%% 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[submission]{FPSAC2019}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

\articlenumber{90}
\addbibresource{90.bib}

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

\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}

\usepackage{lipsum}

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

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%\author[Optional shorter names]{Longer names me\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}\addressmark{1}, \and you\addressmark{2}}

\author{Rebecca Goldin\addressmark{1}\thanks{\href{mailto:rgoldin@gmu.edu}{rgoldin@gmu.edu}} and Allen Knutson\addressmark{2}\thanks{\href{mailto:allenk@math.cornell.edu}{allenk@math.cornell.edu}}}
\address{\addressmark{1}George Mason University, Fairfax, VA 22030, USA\\
  \addressmark{2}Cornell University, Ithaca, NY 14853, USA}


%% 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{We use operators to reformulate the Andersen-Jantzen-Soergel/Billey
  formula for the point restrictions of 
  equivariant Schubert classes of the cohomology of $G/B$.  
  We introduce new operators whose coefficients compute
  Schubert structure constants (in a manifestly polynomial, but not
  positive, way), resulting in a formula much like and generalizing
  the positive AJS/Billey formula.  Our proof involves Bott-Samelson
  manifolds, and in particular, the cohomology basis dual
  to the homology basis of classes of sub-Bott-Samelson manifolds.
}

%% 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{Schubert calculus, nil Hecke algebra}

%% 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, ie. [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]{biblatex}
%\addbibresource{ops.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands


% if your fonts are screwed up, remove the first "renewcommand".

\usepackage{amsopn,amsfonts}%,pb-diagram}
\usepackage{hyperref,mathtools,blkarray}
%\usepackage[all]{xy}
% ^C ^T ^P to put AucTeX in Tex-PDF mode

\usepackage{amsfonts, latexsym, epsfig, xcolor}%,epic}
\renewcommand{\familydefault}{ppl}

%\setlength{\oddsidemargin}{0in} \setlength{\evensidemargin}{0in}
%\setlength{\marginparwidth}{0in} \setlength{\marginparsep}{0in}
%\setlength{\marginparpush}{0in} \setlength{\topmargin}{-.5in}
%\setlength{\headheight}{6pt} \setlength{\headsep}{20pt}
%\setlength{\footskip}{.3in} \setlength{\textheight}{9.2in}
%\setlength{\textwidth}{6.5in} \setlength{\parskip}{4pt}
%\usepackage{hyperref}

\newtheorem*{poetry}{}

%\input linkpattern.tex

\newcommand\MMN{{\mathcal M}_N}
\newcommand\A{\textsc{a}}
\newcommand\B{\textsc{b}}
\newcommand\notdivides{\,\not\!\divides\,}
\newcommand\divides{\,{\mid}\,}
\newcommand\lie[1]{{\mathfrak{#1}}}
\newcommand\lien{{R_N^{\Delta=0}}}
\newcommand\Pf{{\em Proof. }}
\newcommand\Tr{{\rm Tr\,}}
\newcommand\Proj{{\rm Proj\,}}
\newcommand\iso{{\ \cong\ }}
\newcommand\tensor{{\otimes}}
\newcommand\Tensor{{\bigotimes}}
\newcommand\QED{\hfill $\Box$} %{\bf QED}}
\newcommand\calO{{\mathcal O}}
\newcommand\calC{{\mathcal C}}
%\newcommand\cal{\mathcal}
\newtheorem{Theorem}{Theorem} 
\newtheorem{Proposition}{Proposition} 
\newtheorem{Lemma}{Lemma}
\newtheorem{Fact}{Fact}
\newtheorem{Sublemma}{Sublemma}
\newtheorem{Corollary}{Corollary}
\newtheorem*{Corollary*}{Corollary}
\newtheorem*{Definition}{Definition} 
\newtheorem*{Theorem*}{Theorem}
\theoremstyle{remark}
\newtheorem{Example}{Example}
%\newcommand\Case{\em Case}}
\newcommand\frob{\bowtie}
\newcommand\onto{\mathop{\twoheadrightarrow}}
\newcommand\into{\operatorname*{\hookrightarrow}}
\newcommand\Ex{\noindent{\em Example. }}
\newcommand\union{\bigcup}
\newcommand\Pone{{\mathbb P}^1}
\newcommand\pt{pt}
\newcommand\calL{{\mathcal L}}
\newcommand\Id{{\bf 1}}
\newcommand\Td{{\rm Td}}
\newcommand\vol{{\rm vol}}
\newcommand\CP{{\mathbb C \mathbb P}}
\newcommand\PP{{\mathbb P}}
\newcommand\ind{{\rm index}}
\newcommand\reals{{\mathbb R}}
\newcommand\complexes{{\mathbb C}}
\newcommand\CC{{\mathbb C}}
\newcommand\integers{{\mathbb Z}}
\newcommand\rationals{{\mathbb Q}}
\newcommand\onehalf{\frac{1}{2}}
\newcommand\naturals{{\mathbb N}}
\newcommand\Star{\operatornamewithlimits\bigstar}
\newcommand\dec{\hbox{\ \rm\tiny decreasing}}
\newcommand\area{\operatorname{area}}
\newcommand\ol{\overline}
\newcommand\Grmn{{\rm Gr}^m(\complexes^n)}
\newcommand\<{\langle}
\renewcommand\>{\rangle}
\newcommand\junk[1]{}
\newcommand\To{\longrightarrow}
\newcommand\Coprod{\coprod}
\newcommand\Union{\bigcup}
\newcommand\dash{\text{--}}

