%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2025}
\articlenumber{46}
%% but DO NO,T pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}

\input{preamble.tex}
\def\iG{\widehat{\cG}}
\def\ellfpf{\ell_{\mathsf{fpf}}}
\def\ellhat{\ell_{\mathsf{inv}}}
\def\last{\operatorname{end}}
\def\ODes{\mathrm{ODes}}
\def\EDes{\mathrm{EDes}}

\def\forced{\mathsf{F}}
\def\prohib{\mathsf{P}}
\def\simplus{\xrightarrow{[n]}}
\def\NN{\ZZ_{\geq 0}}

% \def\GC{\mathsf{OGC}}
\def\GC{\operatorname{GC}^{\mathsf{O}}}
\def\GQ{GQ}
\def\Sh{\mathsf{Sh}}

\def\cBfpf{\mathcal{B}_\fpf}
\def\cB{\mathcal{B}_\inv}

\def\SubDiag{\mathsf{SubDiag}}
\def\word{\mathsf{word}}

\newcommand{\Ivex}{I^{\mathsf{vex}}_\infty}

\newcommand{\eric}[1]{\todo[inline,size=\tiny,color=green!30]{#1 \\ \hfill --- E.}}
\newcommand{\Eric}[1]{\todo[size=\tiny,color=green!30]{#1 \\ \hfill --- E.}}
\newcommand{\Jiayi}[1]{\todo[size=\tiny,color=blue!30]{#1 \\ \hfill --- J.}}



% \usepackage{lipsum}

%% define your title in the usual way
% \title[Optional shorter title]{Something to do with math}
\title{On some Grothendieck expansions}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author
\author[Eric Marberg and Jiayi Wen]{\texorpdfstring{Eric Marberg\thanks{\href{mailto: emarberg@ust.hk}{emarberg@ust.hk}. This work was supported by Hong Kong RGC grants 16306120 and 16304122.}\addressmark{1} \and Jiayi Wen\thanks{\href{mailto: jiayiwen@usc.edu}{jiayiwen@usc.edu}.} \addressmark{2}}{Eric Marberg and Jiayi Wen }}




%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, Hong Kong University of Science and Technology \\ \addressmark{2}Department of Mathematics, University of Southern California}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{
The complete flag variety admits a natural action by both the orthogonal group and the symplectic group.
Wyser and Yong defined orthogonal Grothendieck polynomials $\fkGO_z$ and symplectic Grothendieck polynomials $\fkGSp_z$ as the $K$-theory classes of the corresponding orbit closures. There is an explicit formula to expand $\fkGSp_z$ as a nonnegative sum of Grothendieck polynomials $\fkG$, which represent the $K$-theory classes of Schubert varieties. Although the constructions of $\fkGSp_z$ and $\fkGO_z$ are similar, finding the $\fkG$-expansion of $\fkGO_z$ or even computing $\fkGO_z$ is much harder. 
When $z$ is vexillary, it has been shown that $\fkGO_z$ has a nonnegative $\fkG$-expansion, but the $\fkG$-coefficients are mostly unknown. This paper derives several new formulas for $\fkGO_z$ and its $\fkG$-expansion when $z$ is vexillary. In particular, we prove that the latter expansion has a nontrivial stability property when $z(1)=1$.
}

%% put your French abstract here, or comment this out if you don't have one
% \resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{K-theory, Grothendieck polynomials, matrix Schubert varieties}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
% \usepackage[backend=bibtex,doi=false,isbn=false,url=false,eprint=false]{biblatex}
\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{ogroth.bib}

%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above







%\tableofcontents
%\section{Introduction}

% \section{Matrix Schubert varieties}

\section{Introduction}

This article is concerned with combinatorial formulas for expansions of three different families of 
\defn{Grothendieck polynomials} related to $K$-theory classes of Schubert varieties.
We first briefly introduce these polynomials in the context of \defn{matrix Schubert varieties}.
We will then discuss the central problem of interest and some of our partial solutions.

\subsection{Grothendieck polynomials}


Let $n$ be a positive integer. Write $S_n$ for the group of permutations of the integers $\ZZ$ with support in $[n]:=\{1,\dots,n\}$.
For $w \in S_n$,
the \defn{matrix Schubert variety} $\MX_w$ is the set of complex $n\times n$ matrices $M \in \Mat_{n\times n}$
whose upper $i\times j$ submatrices $M_{[i][j]}$ all satisfy 
$ \rank(M_{[i][j]}) \leq |\{ t \in [i]  : w(t) \leq j\}|$. % for all $i,j\in [n]$.
Results in \cite{KnutsonMiller} identify an equivariant $K$-theory class 
\[[\MX_w] \in \ZZ[a_1^{\pm1},a_2^{\pm1},\dots,a_n^{\pm1}] \cong K_T(\Mat_{n\times n }). \]
The \defn{Grothendieck polynomial} $\fkG_w$ introduced in \cite{FominKirillov} can be formed from
$[\MX_w]$
 by making the variable substitutions $a_i \mapsto 1+\beta x_i$ for all $i \in [n] $
and dividing by $(-\beta)^{\codim(\MX_w)}$. This function always belongs
to
 $\NN[\beta][x_1,x_2,\dots,x_n]$ and if $w \in S_\infty := \bigcup_{n\geq 1} S_n$
 then the value of $\fkG_w$ does 
 not 
depend on the choice of $n$ with $w \in S_n$. 
Moreover,
it is well-known \cite[Corollary~3.3]{MP2021Sp} that the family  $\{\fkG_w : w \in S_\infty \}$ is a $\ZZ[\beta]$-basis for $ \ZZ[\beta][x_1,x_2,\dots]$.



\subsection{Orthogonal Grothendieck polynomials}

Let $I_n = \{ z \in S_n : z=z^{-1}\}$ and  $I_\infty = \{ z \in S_\infty : z=z^{-1}\}$.
For $z \in I_n$ define
\[ \Mat^\O_{n\times n} = \left\{ X \in \Mat_{n\times n} : X^\top = X\right\}
\quand  \MX^\O_z = \MX_z \cap \Mat^\O_{n\times n}.\]
Results in \cite{MP2020} identify an equivariant $K$-theory class 
\[[\MX^\O_z] \in \ZZ[a_1^{\pm1},a_2^{\pm1},\dots,a_n^{\pm1}] \cong K_T(\Mat^\O_{n\times n }). \]
The \defn{orthogonal Grothendieck polynomial} $\fkGO_z$ introduced in \cite{WyserYong}  can be formed from 
$[\MX^\O_z]$
 by substituting $a_i \mapsto 1+\beta x_i$ for all $i \in [n]$
and dividing by $(-\beta)^{\codim(\MX^\O_z)}$. As with $\fkG_w$, this function always belongs to  $\NN[\beta][x_1,x_2,\dots,x_n]$ and 
 does not 
depend on the choice of $n$ with $z \in I_n$. 



 

%The \defn{visible descent set} of $z \in I_\infty$ is $\DesV(z) = \{i \in \PP : z(i) >  z(i+1) \leq i\}$.
When $z \in I_\infty$ is \defn{vexillary} in the sense of being $2143$-avoiding, it is known \cite[Proposition~3.29]{MS2023} that 
$ \fkGO_z \in \NN[\beta]\spanning\{\fkG_w : w\in S_\infty\}.$
It is an open problem to describe
the $\fkG$-expansion of $\fkGO_z$ explicitly. This problem is the focus of this article.

%
%\begin{example}
%The involution $z = (1,4)(2,5)$ is dominant of shape $ (3,3,2)$ 
%with  
%\[\ba
% \fkGO_z&=  (x_1\oplus x_1)(x_1\oplus x_2)(x_1\oplus x_3)(x_2\oplus x_2)(x_2\oplus x_3) \\
%&=4   \fkG_{25314} +
%4   \fkG_{35124} +
%4   \fkG_{24513} \\&\quad+
%\beta\(2    \fkG_{45123} +
%2    \fkG_{361245} +
%2    \fkG_{34512} +
%2    \fkG_{263145} +
%6   \fkG_{25413} +
%6    \fkG_{35214} \)\\&\quad+
% \beta^{2} \(\fkG_{461235} +
%2   \fkG_{264135} +
%3  \fkG_{35412} +
%3   \fkG_{362145} +
%3  \fkG_{45213}\)\\&\quad +
% \beta^{3} \(\fkG_{364125} +
%  \fkG_{45312} +
%2  \fkG_{462135}\) \\&\quad+
% \beta^{4} \fkG_{463125}.
% \ea
% \]
% \end{example}
 
 

 \subsection{Symplectic Grothendieck polynomials}

 It is instructive to contrast this open problem with what is known about the formally similar \defn{symplectic Grothendieck polynomials}. 
 Let $\Ifpf_n$ be the set of fixed-point-free involutions of $\ZZ$
 sending $i \mapsto i-1$ for all even integers $i \notin [n]$.
  %Observe that $\Ifpf_n$ is empty if $n$ is odd. % and define $\Ifpf_\infty = \bigcup_{n\geq 0} \Ifpf_n$.
 %In examples, we write elements of $\Ifpf_n$ as permutations of $[n]$. % and omit the trailing cycles $(2i-1,2i)$ for $i>n$.
 %
 For $z \in\Ifpf_n$ define
 \[ \Mat^\Sp_{n\times n} = \left\{ X \in \Mat_{n\times n} : X^\top = -X\right\}
 \quand  \MX^\Sp_z = \MX_z \cap \Mat^\Sp_{n\times n}.\]
 Just as in the previous two cases, there is an equivariant $K$-theory class \cite{MP2020}
 \[ [\MX^\Sp_z] \in \ZZ[a_1^{\pm1},a_2^{\pm1},\dots,a_n^{\pm1}] \cong K_T(\Mat^\Sp_{n\times n }). \]
 The \defn{symplectic Grothendieck polynomial} $\fkGSp_z$ introduced in \cite{WyserYong} can be formed from 
 $[\MX^\Sp_z]$ % \in K_T(\Mat^\Sp_{n\times n }) = \ZZ[a_1^{\pm1},a_2^{\pm1},\dots,a_n^{\pm1}]\]
  by substituting $a_i \mapsto 1+\beta x_i$ for all $i \in [n]$
 and dividing by $(-\beta)^{\codim(\MX^\Sp_z)}$. Once again
%  the resulting polynomial belongs to
  $\fkGSp_z\in\NN[\beta][x_1,x_2,\dots]$ and if $z \in \Ifpf_\infty:= \bigcup_{n\geq 1} \Ifpf_n$ then $\fkGSp_z$
  does not 
 depend on the choice of $n$ with $z \in \Ifpf_n$. 

