$ and the \defn{weight} of $T$ is
 \begin{center}$\displaystyle
x^T = \prod_{(i,j) \in \lambda(T)} x^{T_{ij}} \quad \mbox{where} \quad x^{S} = x_{|s_1|} \dots x_{|s_k|} \quad \mbox{for} \quad S = \{s_1,\dots,s_k\} \subseteq \Z.
$\end{center}


\begin{example}
For $\lambda = (4,3,1)$ we depict, from left to right,  $D_{\lambda}$ and shifted Young tableaux of shape $\lambda$ that are standard, set-valued standard and set-valued semistandard:
    \begin{center}$\displaystyle
    \begin{ytableau}
        \; & \; & \; & \;\\
         \none & \; & \; & \;\\
         \none & \none & \;
    \end{ytableau}\ , \quad \quad
    \begin{ytableau}
        1 & 2 & 4 & 5 \\
         \none & 3 & 6 & 8\\
         \none & \none & 7
    \end{ytableau}\ ,
    \quad \quad
    \begin{ytableau}
        1 & \textrm{\tiny 2,3} & \textrm{\tiny 4,6} & 8 \\
         \none & \textrm{\tiny 5,7} & 9 & 10\\
         \none & \none & 11
    \end{ytableau}\ ,
    \quad \quad
    \begin{ytableau}
        1' & \textrm{\tiny 1,2} & \textrm{\tiny 2,4'} & 6'\\
        \none & \textrm{\tiny 3,4'} & 6' & 7\\
        \none & \none & 10
    \end{ytableau}\ .
    $\end{center}
The rightmost tableau has size 11 and weight $x_1^2x_2^2x_3x_4^2x_6^2x_7x_{10}$.
\end{example}
Let $\ShSet^*(\lambda/\mu)$ be the set of shifted set valued semistandard Young tableaux of shape $\lambda/\mu$.
Then
\begin{equation}
\label{eq:GQ}
GQ_{\lambda/\mu} = \sum_{T\in \ShSet^*(\lambda/\mu)} (-1)^{|T| - |\lambda/\mu|} x^T.    
\end{equation}
We write $GQ_\lambda := GQ_{\lambda/\varnothing}$.
For $\mu = \varnothing$, the $GQ$'s were first introduced in~\cite{ikeda2013k} as $K$-theory representatives for Schubert classes in the Lagrangian Grassmannian.
Though it is not obvious, $GQ_{\lambda/\mu}$ is a symmetric in the variables $\bfx$.
For $\lambda$ a strict partition with $\ell(\lambda) = k$, $\mu = (k-1,\dots,1)$ and $\nu = (\lambda_1 - (k{-1}), \dots \lambda_k)$, define $GS_{\nu} = GQ_{\lambda/\mu}$.
Note $GS$ functions are generating functions for unshifted marked set-valued tableaux.


\section{Type \texorpdfstring{$C$}{C} \texorpdfstring{$K$}{K}-Stanley symmetric functions}
\label{s:GC}

In~\cite{kirillov2017construction}, Kirillov and Naruse introduce Type $C$ $K$-Stanley symmetric functions $G^C_w$ as part of their construction of Type C double Grothendieck polynomials.

\begin{definition}
The \defn{Type C $K$-Stanley symmetric function} for $w \in W_n$ is
\[
G^C_w(\bfx) = \sum_{p \geq 0} \sum_{\bfa \in \HH_p(w)} \sum_{\bfi \in \kappa(\bfa)} (-1)^{p - \ell(w)} x_{|i_1|} \dots x_{|i_p|} \quad \mbox{where}
\]
\[\kappa(\bfa) = \left\{(i_1 \preceq \dots \preceq i_p) \in (\Z-\{0\})^p: i_k = i_{k+1} \ \mbox{implies}\quad \begin{cases}
     a_k > a_{k+1} & i_k < 0\\
     a_k < a_{k+1} & i_k > 0
\end{cases}\right\}.
\]
We call the pair $(\bfa,\bfi)$ a \defn{compatible sequence}.
\end{definition}

\begin{proposition}
    \label{p:GC-properties}
    Let $w \in W_n$ and $v \in S_{n+m}$ with $v(i) = i$ for $i \in [n]$.
    Then:
    \begin{enumerate}
        \item $G^C_w \in \Z[GQ_\lambda: \lambda\ \mbox{strict}]$;
        \item The coefficient of $x_1\dots x_p$ in $G^C_w$ is $(-1)^{p-\ell(w)}\cdot 2^n \cdot |\HH_p(w)|$;
        \item $G^C_{wv} = G^C_u \cdot G^C_v$;
        \item If $w$ is vexillary, then $G^C_w = GQ_{\lambda(w)}$.
    \end{enumerate}
\end{proposition}


These properties are not stated in~\cite{kirillov2017construction}.
The first is an easy consequence of their work and~\cite{chiu2023expanding}, while the second and third follow easily from the definition.
The fourth follows by specializing a Pfaffian formula of Anderson~\cite{anderson2019k}.