\theoremstyle{plain}
\newtheorem*{prequel}{Theorem \ref{bigthm}}
\newtheorem*{Conjecture}{Conjecture}
                  
%\renewcommand{\familydefault}{ccr}

\newcommand\dfn{\bf} % maybe should be \em
\newcommand\indentme{\parindent 2 cm}
%\newcommand\indentme{{\bf (Indent this)\ }}
%\renewcommand\caption{{\bf PICTURE:\ }}

%\newcommand\GLN{{{GL_N(\complexes)}}}
\newcommand\GLN{{{GL_N}}}
\newcommand\glN{\lie{gl}_N}
\newcommand\SLN{{{SL_N}}}%(\complexes)}}}
\newcommand\FlN{{Flags(\vec N)}}

\newcommand\thmlabel[1]{\label{thm:#1}}
\newcommand\proplabel[1]{\label{prop:#1}}
\newcommand\seclabel[1]{\label{sec:#1}}
\newcommand\sseclabel[1]{\label{ssec:#1}}
\newcommand\lemlabel[1]{\label{lem:#1}}
\newcommand\figlabel[1]{\label{fig:#1}}

\newcommand\thmref[1]{theorem \ref{thm:#1}}
\newcommand\propref[1]{proposition \ref{prop:#1}}
%\newcommand\secref[1]{section \ref{sec:#1}}
\newcommand\secref[1]{\S\ref{sec:#1}}
%\newcommand\ssecref[1]{subsection \ref{ssec:#1}}
\newcommand\ssecref[1]{\S\ref{ssec:#1}}
\newcommand\lemref[1]{lemma \ref{lem:#1}}
\newcommand\figref[1]{figure \ref{fig:#1}}

\newcommand\Thmref[1]{Theorem \ref{thm:#1}}
\newcommand\Propref[1]{Proposition \ref{prop:#1}}
%\newcommand\Secref[1]{Section \ref{sec:#1}}
\newcommand\Secref[1]{\S\ref{sec:#1}}
%\newcommand\Ssecref[1]{Subsection \ref{ssec:#1}}
\newcommand\Ssecref[1]{\S\ref{ssec:#1}}
\newcommand\Lemref[1]{Lemma \ref{lem:#1}}
\newcommand\Figref[1]{Figure \ref{fig:#1}}