% As with $\fkG_w$ but unlike $\fkGO_z$, the polynomials $\fkGSp_z$ are completely determined by a product formula
%  and a divided difference recurrence;
% see \cite[Theorem~3.8 and Proposition~3.11]{MP2020}.
% The product formulas applies when 
%  there exists a partition $\lambda$ with
% \[ \{ (i,j) \in D(z) : i < j\} = \{ (i,j) \in \D_\lambda : i < j\},\] in which case we have
% an identity similar to \eqref{o-prod-formula}:
% \be \fkGSp_z = \prod_{\substack{(i,j) \in \D_\lambda \\ i<j}} x_i \oplus x_j.\ee
Like $\fkGO_z$ (at least for vexillary $z$), there is a positive $\fkG$-expansion of each $\fkGSp_z$.
Unlike $\fkGO_z$, this expansion can be explicitly computed in the following way.

 %then 
 %\[ \fkGSp_z = \prod_{\substack{(i,j) \in \D_\lambda \\ i> j}} x_i\oplus x_j 
 %.\]
 %There is a divided difference recurrence . . . 


  Write $w_i=w(i)$ for $w \in S_\infty$ and $i \in \ZZ$. Let $\approx$ be the transitive closure of the relation
  on $S_\infty$ that has $v^{-1} \approx w^{-1}$ if there is an even index $i \in 2\NN$
  and integers $a<b<c<d$ such that 
 $ v_{i+1}v_{i+2}v_{i+3}v_{i+4}$ and $w_{i+1}w_{i+2}w_{i+3}w_{i+4}$ are both in $ \{adbc,bcad,bdac\}$, while $v_j=w_j$ for all $j \notin \{i+1,i+2,i+3,i+4\}$.
For example, 
 in one-line notation, we have
 \[
 (\hs15\underline{2634}\hs)^{-1} \approx (\hs15\underline{3624}\hs)^{-1} \approx (\hs\overline{15\underline{34}}\underline{26}\hs)^{-1} \approx (\hs\overline{3415}26\hs)^{-1} \approx (\hs\overline{3514}26\hs)^{-1}.
 \]
 %Here, the underline and overline annotations indicate the relevant subwords $ v_{i+1}v_{i+2}v_{i+3}v_{i+4}\approx w_{i+1}w_{i+2}w_{i+3}w_{i+4}$ that manifest each equivalence.
 Given $z \in \Ifpf_\infty$ let $a_1<a_2<a_3<\dots$ be the integers with $0<a_i<b_i:=z(a_i ) $
 and define 
 $\alpha_\fpf(z)$ to be inverse of the one-line permutation 
 $a_1 b_1a_2 b_2 a_3b_3\cdots.$
% Set  $ \cBfpf(z) = \{ w \in S_\infty : w \approx \alpha_\fpf(z)\}$.
Then \cite[Theorem~3.12]{MP2020}
\[\textstyle
 \fkGSp_z = \sum_{w\approx \alpha_\fpf(z)} \beta^{\ell(w) - \ell(\alpha_\fpf(z))} \fkG_w.
 \]


\subsection{Grothendieck expansions}

For each $z \in I_\infty$ there exists a  \defn{orthogonal Grothendieck coefficient function} $\GC_{z} : S_\infty \to \ZZ$
such that $\fkGO_z = \sum_{w \in S_\infty} \GC_z(w) \cdot \beta^{\ell(w) - \ellhat(z)} \cdot \fkG_w$.
% \be
% \fkGO_z = \sum_{w \in S_\infty} \GC_z(w) \cdot \beta^{\ell(w) - \ellhat(z)} \cdot \fkG_w.
% \ee
%This holds as the set $\{\fkG_w : w \in S_\infty\}$ is a $\ZZ[\beta]$-basis for $\ZZ[\beta][x_1,x_2,\dots]$
%and since $\fkGO_z$ and $\fkG_w$ are both homogeneous, of respective degrees $\ellhat(z)$ and $\ell(w)$, when $\deg(x_i) = 1$ and $\deg(\beta)=-1$. These facts also imply that 
The support 
$
\supp(\GC_z) := \{ w \in S_\infty : \GC_z(w) \neq 0\}
$ %of the \defn{Grothendieck coefficient function}
must be a finite set of permutations. 
%It is an open problem to determine $\GC_z$ for general $z \in I_\infty$, or even to compute this function in an efficient way.
As noted earlier, when $z \in I_\infty$ is vexillary, it is known that $\GC_z : S_\infty \to \NN$ takes all nonnegative values, but otherwise little is known about this function in the literature to date.

We mention that if one sets $\beta=0$ then $\fkG_w$, $\fkGO_y$, and $\fkGSp_z$
turn into the \defn{(involution) Schubert polynomials} $\fkS_w$, $\iS_y$, and $\Sfpf_z$ studied in \cite{HMP1,Manivel},
for which the relevant expansions are all much simpler: 
both $\iS_y$ and $\Sfpf_z$ are equal to a constant (which is $1$ in the second case) times a multiplicity-free sum of $\fkS_w$'s. Moreover, the terms that appear are predicted by a general formula of Brion \cite{Brion}
and are described combinatorially in \cite{CJW}.
 
This work contains the first explicit results about the coefficient functions $\GC_z$.
Our main theorems can be summarized as follows.
In Sections~\ref{iG-sect} and \ref{qd-sect} we derive an exact, though not obviously positive formula for $\GC_z $ when $z$ is any \defn{quasi-dominant} involution (see \Cref{dom-thm}). 
Then we explain a new formula for $\fkGO_z$ when $z$ is any vexillary involution, which shows that 
$\GC_z$ is \defn{shift invariant} whenever $z(1)=1$ (see Theorems~\ref{vex-thm} and \ref{gg-thm}).
These results lead to a new proof of the existence of \defn{stable limits} of orthogonal Grothendieck polynomials.
 Finally, we compute $\GC_z$ in some special cases in \Cref{special-sect}.


\section{Product formulas and divided difference operators}

So far we have not discussed any method of computing $\fkG_w$ or $\fkGO_z$ as polynomials,
let alone the expansion of the latter in terms of the former. 
This section quickly reviews an algebraic method to compute $\fkG_w$, which can be adapted to $\fkGO_z$ when $z$ is vexillary.

The group $S_\infty$ acts on $\ZZ[\beta][x_1,x_2,\dots]$ by permuting the $x_i$ variables.
For each $i \in \PP$ the \defn{divided difference operators}  $\partial_i$ and  $\bpartial_i$ act on $ \ZZ[\beta][x_1,x_2,\dots]$
by the formulas
\be
\partial_i f =  \tfrac{f-s_i f}{x_i-x_{i+1}} \quand \bpartial_i f = \partial_i\big((1+\beta x_{i+1}) f\big) = -\beta f + (1+\beta x_i) \partial_i f.
\ee
These operators satisfy the Coxeter braid relations for $S_\infty$ along with %$\partial_i\partial_i = 0$ and 
$\bpartial_i \bpartial_i = -\beta \bpartial_i$.


The \defn{Rothe diagram} of $w \in S_\infty$ is the set
$ D(w)$ of positive integer pairs $ (i, j)$ satisfying both $ i<w^{-1}(j)$ and $j< w(i)$.
A permutation $w \in S_\infty$ is \defn{dominant} 
if
there is a partition $\lambda$ such that 
$D(w)$
coincides with the \defn{Young diagram} $\D_\lambda := \{(i,j)  : 1 \leq j \leq \lambda_i\}$.
% \be \D_\lambda := \{(i,j) \in \PP\times \PP : 1 \leq j \leq \lambda_i\}.\ee
In this case we say that $w$ is of shape $\lambda$. 

There is a unique dominant $w \in S_\infty$ of each partition shape $\lambda$,
and for this permutation $\fkG_w = \prod_{(i,j) \in \D_\lambda} x_i  $ \cite{FominKirillov}.
Moreover, for any $w \in S_\infty$,
one has $\bpartial_i \fkG_w =  \fkG_{ws_i} $ if
 $i \in \DesR(w) := \{ i \in \ZZ  : w(i)>w(i+1)\}$  \cite{FominKirillov}.
%  then
%\be\label{Grothendieck-recursion}
%\bpartial_i \fkG_w = \begin{cases} \fkG_{ws_i} &\text{if }i \in\DesR(w) := \{ i \in \ZZ  : w(i)>w(i+1)\} \\ -\beta \fkG_w&\text{otherwise}.\end{cases}
%\ee
These formulas determine 
$\fkG_w$ for all $w$.