Let
$
w(a,b,n) = 12 \dots k\,\ell{+}1\, \ell{+}2\dots n\,k{+}1\,k{+2}\dots \ell
$
where $k = n - a - b $ and $\ell = a + k$.
Since $w(a,b,n)$ is vexillary of shape $\tau(a,b)$ as asserted in~\cite{billey1998vexillary},  \Cref{p:GC-properties}~(4) implies:
\begin{corollary}
\label{c:GQ-trapzeoid}
    For $a < b$ positive integers and $a + b \leq n$,
\[
G^C_{w(a,b,n)} = GQ_{\tau(a,b)}.
\]
\end{corollary}
Combined with \Cref{t:skew-GQ}, this result implies~\cite[Conjecture~4.23]{lewis2021enriched}.

Say $w \in W_n$ is \defn{fully commutative} if its reduced words contain no braid relations and \defn{top} if its reduced words also avoid the consecutive subword $(1,0,1)$.
Top fully commutative elements are in bijection with skew strict shapes, and there is a transparent bijection between shifted standard tableaux of strict shape $\lambda/\mu$ and reduced words for the top element $w_{(\lambda/\mu)}$ due to Stembridge~\cite{stembridge97}.
We extend this bijection to a weight preserving map $\res$ from  $\ShSSYT_p(\lambda/\mu)$ to compatible sequences for $w(\lambda/\mu)$ of length $p$.

\begin{example}
    \label{ex:res}
    To compute $\res$, label each cell $(i,j)$ in $D_{\lambda/\mu}$ with its content $j-i$.
    Then read the entries of $T$ in increasing order with respect to $\prec$ to form $\bfi$ and the contents to form $\bfa$, breaking ties to enforce compatibility.
    For example, we have
\[
T = \ytableausetup{boxsize=normal} \scalebox{.87}{
\begin{ytableau}
1_{\color{red}{0}} & 1_{\color{red}{1}} & 1_{\color{red}{2}} & 2'_{\color{red}{3}} & 3'_{\color{red}{4}} & 5'_{\color{red}{5}}\\
\none & 2_{\color{red}{0}} & 2_{\color{red}{1}} & 3'_{\color{red}{2}} & 3 4'_{\color{red}{3}}\\
\none & \none & 4_{\color{red}{0}} & 4_{\color{red}{1}} & 4_{\color{red}{2}}\\
\none & \none & \none & 5_{\color{red}{0}} & 5_{\color{red}{1}}
\end{ytableau} }\,, \quad  \res(T) = \binom{\bfi}{\bfa} = \binom{1,1,1,\overline{2},2,2,\overline{3},\overline{3},3,\overline{4},4,4,4,\overline{5},5,5}{0,1,2,3,0,1,4,2,3,3,0,1,2,5,0,1}.
\ytableausetup{smalltableaux}
\]
Here, the contents are depicted using red subscripts inside of $T$ for reference.
\end{example}

As a consequence, we have:
\begin{theorem}
    \label{t:skew-GQ}
    For $\mu \subseteq \lambda$ strict shapes, $G^C_{w(\lambda/\mu)} = GQ_{\lambda/\mu}$.
\end{theorem}

As an important special case, we have:
\begin{corollary}
    \label{c:GQ-GS}
    Let $\mu \subseteq \lambda$ be strict partitions and $\nu$ be a partition (not necessarily strict).
    Then for appropriately chosen strict partitions $\rho,\tau$ the signed permutation $w(\lambda/\mu) \cdot w(\tau/\rho)$ is top and
    \[
G^C_{w(\lambda/\mu) \cdot w(\tau/\rho)} = GQ_{\lambda} \cdot GS_\nu.
    \]
\end{corollary}


\begin{remark}
    A (possibly equivalent) formula for $G^C_w$ using tableaux with straight shapes appears in~\cite{tamvakis2023tableau}, though the connection to $GQ$-functions is not made in that work.
\end{remark}