%tentative defs
\newcommand\cycinside[1]{#1}
\newcommand\cyc[1]{\circlearrowleft (\cycinside{#1})}
\newcommand\cp{\bullet} % new product
\newcommand\der{\partial}
\newcommand\mdeg{{\rm m}\!\deg}
\newcommand\codim{{\rm co}\!\dim}
\newcommand\SN{\mathcal{S}_N}
\newcommand\MNC{{\text{Mat}_N}}%(\complexes)}}
\newcommand\MnC{{M_n}}%(\complexes)}}
\newcommand\rank{{\rm rank\ }}
\newcommand{\comment}[1]{$\star${\sf\textbf{#1}}$\star$}
\newcommand\Br{\mathcal{B}r}
\newcommand\wt[1]{{\widetilde #1}}

\numberwithin{equation}{section}

\begin{document}
%\pagestyle{plain}
%\baselineskip16pt% improves readability for draft

\title{Schubert structure operators}
\date{\today}

\maketitle

{ \small\tableofcontents}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%


\newcommand\ZZ{{\mathbb Z}}

\newcommand\defn[1]{{\bf #1}}

\newcommand\com[1]{{\color{red} #1}}

\section{Introduction and the main theorem}

Fix a complex reductive Lie group $G$ and maximal torus $T\leq G$, for example 
$G=GL_n(\CC)$ and $T$ the diagonal matrices.  
%Let $G= GL_n(\CC)$ and $T\leq G$ denote the diagonal matrices.  
Fix opposed Borel subgroups $B,B_-$ with intersection $T$. This choice
results in a length function $\ell$ on $W = N(T)/T$ and a set
$\{\alpha_i\}$ of simple roots.  The quotient $G/B$ is the associated
\defn{flag manifold} and the left $T$-action on $G/B$ has isolated
fixed points $\{wB/B\ :\ w\in W\}$, where $W:= N(T)/T$ is the \defn{Weyl group}.

In the case that $G = GL_n(\CC)$ and $B =$ upper-triangular matrices,
$G/B$ is (uniquely) $G$-isomorphic to the set of complete flag
manifolds $Fl(\CC^n)$. The fixed points $N(T)B/B$ of the $T$-action
correspond, under that isomorphism, to coordinate flags in
$Fl(\CC^n)$. In particular, there are $n!$ such flags, corresponding
to elements in the Weyl group $W \cong S_n,$ the symmetric group on
$n$ letters.

We denote by $H^*_T$ the $T$-equivariant cohomology of a point with
coefficients in $\ZZ$, and recall that $H^*_T$ is the polynomial ring
$Sym(T^*)$ over $\ZZ$ in the weight lattice $T^* := Hom(T,\CC^\times)$.
The equivariant cohomology $H_T^*(G/B)$ %(over $\ZZ$)
is a free $H^*_T$-module with a basis given by Schubert classes 
(recalled below). Our references for equivariant (co)homology are
\cite{BrionH,Kostant-Kumar,Kumar-Nori}.

Let $\ZZ[\partial]$
denote the \defn{nil Hecke algebra} %{\color{blue} not the zero Hecke algebra}} 
with $\ZZ$-basis $\{\partial_w\ :\ w\in W\}$, whose products are
defined by 
$$
\partial_w \partial_v := \begin{cases}\partial_{wv} &\mbox{if $\ell(vw)=\ell(v)+\ell(w)$}\\
0 & \mbox{otherwise, i.e. if $\ell(vw) < \ell(v)+\ell(w)$}.
\end{cases}
$$
These $\{\partial_w\}$ act on the polynomial ring $H^*_T$ 
%in $T$'s weight lattice $T^*$ 
as follows: for each root $\alpha$ with simple
reflection $r_\alpha$, the \defn{divided difference operator}
$\partial_{r_\alpha} := \partial_\alpha$ is defined by
$$ \partial_\alpha \cdot f := \frac{f-r_\alpha f}{\alpha} $$  
% and $\partial_w := \partial_{\alpha_1}\cdots \partial_{\alpha_k}$ where
% $w=r_{\alpha_1}\cdots r_{\alpha_k}$ is a reduced word expression and
% $r_\alpha f$ is the function obtained by exchanging the variables
% in dicated by $\alpha$.  
The nil Hecke algebra acts on the first factor in the tensor product
$H^*_T \tensor_{\ZZ} H^*_T$, and this action descends to the
quotient $H^*_T \ \tensor_{(H^*_T)^W}\ H^*_T$.
This latter ring has a well-defined map 
$\lambda \tensor \mu \mapsto \lambda c_1(\calL_{\mu})
\in H_T^*(G/B)$
called the \defn{equivariant Borel presentation} of $H^*_T(G/B)$,
which is a rational (and for $G=GL_n$, an integral) isomorphism.
(Here $\calL_\mu$ is the Borel-Weil line bundle $G \times^B \CC_\mu$,
where $\CC_\mu$ is the $1$-dimensional representation of $B$, neither of which
will be using again.)