Suppose $z \in I_\infty$ is dominant of shape $\lambda$. Then 
$z$ is also vexillary and
$\lambda=\lambda^\top$ is necessarily a \defn{symmetric} partition since $D(z) = D(z^{-1}) = D(z)^\top$, and one has  $i<z(i)$ if and only if $(i,i) \in \D_\lambda$.
For dominant involutions, one again has a product formula 
\be\label{idom-eq} \textstyle\fkGO_z = \prod_{\substack{(i,j) \in \D_\lambda \\ i\leq j}} x_i\oplus x_j \quad\text{where $x\oplus y := x+y+\beta xy$ \cite[Theorem~3.8]{MP2020}.}
\ee
Moreover, if $z \in I_\infty$ is vexillary and $i \in \DesR(z)$ is such that $s_i z s_i\neq z$ is also vexillary, then the formula $\bpartial_i \fkGO_z=\fkGO_{s_izs_i}$ also holds \cite[Proposition~3.23]{MP2020}.
This recurrence, combined with the product formula \eqref{idom-eq}
can be used to calculate $\fkGO_z$ for any vexillary $z \in I_\infty$. 


 For involutions $z \in I_\infty$ that are not vexillary,
no simple algebraic formula is known for computing $\fkGO_z$. In particular, these polynomials cannot be expressed using $\bpartial_i$'s. 
We mention that by contrast, the polynomials $\fkGSp_z$ are completely determined by a product formula
  and a divided difference recurrence;
 see \cite[Theorem~3.8 and Proposition~3.11]{MP2020}.

\section{Involution Grothendieck polynomials}\label{iG-sect}

There is another family of polynomials indexed by involutions $z \in I_\infty$ that share several favorable algebraic properties with $\fkGSp_z$,
and will turn out to be closely related to $\fkGO_z$.

%Let $\leq$ denote the \defn{(strong) Bruhat order} on $S_\infty$,
%which is the transitive closure of the relation with $v<w$ whenever $\ell(v) < \ell(w)$ and $w=vt$ for any transposition $t=(i, j) \in S_\infty$.
%For each $u, v \in S_\infty$ there is a unique element $w \in S_\infty$ such that the set-wise product of lower intervals
%\[
%\{ x\in S_\infty : x \leq u\} \{ x \in S_\infty : x \leq v\}  \text{ is equal to } \{x \in S_\infty : x \leq w\}.
%\]
%If we set $u\circ v := w$ then $\circ$ gives an associative operation $ S_\infty \times S_\infty \to S_\infty$, sometimes called the \defn{Demazure product}. 

%Using standard properties of the Bruhat order \cite[Section~2.2]{CCG}, one can show that 

Write $\ell : S_\infty \to \NN$ for the usual Coxeter length function counting the number of inversions of a permutation.
The \defn{Demazure product}  is the unique associative operation $\circ :  S_\infty \times S_\infty \to S_\infty$ with
 $u\circ v = uv$ if and only if $\ell(uv) = \ell(u)+\ell(v)$, and with $s_i\circ s_i = s_i$ for simple transpositions $s_i:=(i,i+1) \in S_\infty$.
 %
The formula $ w\mapsto w^{-1}\circ w$ is a surjective map $S_\infty\to I_\infty$,
so the set $\cB(z) := \{w \in S_\infty : w^{-1} \circ w = z\}$ is nonempty for $z \in I_\infty$.
Define the \defn{involution Grothendieck polynomial} of $z$ to be
\be\textstyle \iG_z := \sum_{w \in \cB(z)} \beta^{\ell(w) - \ellhat(z)} \fkG_w
\quad\text{where } \ellhat(z) :=\min\{\ell(w) : w \in \cB(z)\}.\ee

%\begin{definition} Fix $z \in I_\infty$. Then let \[\cB(z) = \{w \in S_\infty : w^{-1} \circ w = z\}
%\quand \ellhat(z) =\min\{\ell(w) : w \in \cB(z)\}.\]
%The \defn{involution Grothendieck polynomial} of $z$ is finally given by $\iG_z = \sum_{w \in \cB(z)} \beta^{\ell(w) - \ellhat(z)} \fkG_w$.
%% \[ \textstyle \iG_z = \sum_{w \in \cB(z)} \beta^{\ell(w) - \ellhat(z)} \fkG_w.\]
%\end{definition}

The set $\cB(z)$ was extensively studied in \cite{HMP2}
%and the polynomials $\iG_z$ were introduced in \cite[Section 4]{MP2021}. 
% Like $\cBfpf(z)$, 
and can be generated using a certain equivalence relation.
Let $\sim$ be the transitive closure of the relation
 on $S_\infty$ that has $v^{-1} \sim w^{-1}$ if there is an index $i \in \PP$
 and integers $a<b<c$ such that 
$ v_{i}v_{i+1}v_{i+2} $ and $w_{i}w_{i+1}w_{i+2} $ are both in $ \{cba,cab,bca\}$, while $v_j=w_j$ for all $j \notin \{i,i+1,i+2\}$.
%More concisely, this is the transitive relation on $S_\infty$ with 
%\[ (\cdots cab \cdots)^{-1} \sim (\cdots cba \cdots)^{-1} \sim (\cdots bca \cdots)^{-1}\quad\text{for all }a<b<c,\]
%where corresponding ellipses ``$\cdots$'' mask identical subsequences.

For $z \in I_\infty$ let $a_1<a_2<\dots$ be the positive integers with $a_i\leq b_i:=z(a_i ) $.
Define 
$\alpha_\inv(z)$ to be inverse of the permutation whose one-line notation is formed by removing the repeated letters from
$b_1 a_1b_2 a_2 b_3a_3\cdots.$
Then by \cite[Section~6.1]{HMP2} and \cite[Section~3]{HMP1} we have 
\be\label{cB-eq}
\cB(z) = \{ w \in S_\infty : w \sim \alpha_\inv(z)\}
\quand \ellhat(z) = \ell(\alpha_\inv(z))
.\ee
For example, 
 $ \cB( 45312 ) = \{\alpha_\inv =24513,\ 25413,\ 25314,\ 35214,\ 35124\}$.
%\in I_\infty$ then $\alpha_\inv(z) = 41523^{-1} = 24513$ and
%Writing elements of $S_\infty$ in one-line notation, we have
%\[
%\ba \cB(z) &= \{\alpha_\inv(z) =24513,\ 25413,\ 25314,\ 35214,\ 35124\} \\&= \left\{41523^{-1},\ 41532^{-1},\ 41352^{-1},\ 43152^{-1},\ 34152^{-1}\right\}
%.
%\ea
%\]
%Here, the underline and overline annotations indicate the relevant subwords $ v_{i+1}v_{i+2}v_{i+3}v_{i+4}\approx w_{i+1}w_{i+2}w_{i+3}w_{i+4}$ that manifest each equivalence.
%\end{example}
The polynomials $\iG_z$ were previously considered in \cite[Section~4]{MP2021}, but
the following theorem is new.

\begin{theorem}\label{iG-thm}
 Suppose $z \in I_\infty$. Then for any $i \in \PP$  it holds that
\[\bpartial_i \iG_z = \begin{cases} \iG_{zs_i} &\text{if $i \in \DesR(z)$ and $z(i) = i+1$}
\\ 
\iG_{s_i z s_i} &\text{if $i \in \DesR(z)$ and $z(i) \neq i+1$}
\\
-\beta \iG_z &\text{if $i \notin \DesR(z)$.}\end{cases}
\]
Moreover, if $z $ is dominant of shape $\lambda$ then $\ds\iG_z = \prod_{\substack{(i,j) \in \D_\lambda \\ i = j}} x_i \prod_{\substack{(i,j) \in \D_\lambda \\ i < j}} x_i\oplus x_j$.
% \be \iG_z = \prod_{\substack{(i,j) \in \D_\lambda \\ i = j}} x_i \prod_{\substack{(i,j) \in \D_\lambda \\ i < j}} x_i\oplus x_j.\ee
\end{theorem}

% \begin{proof}
% From \Cref{iG-thm2-lem}, this result is a special case of \cite[Theorem~4.5]{MP2021}, which states that for  any $z \in I_\infty$ one has  $ \textstyle \iG_z = \sum_D \(\prod_{(i,j) \in D, i = j} x_i \prod_{(i,j) \in D, i \neq j} x_i\oplus x_j\)$ where $D$ runs over all involution Hecke pipe dreams for $z$.
% \end{proof}

%For a partition $\lambda$ define $k(\lambda) = |\{ (i,j) \in \D_\lambda : i=j\}|$.

\begin{example}
If  $z = 45312=(1,4)(2,5) \in I_\infty$ then $z$ is dominant of shape $(3,3,2)$
and
%so from \Cref{cB-ex2} and \Cref{iG-thm2} we get the two expressions
\[ \ba \iG_z &=
\fkG_{24513} + \beta\fkG_{25413} + \fkG_{25314} +\beta \fkG_{35214} + \fkG_{35124}
= x_1x_2(x_1\oplus x_2)(x_1\oplus x_3)(x_2\oplus x_3)
.\ea
\]
\end{example}




\section{Grothendieck expansions in the quasi-dominant case}\label{qd-sect}

We now explain how to leverage the $\fkG$-expansion of $\iG_z$ to get information about $\fkGO_z$.
Let $\Ivex$ be the set of vexillary (i.e., $2143$-avoiding) involutions in $ I_\infty$.
An element $z \in \Ivex$ is \defn{quasi-dominant} if  $i-1<z(i-1)$ whenever $1<i<z(i)$.
Every dominant involution is quasi-dominant. % since dominant permutations are $132$-avoiding.
Define
$ k(z) :=\min\{ i \in \NN : (j,j) \notin D(z) \text{ for all } j>i\}$ for $z \in I_\infty$.
%The next result, connecting $\fkGO_z$ to $\iG_z$, is almost immediate from \Cref{iG-thm}:

\begin{theorem}\label{qd-thm}
If $z \in \Ivex$ is quasi-dominant with $k=k(z)$
then $\fkGO_z = \iG_z \prod_{i=1}^{k} (2 + \beta x_i).$
\end{theorem}

We can turn this theorem into an exact, though not manifestly positive, formula for the coefficient function $\GC_z : S_\infty \to\NN$ with $\fkGO_z = \sum_{w \in S_\infty} \GC_z(w) \cdot \beta^{\ell(w) - \ellhat(z)} \cdot \fkG_w$.