\section{Kra\'skiewicz--Hecke Insertion}
\label{s:insertion}

In order to define Kra\'skiewicz--Hecke insertion $KH$, we first define strict decomposition tableaux.
These generalize reduced decomposition tableaux from~\cite{kraskiewcz1989reduced,lam1995b} and standard decomposition tableaux from~\cite{serrano2010shifted}\footnote{Confusingly, reduced decomposition tableaux are referred to as `standard' in~\cite{lam1995b}. We introduce the term `reduced' as the Serrano tableaux more closely resemble the conventional meaning of `standard'.}.
These are the insertion tableaux for $KH$.

\begin{definition}
\label{d:unimodal}
Let $R_i= r_i r_{i+1} \dots r_k$ be the $i$th row in a shifted tableau $T$. We say $R_i$ is \defn{(strictly) unimodal} if there exists $i\leq j \leq k$ so that $r_i > r_{i+1} > \dots > r_j < \dots < r_k$.
Depending on context, we refer to both $j$ and $r_j$ as the \defn{dip} of $R_i$.
The \defn{decreasing} and \defn{increasing parts} of $R_i$ are $R_i^\downarrow = r_i \dots r_{j-1}$ and $R_i^\uparrow = r_{j} \dots r_k$, respectively.
Note the increasing part is always non-empty, which is at odds with the conventions in~\cite{lam1995b}.
For a unimodal row $R_{i}$ with dip $j$, it is frequently convenient to view it as
\begin{align*}
\mathsf{T}(R_i) &:= -r_i < \dots < -r_{j-1} < r_j < r_{j+1} < \dots < r_k,\ \text{or}\\
\mathsf{B}(R_i) &:= -r_i < \dots < -r_{j-1} < -r_j < r_{j+1} \dots < r_k,
\end{align*}
which are increasing sequences of integers.
\end{definition}

\begin{definition}
\label{d:strict-decomposition}
    Let $\lambda$ be a strict partition with $\ell(\lambda) = k$.
    A \defn{strict decomposition tableau} is 
    a tableau $T:D_\lambda \rightarrow \mathbb{N}$ with unimodal rows $R_1 \dots R_k$ such that for each $i\in [k-1]$ every entry of $R_{i+1}$ is less than the first entry of $R_i$ and the following \defn{witness condition} is satisfied.
 For increasing sequences $\mathsf{T}(R_i) = a_i \dots a_{i+\lambda_i}$, $\mathsf{B}(R_{i+1}) = b_{i+1} \dots b_{i+1+\lambda_{i+1}}$ and for all
        $i+1 \leq j \leq i+1+\lambda_{i+1}$, 
        \begin{equation}
        \label{eq:row-rule}
        \{ \pm a_i, \dots, \pm a_{j-1},\pm b_{j+1},\dots,\pm b_{i+1+\lambda_{i+1}}\} \cap (b_j,a_{j}] = \varnothing.    
        \end{equation}
    Note that $a_{j}$ appears immediately above $b_j$ in $T$.
    If $a_{j} \leq b_j$, then $(b_j,a_{j}] = \varnothing$, so Equation~\eqref{eq:row-rule} is satisfied vacuously.
If $x$ is an element of the LHS of Equation~\eqref{eq:row-rule}, we say $x$ \defn{witnesses} $b_j < a_{j}$, showing $T$ is not a strict decomposition tableau.
\end{definition}

\noindent The \defn{reading word} of a strict decomposition tableau $T$ is $\rho(T)=R_{\ell} R_{\ell-1}\ldots R_1$.

\begin{lemma}\label{lem:config}
    A shifted tableau with unimodal rows is a strict decomposition tableau if and only if the tableau avoids the following five configurations:
\[
(i)\;\begin{ytableau}
    a & \cdots & \\
    \none & \cdots & b
\end{ytableau},\;
(ii)\;\begin{ytableau}
    \; & \cdots & a & \cdots & \\
    \none & \cdots & c & \cdots & b
\end{ytableau},\;
(iii)\;\begin{ytableau}
    \; & \cdots & v & z & \cdots & \\
    \none & \cdots & \cdots & x & \cdots & y
\end{ytableau},\;
(iv)\;\begin{ytableau}
    y & \cdots & z\\
    \none & \cdots & x
\end{ytableau},\;
(v)\;\begin{ytableau}
    \; & \cdots & y & \cdots & z\\
    \none & \cdots & \; & \cdots & x
\end{ytableau}
\]
with $a\leq b < c$, $x < y \leq z$, and $v < z$.
\end{lemma}