%{\color{blue} Finally, we define $H_T^*[\partial]$ as the smash product of $H^*_T$ and $\ZZ[\partial]$.}

%{\bf but who cares. This'll be read by someone who's seen them before.
%  We can set up notation but not write a survey paper.}
%where $G=Gl(n,\CC)$ and $B$
%is a choice of Borel, as their kernels consists of symmetric
%polynomials in the variables permuted by $W$.
% ---- I don't think we need this stuff. 


%In between these two,
%let $H^*_T[\partial]$ denote the sub-$H^*_T$-algebra generated by

Since our interest is in cohomology not homology, 
we privilege codimension over dimension and define
$X^v := \overline{BvB}/B \subseteq G/B$ to be an
\defn{opposite Schubert variety} with equivariant homology class
$[X^v] \in H^T_*(G/B)$. As these $\{[X^v]\}$ form an $H^*_T$-basis and
$G/B$ enjoys equivariant Poincar\'e duality,
we can define the dual basis $\{S_w \in H^*_T(G/B)\}$ 
of \defn{Schubert classes} by $\langle S_w, [X^v] \rangle = \delta_{wv}$.
Here $\langle,\rangle$ denotes the Alexander pairing, of (equivariant)
cap-product followed by pushforward to a point.
In fact $S_w$ is the Poincar\'e dual to the subvariety $\overline{B_- w B}/B$.

\junk{
  Let $S_v \in H_T^*(G/B)$ denote the Poincar\'e dual to 
  $X_v := \overline{B_- v B}/B$, and similarly let $S^v\in H_T^*(G/B)$
  denote the dual to $X^v := \overline{B v B}/B$. 
In particular, $S^{w_0}= S_1 =1$, where $w_0$
is the longest word of $W$. We call $\{S^v\}_{v\in W}$ the 
\defn{Schubert classes} and $\{S_v\}_{v\in W}$ the 
\defn{opposite Schubert classes}. 
{\color{red} Do we like this naming convention? AK prefers the opposite,
  with a caveat ``As our focus is cohomology we give preference
  to codimension over dimension, and so choose ...}
}

%of $S_v$ to the point $wB/B$.
The nil Hecke algebra $\ZZ[\partial]$ acts on % $H_T(G/B)$ as well as on
\junk{ each of the bases $\{S_v\}_{v\in W}$ and $\{S^v\}_{v\in W}$; in particular,
  $\partial_w \cdot S^1 = S^{w}$ for each $w\in W$ 
  (since we act on the left factor in the Borel presentation).}
the basis $\{S_v\}_{v\in W}$: in particular,
$\partial_w \cdot S_{w_0} = S_{w w_0}$ for each $w\in W$ 
(since we act on the left factor in the Borel presentation),
though we won't use this recursion.

% Choose simple roots $\alpha_1,\dots, \alpha_n\in \mathfrak{t}$ so
% that the roots of the Lie algebra of $B$ are positive.