 
 

 
 
 Following the notation in \cite{LenartSottile}, we write 
$v\xrightarrow{(a,b)}w$ for $v,w \in S_\infty$ and positive integers $a<b$
 to indicate that $w = v(a,b)$ and $\ell(w) = \ell(v)+1$, meaning that $w$ covers $v$ in the Bruhat order on $S_\infty$.
  The length condition holds precisely when $v(a) < v(b)$ and 
 no $i$ with $a<i<b$ has $v(a) < v(i) <v(b)$. 
%
 Fix a positive integer $k$. The \defn{$k$-Bruhat order} on $S_\infty$  is
 transitive closure of the relation  with $v<_k w$ whenever $v \xrightarrow{(a,b)}w$ and $a\leq k <b$.
 %For integer pairs $(a,b)$ and $(c,d)$ we write 
 %\be 
 %(a,b) \prec (c,d)\quad\text{to mean that either}\quad (b>d) \text{ or }(b=d\text{ and }a<c).
 %\ee
 \begin{definition}\label{kpieri-def}
 An \defn{unmarked $k$-Pieri chain} between from $v\in S_\infty$ to $w \in S_\infty$ is a saturated chain in $k$-Bruhat order of the form 
$v=v_0 \xrightarrow{(a_1,b_1)} v_1 \xrightarrow{(a_2,b_2)} \cdots \xrightarrow{(a_q,b_q)} v_q = w$
 satisfying %$1\leq a_i \leq k < b_i$ for all $i \in [q]$ along with the following two conditions:
%  \ben
% %  \item[(P0)]  if $a_j =a_i$ for some $1 \leq j <i < q$ then $(a_i,b_i) \prec (a_{i+1},b_{i+1})$; and
%  \item[(P0)]  If $a_j =a_i > a_{i+1}$ for some $1 \leq j <i < q$ then $b_i>b_{i+1}$. 
%  \item[(P1)] $b_1\geq b_2\geq \dots \geq b_q$.
%  \een 
 $b_1\geq b_2\geq \dots \geq b_q$ and $b_i>b_{i+1}$ if $a_j =a_i > a_{i+1}$ for some $1 \leq j <i < q$.% then $b_i>b_{i+1}$. 
 \end{definition}
We write $v \xrightarrow{c(k)} w$ if such a chain exists.
 An essential and non-obvious property of this definition is 
 that for any permutations $v,w \in S_\infty$ at most one unmarked $k$-Pieri chain exists from $v$ to $w$. See \cite[Theorem~2.2]{LenartSottile}, which also explains how to construct this chain.

 Suppose  $v=v_0 \xrightarrow{(a_1,b_1)} v_1 \xrightarrow{(a_2,b_2)} \cdots \xrightarrow{(a_q,b_q)} v_q = w$
is the unique unmarked $k$-Pieri chain from $v\in S_\infty$ to $w \in S_\infty$.
 Define $\forced_k(v,w)$ to be the number of indices $i \in [q]$ such that either 
$b_1=\dots=b_i$ and $a_1>\dots>a_i$, or
 $b_i=b_{i+1}$ and $a_i>a_{i+1}$.
 % NOT ($b_i>b_{i+1}$, or $b_i=b_{i+1}$ and $a_{i}<a_{i+1}$)
 %$(a_i,b_i) \not \prec (a_{i+1},b_{i+1})$.
Also 
let $\prohib_k(v,w)$ be the number of indices $i \in [q]$ such that 
 $a_j=a_i$ for some $1\leq j <i$.

% Notice that we have $\forced_k(v,w)\geq1$ if $v\neq w$ and that
% \be\label{fp-obs}
% \forced_k(v,w) + \prohib_k(v,w) \leq   \ell(w) - \ell(v)
% \quand
% k  \leq   \ell(w) - \ell(v) - \prohib_k(v,w).
% \ee
%For permutations $v,w \in S_\infty$, recall the definitions of $\forced_k(v,w)$ and $\prohib_k(v,w)$ from \Cref{lensot-thm1} and 
Now, for $v=w$ set $\epsilon_k(v,w) = 1$ and $\rho_k(v,w)=2^k$, and for $v \xrightarrow{c(k)} w\neq v$ define
\be
\epsilon_k(v,w) = (-1)^{1+\forced_k(v,w)}
\quand
\rho_k(v,w) = 2^{k + \ell(v) - \ell(w) + \prohib_k(v,w)}.
\ee
Set $\epsilon_k(v,w)= \rho_k(v,w)=0$ when we do not have $v \xrightarrow{c(k)} w$. 

\begin{theorem}\label{dom-thm}
Suppose $z \in \Ivex$ is quasi-dominant   and $k=k(z)$.  Then
\[\textstyle \GC_z(w) = \sum_{v \in \cB(z)} 
\epsilon_k(v,w)   \rho_k(v,w) \geq 0\quad\text{for all $w \in S_\infty$}.\]
\end{theorem}








Fix $1\neq z \in I_\infty$, define $k=k(z)=\min\{ i \in \NN : (j,j) \notin D(z) \text{ for all } j>i\}$ as above, and  
 let $j=j(z)$ be the largest integer with $z(i) = i$ for all $1\leq i\leq j$. 
For  $v,w \in S_\infty$ we write $v \xrightarrow{[z]}w$ if there exists an unmarked $k$-Pieri chain
%$ v=v_0 \xrightarrow{(a_1,b_1)} v_1 \xrightarrow{(a_2,b_2)} \cdots \xrightarrow{(a_q,b_q)} v_q = w$
as in \Cref{kpieri-def} that has
$ j \leq a_i \leq k < b_i $ and also either $a_i<z(a_i)$ or $z(b_i)<b_i$ for each $i\in[q]$.
Finally, 
let 
\be\cB^+(z) := \left\{ w \in S_\infty : v\xrightarrow{[z]} w \text{ for some }v \in \cB(z)\right\}.\ee
Also define $\cB^+(1) = \cB(1) = \{1\}$.
Based on computations and the preceding theorem, the set $\cB^+(z)$ appears to give a good approximation for $\supp(\GC_z)$. In particular:

%\begin{example}
%If $z=1 \in I_\infty$ then $j(z) = 1>0=k(z)$ so only empty chains satisfy the conditions in \Cref{zpieri-def},
%and thus $\cB^+(1) = \cB(1) = \{1\}$.
%\end{example}

%There is a useful case when $\cB^+(z)$ is equal to the set in \eqref{dom-supp-eq}.

\begin{corollary}
If $z \in \Ivex$ is quasi-dominant and $k=k(z)$ then
\[
\supp(\GC_z) \subseteq \cB^+(z)
=\left\{ w \in S_\infty : v\xrightarrow{c(k)} w \text{ for some }v \in \cB(z)\right\}.
\]
\end{corollary}

% \begin{proof}
% In this case $j(z)=1$ and $k(z)=k$, so $a<z(a)$ for all $j\leq a \leq k$.
% Thus all unmarked $k$-Pieri chains from $v$ to $w$ satisfy the conditions in \Cref{zpieri-def}, 
% \end{proof}

%For $a\leq b$  let $[a,b] = \{ i \in \ZZ : a\leq i \leq b\}$.
%For $z \in I_\infty$ let 
%$\supp(z) = \{ i \in \ZZ : z(i) \neq i\} .$
%
%\begin{proposition}
%If $z \in I_\infty$ has $\supp(z)  \subseteq [a,b]$ for integers $a\leq b$,
%then 
%$\supp(v) \subseteq [a,b]$
%for all $v \in \cB(z)$ and $ \supp(w) \subseteq [a-1,b+1]$ for all $w \in \cB^+(z)$.
%\end{proposition}

We have used a computer to verify the following for all vexillary $z \in I_{11}$:



\begin{conjecture}\label{b+conj}
If $z \in \Ivex$ then $\cB(z)\subseteq \supp(\GC_z) \subseteq \cB^+(z)$.%$\fkGO_z \in \NN[\beta]\spanning\left\{\fkGO_w : w \in \cB^+(z) \right\}$.
\end{conjecture}

%It frequently happens that   the set $\cB^+(z)$ is equal to or  overcounts $\supp(\GC_z)$ by just a few elements. For example . . . TODO
% \eric{todo: can this be proved?}