\begin{example}
    The leftmost tableau is a strict decomposition tableau, the rest are not:
\begin{equation}
    \label{eq:dec-tab}
    \begin{ytableau}
        5 & 2 & 1 & 0 & *(cyan)3 & 4\\
        \none & 2 & 0 & 1 & *(cyan)2 & 4
    \end{ytableau}\;\;\;
    \begin{ytableau}
        3 & 0 & 2\\
        \none & 0 & 4
    \end{ytableau}\;\;\;
    \begin{ytableau}
        5 & *(cyan) 2 & 1 & 0 & 2 & 4\\
        \none & *(cyan) 3 & 0 & 1 & *(red)2 & 4
    \end{ytableau}\;\;\;
    \begin{ytableau}
        7 & 2 & 1 & 0 & *(cyan)5 & 6\\
        \none & 2 & 0 & 1 & *(cyan)2 & *(red) 5
    \end{ytableau}\;\;\;
    \begin{ytableau}
        *(red) 5 & 4 & 1 & 0 & *(cyan)5 & 6\\
        \none & 2 & 0 & 1 & *(cyan)4 & 6
    \end{ytableau}\;\;\;
    \begin{ytableau}
        5 & *(red) 3 & 1 & 0 & *(cyan)3 & 4\\
        \none & 2 & 0 & 1 & *(cyan)2 & 4
    \end{ytableau}
    \end{equation}
       The entries highlighted in blue are $a_j, b_j$ so that $a_j < b_j$, while their witnesses, if they exist, are in red.
    Further notice each of the non-examples exhibits the configurations in \Cref{lem:config} from left to right.
\end{example}

For $\lambda$ a strict partition with $\ell(\lambda) = k$, a \defn{reduced decomposition tableau} of shape $\lambda$ is a tableau $T:D_\lambda \rightarrow \mathbb{N}$ with rows $R_1 \dots R_k$ so that $R_i$ is a unimodal subsequence of maximal length in $R_k R_{k-1} \ldots R_{i}$ for all $i\in [k]$ and whose reading word $\rho(T)$ is a reduced word for some signed permutation.
For $KH$ to generalize Kra\'skiewicz insertion, we require:

\begin{proposition}
    Every reduced decomposition tableau is also a strict decomposition tableau.
\end{proposition}

We will now construct Kra\'skiewicz--Hecke insertion, which maps words in the alphabet $\mathbb{N} = \{0,1,2,\dots\}$ to pairs of tableaux $P,Q$ of the same shape where $P$ is a \emph{strict decomposition tableau} and $Q$ is a \emph{standard set-valued shifted tableau}.
We first present the row insertion rule, which is very similar to Kra\'skiewicz row insertion but requires a plethora of additional cases to ensure the insertion tableau is a strict decomposition tableau.

\begin{definition}
    \label{d:row-insertion}
    \defn{(Kra\'skiewicz--Hecke) row insertion} is an algorithm with inputs $a \in \mathbb{N}$ and a two-row strict decomposition tableau $RS$ that outputs $a' \in \mathbb{N} \cup \{\infty\}$ and two-row strict decomposition tableau $R'S$.
    Note $S$ is unchanged and may be empty.
    Row insertion is a two step procedure, first applying \defn{right insertion} and then \defn{left insertion}.

    Let $R = r_1 \dots r_\ell$ with dip $r_q$ and $S = s_2 \dots s_k$ with dip $s_p$ and $k \leq \ell$.
    Set $r_0, s_1 = -\infty$, $r_{\ell+1} = \infty$ and $s_m = \infty$ for $m > k$.
    Note $r_i$ appears immediately above $s_i$ in $RS$.
    