%  \eric{the following propositions are maybe superfluous, will be generalized and reformulated in \Cref{special-sect}}
Both containments in this conjecture can be strict, 
but in some notable cases we actually have equality $\supp(\GC_z) = \cB^+(z)$.
Below are some relevant examples.
%First consider the transposition
%$
%t_{ij} := (i, j) \in I_\infty$ 
%for $i<j$.



 \begin{example}\label{t-ex}
 Suppose $t=t_{n}:=(1, n)\in I_n$ is a transposition.
 Then $t$ is dominant of shape $\lambda=(n-1,1^{n-2})$ and $ \cB(t)$ consists of the permutations in $S_n$
 whose \textbf{inverses} in one-line notation are the shuffles of $n1$ and $234\cdots(n-1)$
 with at most one letter between $n$ and $1$.
 The larger set $\cB^+(t)$ consists of the permutations in $S_{n+1}$
 whose \textbf{inverses} in one-line notation are the shuffles of the words $n1$ and $234\cdots(n-1)(n+1)$
 with at most two letters between $n$ and $1$, excluding the inverse of $234\cdots(n-1)(n+1)n1$.
 In this case it can be proved that $\supp(\GC_t) = \cB^+(t)$.
 \end{example}



 It is useful to represent $\cB^+(z)$ as the following directed graph.
For  $v,w \in S_\infty$ we write $v \lessdot_L w$ when
 $\ell(w) = \ell(v)+1$ and $ w=s_i v$ for some $ i\in \PP$.
We then turn  $\cB^+(z)$ into a directed graph by adding edges $v \to w$
whenever   $v \lessdot_L w$. 
\Cref{t1n-fig} shows some instances of this graph corresponding to
the previous and next two examples.


%The permutation 
%$g_{1,n} := [n^n; n]%= (n+1)(n+2)\cdots 2n, 1,2\cdots n 
%= (1, n+1)(2, n+2)\cdots(n, 2n) \in I_{2n}$
%is both $n$-Grassmannian and dominant of shape $\lambda=(n^n)$. 

 \begin{example}\label{g-ex}
Suppose $g=g_n:=  (1, n+1)(2, n+2)\cdots(n, 2n) \in I_{2n}$. Then $g$ is dominant of shape $\lambda=(n^n)$ and 
 the set $ \cB(g)$ consists of the single element whose \textbf{inverse} is 
 $
 (n+1)1(n+2)2\cdots  (2n)n\in S_{2n}.$
 The larger set 
 $\cB^+(g_{1,n})$ consists of the $2^n$ permutations   whose \textbf{inverses} have the form 
 $ (n+1)a_1b_1a_2b_2\cdots a_nb_n \in S_{2n+1}$
where $\{a_i,b_i\} = \{ i, n+1+i\}$ for each $i \in [n]$.
 %See \Cref{gn-fig}.
It again can be proved that $\supp(\GC_g) = \cB^+(g)$.
 \end{example}
 
 

% \begin{figure}[ht]
%\centerline{\includegraphics[scale=0.3]{g12.png}\qquad \includegraphics[scale=0.3]{g13.png}}
%\caption{The directed graphs $\cB(z)  \subseteq \cB^+(z)$ when $z \in I_\infty$ is $g_{2}=(1,3)(2,4)$ and $g_{3}=(1,4)(2,5)(3,6)$, presented using the conventions as in \Cref{t1n-fig}.}\label{gn-fig}
%\end{figure}


\begin{example}
Finally let $w_0 = n\cdots 321\in I_n$ be the longest element of $S_n$. Then $w_0$
is dominant of shape $\lambda=(n-1,\dots,3,2,1)$.
The set $\cB(w_0)$ does not have any description simpler than \eqref{cB-eq}.
One can show that
$ \cB^+(w_0)$ is  the set of the permutations in $S_{n+1}$ whose \textbf{inverses} in one-line notation have the form 
$
u_1\cdots u_i(n+1) u_{i+1}\cdots u_n 
$
where $u=u_1u_2\dots u_n $ is the \textbf{inverse} of an element of $ \cB(w_0)$ and $i \in [n]$ has $n \geq 2 u_j$ for each $i<j\leq n$. %See \Cref{w0-fig}.
We conjecture, but do not know how to prove, that $\supp(\GC_{w_0}) = \cB^+(w_0)$ for all $n$.
This has been checked by computer for $n\leq 11$.
\end{example}


 

Extending this example,
let $w_{ij} := (i, j)(i+1,j-1)(i+2,j-2)\cdots (i+k,j-k) \in \Ivex$ for any integers $1\leq i<j$ where $k=\lfloor\tfrac{j-i-1}{2}\rfloor$. Computations support the following:

 \begin{conjecture} 
It holds that
$\supp(\GC_{w_{ij}}) = \cB^+(w_{ij})$ if and only if $i=1$ or $j-i$ is odd.
\end{conjecture}


 

\definecolor{lightskyblue}{RGB}{176,226,255}
\begin{figure}[ht]
    \centerline{
        \resizebox{\textwidth}{!}{
      \begin{tikzpicture}[xscale=2, yscale=1.68,>=latex,baseline=(z.base)]
      \node at (0,1.0) (z) {};
        \node[draw] at (0.5,0) (11) {\tiny $4231^{-1}:1$};
        \node[draw] at (1.5,0) (12) {\tiny $24351^{-1}:1$};
        \node[draw,fill=lightskyblue] at (0,1) (21) {\tiny $4213^{-1}:3$};
        \node[draw,fill=lightskyblue] at (1,1) (22) {\tiny $2431^{-1}:3$};
        \node[draw] at (2,1) (23) {\tiny $23451^{-1}:1$};
        \node[draw,fill=lightskyblue] at (0,2) (31) {\tiny $4123^{-1}:2$};
        \node[draw,fill=lightskyblue] at (1,2) (32) {\tiny $2413^{-1}:2$};
        \node[draw,fill=lightskyblue] at (2,2) (33) {\tiny $2341^{-1}:2$};
        %
        \draw[->,thick]  (31) -- (21);
        \draw[->,thick]  (32) -- (21);
        \draw[->,thick]  (32) -- (22);
        \draw[->,thick]  (33) -- (22);
        \draw[->,thick]  (33) -- (23);
        \draw[->,thick]  (21) -- (11);
        \draw[->,thick]  (22) -- (11);
        \draw[->,thick]  (22) -- (12);
        \draw[->,thick]  (23) -- (12);
       \end{tikzpicture}
       \quad
       \begin{tikzpicture}[xscale=2, yscale=1.25,>=latex,baseline=(z.base)]
        \node at (0,1.5) (z) {};
        \node[draw] at (1,0) (00) {\tiny $4516273^{-1}:1$};
        \node[draw] at (0,1) (21) {\tiny $4156273^{-1}:2$};
        \node[draw] at (1,1) (22) {\tiny $451623^{-1}:2$};
        \node[draw] at (2,1) (23) {\tiny $4512673^{-1}:2$};
        \node[draw] at (0,2) (31) {\tiny $415623^{-1}:4$};
        \node[draw] at (1,2) (32) {\tiny $4152673^{-1}:4$};
        \node[draw] at (2,2) (33) {\tiny $451263^{-1}:4$};
        \node[draw, fill=lightskyblue] at (1,3) (41) {\tiny $415263^{-1}:8$};
          %
          \draw[->,thick]  (21) -- (00);
          \draw[->,thick]  (22) -- (00);
          \draw[->,thick]  (23) -- (00);
          \draw[->,thick]  (31) -- (21);
          \draw[->,thick]  (31) -- (22);
          \draw[->,thick]  (32) -- (21);
          \draw[->,thick]  (32) -- (23);
          \draw[->,thick]  (33) -- (22);
          \draw[->,thick]  (33) -- (23);
          \draw[->,thick]   (41) -- (31);
          \draw[->,thick]   (41) -- (32);
          \draw[->,thick]   (41) -- (33);
         \end{tikzpicture}
         \quad
         \begin{tikzpicture}[xscale=1.8, yscale=1,>=latex,baseline=(z.base)]
            \node at (0,2.0) (z) {};
            \node[draw] at (2,0) (00) {\tiny $43521^{-1}:1$};
            \node[draw] at  (1,1) (11)   {\tiny $43251^{-1}:3$};
            \node[draw] at  (2,1) (12)   {\tiny $34521^{-1}:1$};
            \node[draw] at  (3,1) (13)   {\tiny $43512^{-1}:1$};
            \node[draw] at (0,2) (21) {\tiny $42351^{-1}:2$};
            \node[draw] at (1,2) (22) {\tiny $34251^{-1}:3$};
            \node[draw, fill=lightskyblue] at (2,2) (23) {\tiny $4321^{-1}:6$};
            \node[draw] at (3,2) (24) {\tiny $34512^{-1}:1$};
            \node[draw] at (4,2) (25) {\tiny $43152^{-1}:3$};
            \node[draw,fill=lightskyblue] at (0,3) (31) {\tiny $4231^{-1}:4$};
            \node[draw] at (1,3) (32) {\tiny $32451^{-1}:2$};
            \node[draw,fill=lightskyblue] at (2,3) (33) {\tiny $3421^{-1}:6$};
            \node[draw] at (3,3) (34) {\tiny $34152^{-1}:2$};
            \node[draw,fill=lightskyblue] at (4,3) (35) {\tiny $4312^{-1}:6$};
            \node[draw] at (5,3) (36) {\tiny $41352^{-1}:2$};
            \node[draw, fill=lightskyblue] at (1.5,4) (41) {\tiny $3241^{-1}:4$};
            \node[draw, fill=lightskyblue] at (3,4) (42) {\tiny $3412^{-1}:4$};
            \node[draw, fill=lightskyblue] at (4.5,4) (43) {\tiny $4132^{-1}:4$};
              %
              \draw[->,thick]  (11) -- (00);
              \draw[->,thick]  (12) -- (00);
              \draw[->,thick]  (13) -- (00);
              \draw[->,thick]  (21) -- (11);
              \draw[->,thick]  (22) -- (11);
              \draw[->,thick]  (22) -- (12);
              \draw[->,thick]  (23) -- (11);
              \draw[->,thick]  (24) -- (12);
              \draw[->,thick]  (24) -- (13);
              \draw[->,thick]  (25) -- (13);
              \draw[->,thick]  (31) -- (21);
              \draw[->,thick]  (31) -- (23);
              \draw[->,thick]  (32) -- (22);
              \draw[->,thick]  (33) -- (22);
              \draw[->,thick]  (33) -- (23);
              \draw[->,thick]  (34) -- (24);
              \draw[->,thick]  (34) -- (25);
              \draw[->,thick]  (35) -- (23);
              \draw[->,thick]  (35) -- (25);
              \draw[->,thick]  (36) -- (25);
              \draw[->,thick]   (41) -- (32);
              \draw[->,thick]   (41) -- (33);
              \draw[->,thick]   (42) -- (33);
              \draw[->,thick]   (42) -- (34);
              \draw[->,thick]   (42) -- (35);
              \draw[->,thick]   (43) -- (35);
              \draw[->,thick]   (43) -- (36);
             \end{tikzpicture}      }     
  }