    We first define right insertion, which outputs $a' \in \mathbb{N} \cup \{\infty\}$ and row $R''$, which is an input for left insertion.
    If $a$ equals the dip $r_q$, set $i=q$.
    Otherwise, let $q < i \leq \ell+1$ be minimal such that $r_i \geq a$.
    Set $A_i = \{s_{i+1},\dots,s_k,r_1,\dots,r_{i-1}\}$.
    \begin{enumerate}%[(R1)]
        \item If $a \neq r_i$, then set $a' = r_i$ and create $R''$ by changing $r_i$ to  $a$.
        \item If $a = r_i$ and $r_{i-1} > r_{i+1}$, then set $a' = r_{i+1}$ and create $R''$ by setting $r_{i+1} = r_i$. Note the resulting tableau will not have unimodal rows, however the left insertion step afterwards will always restore unimodal rows.
        \item If $a = r_i$ and $r_{i-1} \leq r_{i+1}$, then let $R''=R$ and set $a' = \min ( \{r_{i+1}\} \cup [ (s_i, r_{i+1}) \cap A_i])$.
    \end{enumerate}

    We now define left insertion. If $a'=\infty$, then set $b=\infty$ and $R'=R''$. Otherwise, let $1\leq j \leq q$ be minimal such that $r_j \leq a'$ where here $r_j$ indicates the $j$\textsuperscript{th} entry of $R''$.
    \begin{enumerate}%[(L1)]
        \item If $a' \neq r_j$, set $b = r_j$ and create $R'$ by changing $r_j$ to $a'$.
        \item If $a' = r_j$, then set $R' = R''$.
        \begin{enumerate}%[(I)]
            \item If $j < p$, we define 
            \begin{center}$
            b = \begin{cases}
            \max(A_j \cap (r_{j+1}, s_j)), & \text{if } j+1 < q \text{ and } A_j \cap (r_{j+1},s_j) \neq \emptyset\\
            \max(A_j\cap (-\infty, s_j)), & \text{if } j+1 \geq q \text{ and } A_j \cap (-\infty,s_j) \neq \emptyset\\
            r_{j+1}, & \text{otherwise.}
            \end{cases}
        $\end{center}
            
            \item If $j \geq p$, then define
            \begin{center}$
            b = \begin{cases}
            s_{j+1}, & \text{ if } (a)\; j+1 \geq q\ \text{and}\  (b)\; r_{j+1} > s_{j+1} \text{ or } r_{j+2}, s_{j+2} > r_j > s_{j+1}\\
            r_{j+1}, & \text{otherwise.}
            \end{cases}
        $\end{center}
        \end{enumerate}
\end{enumerate}
\end{definition}
We extend row insertion to an insertion algorithm by repeated application.
\begin{definition}
    \label{d:full-insertion}
For $P = R_1 \dots R_\ell$ a strict decomposition tableau and $a\in \mathbb{N}$, we 
{\parfillskip=0pt \par} \noindent
\defn{Kra\'skiewicz--Hecke insert} $a$ into $P$ by row inserting $a$ into $R_1R_2$, updating $R_1$ and inserting the output $b$ into $R_2R_3$ and so on until the output is $\infty$.
The insertion \defn{terminates} in row $i$ where $i$ is the row whose output from row insertion is $\infty$.
    
    For $\bfa = (a_1,\dots,a_p) \in \mathbb{N}^p$ and $\bfa'=(a_1,\dots,a_{p-1})$, we define the \defn{Kra\'skiewicz--Hecke insertion tableau} $P_{HK}(\bfa)$ recursively by row inserting $a_p$ into $P_{HK}(\bfa')$.
    The \defn{Kra\'skiewicz--Hecke recording tableau} $Q = Q_{HK}(\bfa)$ is also constructed recursively from $Q' = Q_{HK}(\bfa')$.
    Let $\lambda$ and $\lambda'$ be the shapes of $P_{HK}(\bfa)$ and $P_{HK}(\bfa')$, respectively.
\begin{enumerate}
    \item If $\lambda \neq \lambda'$, they differ by a single cell $(i,j)$.
    We obtain $Q$ from $Q'$ by setting $Q_{ij} = \{p\}$.
    \item If $\lambda = \lambda'$, then construct $Q$ from $Q'$ by adding $p$ to $Q'_{k\,(\lambda_k{+}k)}$ where $k$ is the row where insertion of $a_p$ terminated.
    Note by (R3) that insertion can only terminate in row $k$ if $\lambda_k > \lambda_{k+1} + 1$ or $\lambda_{k+1}=0$.
\end{enumerate}
\end{definition}