% \centerline{
% %\includegraphics[scale=0.3]{t12.png}\qquad 
% \includegraphics[scale=0.3]{t14.png}
% \qquad
% \includegraphics[scale=0.3]{g13.png}
% \quad
%  \includegraphics[scale=0.25]{w14.png}}
\caption{The directed graphs $ \cB^+(z)$ when $z$ is $t_4=(1,4)$ (left), $g_{3}=(1,4)(2,5)(3,6)$ (middle),
or $w_0=4321$ (right).
The data in each box is $\boxed{w:\GC_z(w)}$, with $w$ given in (inverse) one-line notation.
The blue vertices correspond to elements of $\cB(z)$.}\label{t1n-fig}
\end{figure}


%
%\begin{figure}[ht]
%\centerline{\includegraphics[scale=0.3]{w13.png}\qquad \includegraphics[scale=0.3]{w14.png}}
%\caption{The directed graphs $\cB(z)  \subseteq \cB^+(z)$ when $z \in I_\infty$ is $321$ and $4321$, presented using the same conventions as in \Cref{t1n-fig}.}\label{w0-fig}
%\end{figure}




\section{Shift invariance and stable limits}

This section contains a new formula for $\fkGO_z$ that holds for all vexillary $z \in \Ivex$ and 
which will lead to a nontrivial shift invariance property of the coefficient function $\GC_z$.

%A permutation $w\in S_\infty$ is \defn{vexillary} if it is $2143-$avoiding. 
A generic vexillary involution $z \in \Ivex:=\{z\in I_\infty: z\text{ is $2143$-avoiding}\}$
has cycle notation $z = (a_1, b_1)(a_2, b_2)\cdots (a_q, b_q)$
where $1\leq a_1 <a_2 < \dots <a_q < \min\{b_1,b_2,\dots,b_q\}$. 
We refer to the numbers $a_i$ as \defn{left endpoints}, to the numbers $b_i$ as \defn{right endpoints},
and to the ordered pairs $(a_i,b_i)$ as \defn{cycles}.

%If $1\leq i<j\leq q$, then the cycles $(a_i,b_i)$ and $(a_j,b_j)$ are said to be \defn{crossing} 
%when $ b_i<b_j$ or \defn{nesting} when $b_j < b_i$.
%By extension, we will say that $a_i$ and $a_j$ are \defn{crossing} or \defn{nesting} 
%according to whether  $(a_i,b_i)$ and $(a_j,b_j)$ are crossing or nesting.

The \defn{left segments} of $z$ are the maximal subsets of consecutive left endpoints,
that is, the equivalence classes in $\{a_1,a_2,\dots,a_q\}$ under the transitive closure of the relation with $a_i \sim a_j$ if $|a_j-a_i|\leq 1$.
There is at most one left segment containing $1$, which we refer to as the \defn{immobile segment}.
All other left segments are \defn{mobile}.

Suppose $L$ is a mobile left segment of $z$ and define $c_0 = \min(L) - 1$. Notice that we must have $c_0 = z(c_0) \in \PP$.
Now, for any subset $  S \subseteq \{a_1,a_2,\dots, a_q\}$ define $\sigma_{S,L} \in S_\infty$ to be the cyclic permutation
$ \sigma_{S,L} = (c_0,c_1,c_2,\dots,c_k) $ where $S\cap L = \{c_1<c_2<\dots<c_k\}$.
This is the identity element when $S\cap L$ is empty. 
%The conjugated permutation 
%\[  (\sigma_{S,L})^{-1} \cdot z \cdot \sigma_{S,L} \] 
% is not always vexillary, but
%its arc diagram is obtained from $z$ by simply replacing each edge $\{c_i < z(c_i)\}$ for $i \in [k]$ with $\{ c_{i-1} < z(c_i)\}$.
Also define $\textstyle \sigma_{S} = \prod_L \sigma_{S,L}$ where the product is over all mobile left segments of $z$ in any order.
%This product can be taken in any order since as $L$ varies the permutations $\sigma_{S,L}$ have disjoint supports.

\begin{example} \label{sigma-ex}
We often draw $z \in I_n$ as an \defn{arc diagram}, that is, as the graph with vertex set $[n]$ having edges $\{i,z(i)\}$ for $i \in \supp(z)$.
Suppose our vexillary involution is 
\[z = (2, 7)(3, 8)(4, 6)(5, 9)
=
\arcstart
{
*{\bullet}     & *{\bullet}  \arc{1.2}{rrrrr}   & *{\bullet} \arc{1.2}{rrrrr}  & *{\bullet}  \arc{0.6}{rr}   & *{\bullet} \arc{1}{rrrr}  & *{\bullet}    & *{\bullet}  & *{\bullet}  & *{\bullet}  \\
1 & 2 & 3 & 4 & 5  & 6 & 7 & 8 & 9      
} 
\arcstop.
\]
This involution has a unique left segment $L=\{2,3,4,5\}$, which is mobile.
For the subset $S = \{2,4,5\}$ we have $\sigma_{S,L} = (1,2,4,5)$ and 
\[  (\sigma_{S,L})^{-1} \cdot z\cdot \sigma_{S,L} =
(1,7)(2,6)(3,8)(4,9)
= \arcstart
{
*{\color{red}\bullet}     \arc{1.4}{rrrrrr}   & *{\color{red}\bullet}  \arc{1}{rrrr}   & *{\bullet} \arc{1.2}{rrrrr}  & *{\color{red}\bullet}  \arc{1.2}{rrrrr}   & *{\color{red}\bullet}  & *{\bullet}    & *{\bullet}  & *{\bullet}  & *{\bullet}  \\
1 & 2 & 3 & 4 & 5  & 6 & 7 & 8 & 9      
} 
\arcstop.
\]
\end{example}


Suppose $a_i$ and $a_j$ are left endpoints of $z$ in the same left segment with $i<j$.
We say that $a_j$ is a \defn{crossing bound} of $a_i$ if $\{i\} = \{ t : i\leq t <j\text{ and }b_t<b_j\}$.
Now, given  a subset $  S \subseteq \{a_1,a_2,\dots, a_q\}$ we define  $\varpi_{z,S} = \prod_{i \in [q]}\varpi_{z,S}^{(a_i)} $ where
\be
\varpi_{z,S}^{(a)} = \begin{cases}
-1&\text{if $S$ contains any crossing bound of $a$} \\
2+\beta x_{a} &\text{if }a \notin S \\
1 + \beta x_{a} &\text{otherwise}.
\end{cases}
\ee

\begin{example} \label{sigma-ex2}
If $z=(2, 7)(3, 8)(4, 6)(5, 9)$ is as in \Cref{sigma-ex} then 
\[\ba \varpi_{z,\varnothing} &= (2+\beta x_2)(2+\beta x_3)(2+\beta x_4)(2+\beta x_5),\\
 \varpi_{z,\{2,4,5\} } &= -(1+\beta x_2)(2+\beta x_3)(1+\beta x_5).\ea\]
\end{example}

Finally, we define $  S \subseteq \{a_1,a_2,\dots, a_q\}$   to be \defn{shiftable} if
(1) no element of $S$ is in the left segment of $z$ containing $1$, if this exists,
and
(2) if some $a_i \notin S$ then $S$ does not contain any crossing bound of $a_i$.
Given such a subset, write $\iG_{z,S} = \iG_v$ where $v = (\sigma_S)^{-1} \cdot z\cdot \sigma_S$.



\begin{example}
When $z=(2, 7)(3, 8)(4, 6)(5, 9)$ there are 9 shiftable subsets of left endpoints,
given by $\emptyset$, $\{2\}$, $\{4\}$, $\{2,3\}$, $\{2,4\}$, $\{4,5\}$,
$\{2,3,4\}$, $\{2,4,5\}$, and $\{2,3,4,5\}$.
%, and it turns out that $\fkGO_z $ can be expressed as a sum of 9 terms
%\[
%\ba \fkGO_z &= 
%   (2 + \beta x_2)(2 + \beta x_3)(2 + \beta x_4)(2 + \beta x_5) \cdot \iG_{(2, 7)(3, 8)(4, 6)(5, 9)}  \\&\quad +
%   \beta\cdot (1 + \beta x_2)(2 + \beta x_3)(2 + \beta x_4)(2 + \beta x_5)\cdot \iG_{(1, 7)(3, 8)(4, 6)(5, 9)}   \\&\quad +
%   \beta\cdot(2 + \beta x_2)(2 + \beta x_3)(1 + \beta x_4)(2 + \beta x_5)\cdot \iG_{(1, 6)(2, 7)(3, 8)(5, 9)} \\&\quad -
%    \beta^2\cdot(1 + \beta x_3)(2 + \beta x_4)(2 + \beta x_5)\cdot \iG_{(1, 7)(2, 8)(4, 6)(5, 9)} \\&\quad +
%   \beta^2\cdot(1 + \beta x_2)(2 + \beta x_3)(1 + \beta x_4)(2 + \beta x_5)\cdot \iG_{(1, 7)(2, 6)(3, 8)(5, 9)} \\&\quad -
%   \beta^2\cdot(2 + \beta x_2)(2 + \beta x_3)(1 + \beta x_5)\cdot \iG_{(1, 6)(2, 7)(3, 8)(4, 9)} \\&\quad -
%   \beta^3\cdot(1 + \beta x_3)(1 + \beta x_4)(2 + \beta x_5)\cdot \iG_{(1, 7)(2, 8)(3, 6)(5, 9)} \\&\quad -
%   \beta^3\cdot(1 + \beta x_2)(2 + \beta x_3)(1 + \beta x_5)\cdot \iG_{(1, 7)(2, 6)(3, 8)(4, 9)}        \\&\quad +
%   \beta^4\cdot(1 + \beta x_3)(1 + \beta x_5)\cdot\iG_{(1, 7)(2, 8)(3, 6)(4, 9)}.
%\ea
%\]
%Not coincidentally, this expression is exactly $\fkGO_z = \sum_S \beta^{|S|}\cdot \varpi_{z,S} \cdot \iG_{z , S}$
%where the sum is over our 9 shiftable subsets $S$.
\end{example}
%\eric{todo: better example, like $(2,5)(3,7)(4,6)$}

The following theorem significantly generalizes \Cref{qd-thm}:

\begin{theorem}\label{vex-thm}
If $z \in \Ivex$ is any vexillary involution then $\fkGO_z = \sum_S  \beta^{|S|}\cdot\varpi_{z,S} \cdot \iG_{z , S}$ where the sum is over all shiftable subsets of left endpoints of $z$.
\end{theorem}

% \begin{proof}
% \eric{TODO, probably can be shown by a careful induction}
% \end{proof}

Recall that our permutations $w \in S_\infty$ are maps $w : \ZZ \to \ZZ$ with $w(i)=i$ for $i\leq 0$.
Given any integer $n \in \ZZ$, define $w\downarrow n$ to be the permutation of $\ZZ$
with the formula 
\be(w\downarrow n)(i) = w(i+n)-n\quad\text{for $i \in \ZZ$.}\ee
Notice that if $n\leq 0$ then $w\downarrow n \in S_\infty$, but if $w(m) \neq m$ then $w \downarrow n \notin S_\infty$ for all $n\geq m$.
Define $1^n \times w := w\downarrow(-n)$ for $n\in \NN$.

Now for $z \in I_\infty$ we 
 extend the domain of the Grothendieck coefficient function $\GC_z$ by setting $\GC_z(w) = 0$ if $w \notin S_\infty$. Our main application of \Cref{vex-thm} is the following:


\begin{theorem}\label{gg-thm} Suppose $n \in \NN$ and  $z \in \Ivex$.
\ben
\item[(a)] If $z(1) = 1$ then $GC_{1^n\times z}(w) = \GC_z(w \downarrow n)$ for all $w \in S_\infty$.
\item[(b)] If $z\downarrow n  \in S_\infty$ then $\GC_{z\downarrow n} (w)= \GC_{z}(1^n\times w)$ for all $w \in S_\infty$.
\een
% \[\GC_{1^n\times z}(w) = \GC_z(w \downarrow n) \quand \GC_{z\downarrow n} (w)= \GC_{z}(1^n\times w)\quad\text{for all $w \in S_\infty$.}\]
\end{theorem}
The hypothesis $z(1)= 1$  is necessary for the first identity. For example, 
$\fkGO_{(1, 2)} = 2 \fkG_{21} + \beta \fkG_{312}$
but the $\fkG$-expansion of $\fkG_{(n+1,n+2)} = \fkG_{1^n\times (1,2)}$ has 4 terms if $n>0$.

\begin{corollary}
If $z \in \Ivex$ then $\GC_z(w) = \GC_{1^n\times z}(1^n \times w)$ for all $n\in\NN$ and $w \in S_\infty$.
\end{corollary}



%\begin{example}
%The hypothesis $z(1)= 1$ in the theorem is necessary. For example, we have  ...
%\[
%\fkGO_{(1, 2)} = 2 \fkG_{21} + \beta \fkG_{312}
%\]
%compared to
%\[\ba
%\fkGO_{(1, 2)\ll 1}= \fkGO_{(2, 3)} &= 2 \fkG_{132} + \beta \fkG_{1423} + \beta \fkG_{231} + \beta^2 \fkG_{2413}
% \\&\neq 2 \fkG_{21 \ll 1} + \beta \fkG_{312 \ll 1}
% \ea\] and \[ \ba
%\fkGO_{(2, 3)\ll 1}= \fkGO_{(3, 4)} &= 2 \fkG_{1243} + \beta \fkG_{12534} + \beta \fkG_{1342} + \beta^2 \fkG_{13524} 
% \\
% &=  2 \fkG_{132 \ll 1} + \beta \fkG_{1423\ll 1} + \beta \fkG_{231\ll 1} + \beta^2 \fkG_{2413\ll 1}.
% \ea\]
% \end{example}



These shift invariance properties of $\GC_z$ are consistent with \Cref{b+conj} since 
one can show that 
if $n \in \ZZ$ and  $z \in \Ivex$ are such that  $z(1)=1$ and $1^n\times z \in S_\infty$ then
\[\ba
\cB(1^n\times z) = \left\{ w \in S_\infty : w \downarrow n \in \cB(z)\right\},
\;
\cB^+(1^n\times z) = \left\{ w \in S_\infty : w \downarrow n \in \cB^+(z)\right\}.
\ea
\]

\Cref{gg-thm} has an application concerning the \defn{stable limit} of $\fkGO_z$.
The \defn{symmetric Grothendieck function} of $w \in S_\infty$ 
is
$\textstyle G_w := \lim_{n\to \infty} \fkG_{1^n\times w}$ where the limit is taken in the sense of formal power series. This means the limit exists precisely when the coefficients of any fixed monomial 
in $\fkG_{1^n\times w}$ is eventually a constant sequence.
It is known \cite{Buch2002} that  $G_w$ always exists and is a formal power series that is symmetric in the $x_i$ variables.


For $z \in I_\infty$ let
$\label{gq-eq}\textstyle \GQ_z := \lim_{n\to \infty} \fkGO_{1^n \times z }.$
 When $z\in \Ivex$, it is known that $\GQ_z$ also exists and is a symmetric formal power series; in fact, it is equal to the \defn{$K$-theoretic Schur $Q$-function} of Ikeda and Naruse \cite{Ikeda.Naruse2013} indexed by the \defn{involution shape} of $z$ \cite[Theorem~4.11]{MP2020}.

 This was proved in \cite{MP2020} by a difficult geometric argument. 
 % using results in \cite{Anderson,NakagawaNaruse}.
 \Cref{gg-thm} leads to a much simpler derivation of the fact that $\GQ_z$ is a symmetric formal power series. We also get a new formula relating the $G$-expansion of $\GQ_z$ to the $\fkG$-expansion of $\fkGO_z$:

 

\begin{corollary}   If $z \in \Ivex$ has $z(1)=1$ then 
$\textstyle \GQ_z = \sum_{w \in S_\infty} \GC_z(w) \cdot \beta^{\ell(w) - \ellhat(z)} \cdot G_w.$
\end{corollary}

% \begin{corollary}
%If $z \in \Ivex$ has $z(1)=1$ then $\GQ_z = \sum_S  \beta^{|S|}\cdot\varpi_{z,S} \cdot \GP_{z , S}$ where the sum is over all shiftable subsets of left endpoints of $z$.
%\end{corollary}


\section{Special cases}\label{special-sect}
We can compute $\GC_z$ explicitly in a few special cases---namely, when $z$ is any transposition $t_{ij} = (i, j)$ or 
the element
$ g_{ij} := (i, j+1)(i+1,j+2)(i+3,j+4)\cdots(j,2j-i+1)$
where $i$ and $j$ are any positive integers with $i<j$.
These involutions are always vexillary, and 
in the notation of \Cref{t-ex} and \ref{g-ex}
we have $t_n =t_{1,n}$ and $g_n = g_{1,n}$.

Define $t_n^+ := t_{2,n}$ and $g_n^+ := g_{2,n}$.
By \Cref{gg-thm}, 
to compute $\GC_{t_{ij}}$ and $\GC_{g_{ij}}$ for all positive integers $ i <j$,
it suffices just to determine $\GC_z$ when $z$ is $t_n^+$ and $g_n^+$.
We explain these calculations in the following sections.