\begin{example}
We present two examples of Kra\'skiewicz--Hecke insertion: 
\[
\begin{ytableau}\none & 4 & 3 & 2 & 0 & *(green)2 & \none[{\scriptstyle\;\leftarrow 1}]\\ \none &\none & 0 & 1 & 2 & 3 & \none\\ \none &\none &\none & \none\\ \end{ytableau}\quad\ \ \ 
\begin{ytableau}\none[{\scriptstyle2\rightarrow\;}] & 4 & 3 & *(green)2 & *(red)0 & 1 & \none\\ \none &\none & 0 & 1 & *(cyan)2 & 3 & \none\\ \none &\none &\none & \none\\ \end{ytableau}
\begin{ytableau}\none & 4 & 3 & 2 & 0 & 1 & \none\\ \none &\none & *(green) 0 & *(red) 1 & 2 & 3 & \none[{\scriptstyle\;\leftarrow 0}]\\ \none &\none &\none & \none\\ \end{ytableau}
\begin{ytableau}\none & 4 & 3 & 2 & 0 & 1 & \none\\ \none &\none[{\scriptstyle1\rightarrow\;}] & *(green) 1 & *(red) 0 & 2 & 3 & \none\\ \none &\none &\none & \none\\ \end{ytableau}
\begin{ytableau}\none & 4 & 3 & 2 & 0 & 1 & \none\\ \none &\none & 1 & 0 & 2 & 3 & \none\\ \none &\none &\none & \none & \none[{\scriptstyle\;\leftarrow 0}]\\ \end{ytableau} 
\begin{ytableau}\none & 4 & 3 & 2 & 0 & 1 & \none\\ \none &\none & 1 & 0 & 2 & 3 & \none\\ \none &\none &\none & 0 & \none\\ \end{ytableau}
\]\[
\begin{ytableau}\none & 5 & *(red) 2 & *(green) 0 & *(cyan) 4 & \none[{\scriptstyle\;\leftarrow 0}]\\ \none &\none & 1 & *(cyan) 0 & 4 & \none\\ \end{ytableau}\;\;\;\;
\begin{ytableau}\none[{\scriptstyle2\rightarrow\;}] & 5 & *(green) 2 & 0 & 4 & \none\\ \none &\none & *(cyan) 1 & *(red) 0 & 4 & \none\\ \end{ytableau}\;
\begin{ytableau}\none & 5 & 2 & 0 & 4 & \none\\ \none &\none & 1 & *(green) 0 &*(red) 4 & \none[{\scriptstyle\;\leftarrow 0}]\\ \end{ytableau}\;\;\;\;
\begin{ytableau}\none & 5 & 2 & 0 & 4 & \none\\ \none &\none[{\scriptstyle4\rightarrow\;}] & *(green) 1 & 0 & 4 & \none\\ \end{ytableau}
\begin{ytableau}\none & 5 & 2 & 0 & 4 & \none\\ \none &\none & 4 & 0 & 4 \\ 
\none &\none & \none & \none &\none[{\scriptstyle\;\leftarrow 1}] \end{ytableau}
\begin{ytableau}\none & 5 & 2 & 0 & 4 & \none\\ \none &\none & 4 & 0 & 4 \\ 
\none &\none & \none & 1 \end{ytableau}
\]
The arrows on the right side represent right insertion, while the arrows of the left side represent left insertion.
Bumping occurs in the green cells, while red cells highlight the next element to be inserted (if distinct from the green cell).
Blue cells are relevant for determining which element to insert next.  

When inserting $\bfa = (5, 3, 0, 5, 2, 4, 0, 4, 0)$, we present some insertion tableaux starting with $P((5,3,0,5)) = \begin{ytableau}
    5 & 3 & 0 & 5
\end{ytableau}$\, and ending with $P = P_{KH}(\bfa)$ as well as $Q = Q_{KH}(\bfa)$:
    \begin{align*}
    \begin{ytableau}\none & 5 & 3 & 0 & 2 & \none\\ \none &\none & 3 & \none\\ \end{ytableau}
    \begin{ytableau}\none & 5 & 3 & 0 & 2 & 4 & \none\\ \none &\none & 3 & \none\\ \none &\none &\none & \none\\ \end{ytableau}
    \begin{ytableau}\none & 5 & 3 & 2 & 0 & 4 & \none\\ \none &\none & 3 & 0 & \none\\ \none &\none &\none & \none\\ \end{ytableau}
    \begin{ytableau}\none & 5 & 3 & 2 & 0 & 4 & \none\\ \none &\none & 3 & 0 & \none\\ \none &\none &\none & \none\\ \end{ytableau}
    P = 
    \begin{ytableau}\none & 5 & 4 & 2 & 0 & 4 & \none\\ \none &\none & 3 & 0 & 3 & \none\\ \none &\none &\none & \none\\ \end{ytableau}
    Q = 
    \begin{ytableau}\none & 1 & 2 & 3 & 4 & \textrm{\tiny6, 8} & \none\\ \none &\none & 5 & 7 & 9 & \none\\ \none &\none &\none & \none\\ \end{ytableau}
    \end{align*}
\end{example}
The following proposition implies $KH$ is well-defined.
\begin{proposition}
    \label{p:row-insertion}
    Given an $a\in\mathbb{N}$ and $T$ a strict decomposition tableau, the tableau $T' = T\leftarrow a$ is also a strict decomposition tableau.
\end{proposition}