\subsection{Transpositions}

Let $\Sh(n)$ denote the set of words obtained by shuffling
\[n2\quand 1345\cdots (n-3)(n-2)(n-1) (n+1).\]
Define $\cX(n)$ to be the subset of words in $\Sh(n)$ for which
\begin{itemize}
    \item the first letter is $1$ while the last letter is $n+1$; and
\item at most one letter appears between $n$ and $2$.
\end{itemize}
Define $\cY(n)\subset\Sh(n)$ be the set of words with exactly one or exactly two letters between $n$ and $2$.
The sets $\cX(n)$ and $\cY(n)$ are not disjoint, but the following holds:

\begin{proposition}
    Let $z=t^+_{n}=(2,n)$. Then $\cB(z)  = \left\{w^{-1} : w \in \cX(n)\right\} $.
Define  $c\in S_n$ to be  the cycle $c=(1, 2, 5, 4)$ when $n\leq 4$ and  $c=(1, 2, 5,6,7, \cdots, n)$ if $n\geq 5$. Then
    %\begin{cases} (1, 2, 5, 4) &\text{if }n=4 \\ (1, 2, 5,6,7, \cdots, n) &\text{if }n\geq 5.\end{cases}$   
\[
\cB^+(z)=\left\{w^{-1} : w \in \cX(n)\cup \cY(n)\right\} \sqcup \begin{cases} \varnothing &\text{if }n\leq 3 \\  \{c\} &\text{if }n\geq 4.
\end{cases}
\]   
Finally, one has
$\textstyle\fkGO_{z} = 2 \sum_{w^{-1} \in \cX(n)} \beta^{w(2)-w(n)-1} \fkG_w +  \sum_{w^{-1} \in \cY(n)} \beta^{w(2)-w(n)-1} \fkG_w
$.
% so
%\[\GC_z(w) = \begin{cases} 2&\text{if }w^{-1} \in \cX(n)\setminus \cY(n) \\
%         3 &\text{if }w^{-1} \in \cX(n) \cap \cY(n) \\ 
%         1 &\text{if }w^{-1} \in \cY(n) \setminus \cX(n) \\
%         0&\text{otherwise}.
%         \end{cases}\]

         \end{proposition}




See \Cref{g2n-fig} for an illustration of this proposition.


\subsection{Vexillary involutions that are fully commutative}

The involutions $g_{ij}$ are the only elements of $I_\infty$ that are both
$2143$- and $321$-avoiding \cite[Theorem~3.35 and Corollary~3.36]{HMP1},
that is, both vexillary and \defn{fully commutative}. 


\begin{proposition}
Let $z=g_n^+ = (2,n+1)(3,n+2)\cdots(n,2n-1)$ where $n>2$. Then:
   \ben
    \item[(a)] $\cB(z)$ consists of the \textbf{inverse} of
    $1(n+1)2(n+2)3\cdots (2n-1)n(2n)\in S_{2n-1}.$
    
    \item[(b)] $\cB^+(z)$  consists of the $2^{n}$ permutations   whose \textbf{inverses} in one-line notation 
    have the form $a_1b_1a_2b_2\cdots a_{n} b_{n} \in S_{2n}$
    where $\{a_i,b_i\}=\{i,n+i\}$ for each $i\in [n]$.

\item[(c)] If we define $\ODes_L(w) = \{ i \in \DesR(w^{-1}) : i\text{ is odd}\}$ 
 then
 \[\textstyle\fkGO_{z} =   \sum_{w \in \cB^+(z)} 2^{n-1-|\ODes_L(w)|}  \beta^{|\ODes_L(w)|}  \fkG_w +
\tfrac{1}{2}  (-\beta)^n \fkG_{w_{\max}}
\]
where $
w_{\max} \in \cB^+(z)$ is the \textbf{inverse} of $(n+1)1(n+2)2\cdots (2n)n$.
%, and so
% \[\GC_z(w) = \begin{cases} 2^{n-1-|\ODes_L(w)|}&\text{if }w_{\max} \neq w \in \cB^+(z) \\
%1 &\text{if $w_{\max}=w$ and $n$ is even} \\ 
%0&\text{otherwise}.
%\end{cases}\]

    \een




\end{proposition}

See \Cref{g2n-fig} for an illustration of this proposition.

\begin{figure}[ht]
  \centerline{
    \resizebox{\textwidth}{!}{%
\begin{tikzpicture}[xscale=2, yscale=1.875,>=latex,baseline=(z.base)]
\node at (0,1.5) (z) {};
  \node[draw,fill=lightgray] at (1,0) (11) {\tiny $415263^{-1}:0$};
  \node[draw] at (0,1) (21) {\tiny $145263^{-1}:1$};
  \node[draw] at (1,1) (22) {\tiny $412563^{-1}:1$};
  \node[draw] at (2,1) (23) {\tiny $41523^{-1}:1$};
  \node[draw] at (0,2) (31) {\tiny $142563^{-1}:2$};
  \node[draw] at (1,2) (32) {\tiny $14523^{-1}:2$};
  \node[draw] at (2,2) (33) {\tiny $41253^{-1}:2$};
  \node[draw,fill=lightskyblue] at (1,3) (41) {\tiny $14253^{-1}:4$};
  %
  \draw[->,thick]  (41) -- (31);
  \draw[->,thick]  (41) -- (32);
  \draw[->,thick]  (41) -- (33);
  \draw[->,thick]  (31) -- (21);
  \draw[->,thick]  (31) -- (22);
  \draw[->,thick]  (32) -- (21);
  \draw[->,thick]  (32) -- (23);
  \draw[->,thick]  (33) -- (22);
  \draw[->,thick]  (33) -- (23);
  \draw[->,thick]  (21) -- (11);
  \draw[->,thick]  (22) -- (11);
  \draw[->,thick]  (23) -- (11);
 \end{tikzpicture}
 \quad
\begin{tikzpicture}[xscale=2.1, yscale=1.5,>=latex,baseline=(z.base)]
\node at (0,2) (z) {};
  \node[draw] at (2.5,0) (11) {\tiny $51627384^{-1}:1$};
  \node[draw] at (1,1) (21) {\tiny $51623784^{-1}:1$};
  \node[draw] at (2,1) (22) {\tiny $51267384^{-1}:1$};
  \node[draw] at (3,1) (23) {\tiny $15627384^{-1}:1$};
  \node[draw] at (4,1) (24) {\tiny $5162734^{-1}:1$};
  \node[draw] at (0,2) (31) {\tiny $51263784^{-1}:2$};
  \node[draw] at (1,2) (32) {\tiny $15623784^{-1}:2$};
  \node[draw] at (2,2) (33) {\tiny $5162374^{-1}:2$};
  \node[draw] at (3,2) (34) {\tiny $15267384^{-1}:2$};
  \node[draw] at (4,2) (35) {\tiny $5126734^{-1}:2$};
  \node[draw] at (5,2) (36) {\tiny $1562734^{-1}:2$};
  \node[draw] at (1,3) (41) {\tiny $15263784^{-1}:4$};
  \node[draw] at (2,3) (42) {\tiny $5126374^{-1}:4$};
  \node[draw] at (3,3) (43) {\tiny $1562374^{-1}:4$};
  \node[draw] at (4,3) (44) {\tiny $1526734^{-1}:4$};
  \node[draw,fill=lightskyblue] at (2.5,4) (51) {\tiny $1526374^{-1}:8$};
  %
  \draw[->,thick]  (51) -- (41);
  \draw[->,thick]  (51) -- (42);
  \draw[->,thick]  (51) -- (43);
  \draw[->,thick]  (51) -- (44);
  \draw[->,thick]  (41) -- (31);
  \draw[->,thick]  (41) -- (32);
  \draw[->,thick]  (41) -- (34);
  \draw[->,thick]  (42) -- (31);
  \draw[->,thick]  (42) -- (33);
  \draw[->,thick]  (42) -- (35);
  \draw[->,thick]  (43) -- (32);
  \draw[->,thick]  (43) -- (33);
  \draw[->,thick]  (43) -- (36);
  \draw[->,thick]  (44) -- (34);
  \draw[->,thick]  (44) -- (35);
  \draw[->,thick]  (44) -- (36);
  \draw[->,thick]  (31) -- (21);
  \draw[->,thick]  (31) -- (22);
  \draw[->,thick]  (32) -- (21);
  \draw[->,thick]  (32) -- (23);
  \draw[->,thick]  (33) -- (21);
  \draw[->,thick]  (33) -- (24);
  \draw[->,thick]  (34) -- (22);
  \draw[->,thick]  (34) -- (23);
  \draw[->,thick]  (35) -- (22);
  \draw[->,thick]  (35) -- (24);
  \draw[->,thick]  (36) -- (23);
  \draw[->,thick]  (36) -- (24);
  \draw[->,thick]  (21) -- (11);
  \draw[->,thick]  (22) -- (11);
  \draw[->,thick]  (23) -- (11);
  \draw[->,thick]  (24) -- (11);
 \end{tikzpicture}}
  }
% \centerline{\includegraphics[scale=0.35]{t25.png}\qquad \includegraphics[scale=0.35]{g23.png}}
\caption{The directed graphs $ \cB^+(z)$ when $z$ is $t^+_5=(2,5)$ (left) and $g^+_3=(2,4)(3,5)$ (right), presented using the conventions as in \Cref{t1n-fig}.
Here the grey boxes indicate the (in these cases, unique) elements of $\cB^+(z)$ that are not in $\supp(\GC_z)$.}\label{g2n-fig}
\end{figure}






%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way
\printbibliography



\end{document}