Our proof of \Cref{p:row-insertion} proceeds by showing if $T'= T\leftarrow a$ contains one of the configurations in \Cref{lem:config}, then $T$ was not a strict decomposition tableau.
The argument is case by case, some of which rely on a lemma showing that row bumping paths of right and left insertion both move weakly to the left.
Two words with the same Kra\'skiewicz insertion tableaux $P$ are necessarily reduced words for the same signed permutation.
Similarly, the Kra\'skiewicz--Hecke insertion tableau for a word determines its 0-Hecke product.

\begin{proposition}
\label{p:reading-word}
    Let $T$ be a SDT and $w$ a signed permutation so that $\rho(T) \in\HH(w)$.
    For $a\in \mathbb{N}$ and $T' = T \leftarrow a$ we have $\rho(T') \in \HH(w \circ s_a)$.
\end{proposition}

The inverse of Kra\'skiewicz--Hecke insertion is defined similarly, first defining inverses to left and right insertion, then using the recording tableau to determine the row and value to begin inverse insertion with.
For space considerations, we omit the details.

Since $KH$ is invertible, \Cref{p:row-insertion} implies \Cref{t:bijection}.
Then \Cref{p:GC-properties,}~(2) and~\ref{p:reading-word} imply \Cref{c:0-hecke-count}.


\section{Conjectures}
\label{s:conjectures}

Our original goal in this project was to compute the $GQ_\lambda$-expansion of $G^C_w$.
As we will explain in \Cref{rem:no-insertion}, insertion methods exhibit a fundamental inadequacy for the task.
Despite this shortcoming, empirically Kra\'skiewicz--Hecke insertion computes this expansion as asserted in \Cref{conj:expansion}, which has been tested up to degree 9 for signed permutation in $W_4$.
By taking the $x_1\dots x_n$ coefficient on each side, we see \Cref{conj:expansion} and \Cref{p:GC-properties}~(2) would imply \Cref{c:0-hecke-count}:
\begin{center}$ \displaystyle
|\HH_p(w)| = \sum_{\lambda\ \mathrm{strict}} a_w^C(\lambda) \cdot |\ShSet_p(\lambda)|.
$\end{center}
Note we have removed the factor $2^p$ from each side, on the right by not allowing barred entries in our tableaux.
We view this as very strong evidence for \Cref{conj:expansion}.
To see why, assume the conjecture were to fail for $w \in W_n$.
By \Cref{p:GC-properties},~(1) we have
\begin{center}$ \displaystyle
G^C_w = \sum_{\lambda\ \mathrm{strict}} b_w^C(\lambda) \cdot GQ_\lambda
$\end{center}
for some coefficients $b_w^C(\lambda) \in \Z$, and conjecturally in $\N$.
Then for fixed $p$, we have
\begin{equation}
\label{eq:tableaux-relation}
\sum_{\lambda\ \mathrm{strict}} a_w^C(\lambda) \cdot |\ShSet_p(\lambda)|    = \sum_{\lambda\ \mathrm{strict}} b_w^C(\lambda) \cdot |\ShSet_p(\lambda)|.
\end{equation}
Additionally, when $|\lambda| = \ell(w)$, we know from~\cite{lam1995b} that $a_w^C(\lambda) = b_w^C(\lambda)$ for all $\lambda$.
Therefore, and especially assuming the positivity of $b_w^C(\lambda)$, the failure of \Cref{conj:expansion} gives a striking relation on the number of shifted set-valued standard tableaux of given sizes.


By combining \Cref{conj:expansion} with  \Cref{c:GQ-GS}, we have:
\begin{conjecture}
\label{conj:skew}
    For strict partitions $\mu \subseteq \lambda$, partition $\nu$ and appropriate signed permutation $w$, 
    \begin{center}$ \displaystyle
    GQ_{\lambda/\mu} \cdot GS_{\nu} = \sum_{\rho} a^C_{w}(\rho)\cdot GQ_\rho.
    $\end{center}
\end{conjecture}
As special cases, with $\nu = \varnothing$ we would recover~\cite[Conjecture 4.36]{marberg2022shifted} and with $\mu = \lambda$ we would recover~\cite[Conjecture~5.14]{lewis2021enriched}.
Both conjectures are statements of $GQ$-positivity, and do not combinatorial descriptions of the coefficients.
These basic facts about skew $GQ$'s have resisted explanation for several years.
As mentioned in the introduction, it is an open problem to find the $GQ$ expansion for products of $GQ$ functions.
By \Cref{c:GQ-trapzeoid}, \Cref{conj:skew} specializes to a rule for computing $GQ_\lambda \cdot GQ_{\tau(a,b)}$.

\begin{example}
    For $w = \overline{521}34891067$, we have $a^C_w((10,3,2)) = 2$ with tableaux
    \begin{center}
    \begin{ytableau}
        9 & 6 & 4 & 3 & 2 & 1 & 0 & 6 & 7 & 8\\
        \none & 8 & 1 & 0\\
        \none & \none & 0 & 7
    \end{ytableau}\;,\;\;
    \begin{ytableau}
        9 & 6 & 4 & 3 & 2 & 1 & 0 & 6 & 7 & 8\\
        \none & 8 & 1 & 0\\
        \none & \none & 7 & 0
    \end{ytableau}\,.
    \end{center}
     This correctly computes the coefficient of $GQ_{1032}$ in $GQ_{521} \cdot GQ_{42}$.
\end{example}

\begin{remark}
\label{rem:no-insertion}
For Kra\'skiewicz--Hecke insertion to extend to a proof of \Cref{conj:expansion}, we need $\bfa$ and $Q(\bfa)$ to have the same peak sets.
We present an example showing an insertion algorithm generalizing Kra\'skiewicz insertion cannot have this property.

For $u = s_1s_0s_2, \; v = s_1s_0s_2s_1, \; w = s_1s_0s_2s_1s_0$,
\begin{center}$
G^C_u = \GQ_{(3)} + \GQ_{(21)}+\beta\GQ_{(31)}\,, \quad \, G^C_v = \GQ_{(31)}\,, \quad G^C_w = \GQ_{(32)}
$\end{center}
The strict decomposition tableaux witnessing these expansions are:
\[u:\ 
U^1 = \begin{ytableau}
1 & 0 & 2
\end{ytableau}\,, \quad
U^2 = \begin{ytableau}
2 & 0 \\
\none & 1
\end{ytableau}\,, \quad U^3 = \begin{ytableau}
2 & 0 & 2\\
\none & 1
\end{ytableau}\,\quad v:\ 
V = \begin{ytableau}
2 & 0 & 1 \\
\none & 1
\end{ytableau}\, \qquad w:\ 
W = \begin{ytableau}
2 & 1 & 0 \\
\none & 1 & 0
\end{ytableau}\,.
\]
For a word $\bfx = (x_1,\dots,x_k)$ and $j \leq k$, let $\bfx[j] = (x_1,\dots,x_j)$.
Consider the words 
%\begin{center}
    $
\bfa = (1,0,2,0,1,0), \bfb = (1,2,0,2,1,0) \in \HH(w), 
%\quad 
\text{ so } 
%\quad 
P(\bfa) = P(\bfb) = W
$
%\end{center}
by \Cref{p:reading-word}.
Similarly, $P(\bfa[5]) = P(\bfb[5]) = V$ since $v$ is vexillary.
From Kra\'skiewicz insertion, we know $P(\bfa[3]) = U^1$ and $P(\bfb[3]) = U^2$.
Any reasonable generalization of Kra\'skiewicz insertion will have $P(\bfa[4]) = P(\bfb[4]) = U^3$, with the former since the peak in position 3 must be preserved.
However, the associated recording tableaux for these insertions would not preserve peak sets at position 5. 
We will give a more detailed discussion of this example in the complete version of this paper \cite{arroyo2025typeckstanleysymmetric}.
\end{remark}

\acknowledgments{We thank Sara Billey, Eric Marberg and Dave Anderson for helpful conversations.}

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