The structure constants $c_{uv}^w\in H_T^*$ are defined by the relation in $H^*_T(G/B)$
\begin{equation}\label{eq:structureconstants}
S_u S_v = \sum_w c_{uv}^w S_w
\end{equation}
% the latter equality due to the upper triangularity of the supports.
These polynomials $c_{uv}^w$ are known to be positive
in the following sense \cite{Graham}: 
%\cite{Graham:Positivity}: 
when written (uniquely) as a sum of
monomials in the simple roots $\{\alpha_i\}$, 
each monomial has a non-negative coefficient.
It is a very famous problem to compute these in a manifestly positive way,
solved in special cases such as $u,v\in W^P$ where $G/P$ is a
Grassmannian or $2$-step flag manifold \cite{Knutson-Tao,Buch}.
Another solved case is $u=w$, in which case $c_{wv}^w$ is computed
positively by the AJS/Billey formula \cite{AJS,Billey} 
(recalled below)
for the point restrictions $S_w|_v = c_{wv}^v$ of Schubert classes.
In this abstract, we prove a formula for the $\{c_{uv}^w\}$ in terms
of a certain composition of operators in the nil Hecke algebra, applied to $1$.
%\com{Why put it as ``operating'', instead of just living in an algebra?}
Along the way, we reprove the AJS/Billey formula;
more specifically, our nonpositive formula reduces to the 
positive AJS/Billey formula in the special case $u=w$.

\begin{Theorem}\label{thm:main}
  Let $Q$ be a reduced word for $w$. Then
  $$ c_{uv}^w = 
  \sum_{P,R\subseteq Q \text{ reduced}\atop \prod P = u, \ \prod R = v}
  \prod_Q \left(
    {\alpha_q}^{\ [q \in P,R]} \
    { \partial_q}^{\ [q \notin P,R]} \
    r_q \right) \cdot 1 $$
  where the exponent 
  ``$[\sigma]$'' is $1$ if the statement $\sigma$ is true, $0$ if false.
\end{Theorem}

{\em Example.}
Let $Q = 1\,2\,1$ so $w = r_1 r_2 r_1$, $u = r_1$, $v = r_1 r_2$ all in 
$S_3$ the Weyl group of $GL_3$. Then $P \in \{1--,--1\}$, $R = 1\,2\, -$
as subwords of $1\,2\,1$, in our sum
$$
  c_{r_1,\ r_1r_2}^{r_1r_2r_1} 
  = (\alpha_1 r_1\  r_2\  \partial_1 r_1)\cdot 1 + (r_1\  r_2\  r_1) \cdot 1 
  = 0 + 1
$$
whereas if we change $v$ to $r_2r_1$ so $R = -\,2\,1$, then
$$
  c_{r_1,\ r_2r_1}^{r_1r_2r_1} 
  = (r_1\  r_2\  r_1) \cdot 1 + (\partial_1 r_1\  r_2\  \alpha_1 r_1)\cdot 1 
  = 1 + \partial_1\cdot \alpha_2 = 0.
$$

{\em Example.}
Let $Q = 1\,2\,3\,1\,2$, so $w = r_1 r_2 r_3 r_1 r_2 =[3421]$ in one-line
notation, and take $u=r_2r_3r_2=[1432],\ v=r_1r_2r_1=[3214]$. 
Then $P = -23-2$ and $R\in \{12-1-,-2-12\}$ so we have
\begin{align*}
c_{uv}^w &= %(one P, 2 Qs)
(r_1\  \alpha_2 r_2\   r_3\  r_1\  r_2 % P = (-,2,3,-,2), Q = (1,2,-, 1-)
+ \partial_1r_1\  \alpha_2 r_2\   r_3\  r_1\ \alpha_2 r_2 )% P =  (-,2,3,-,2), Q = (-,2,-, 1,2)
\cdot 1\\
&= (\alpha_1+\alpha_2)\cdot 1 + \partial_1(\alpha_1+\alpha_2)(\alpha_2+\alpha_3)\cdot 1\\
& =  \alpha_1+\alpha_2 + \partial_1 (\alpha_1+\alpha_2)\alpha_2 \cdot 1 + \partial_1 (\alpha_1+\alpha_2)\alpha_3\cdot 1\\
&=\alpha_1+\alpha_2 +0+ \alpha_3.
%\partial_1(\alpha_13\alpha_24)= alpha_13 alpha_24  - alpha_23 alpha_14 /alpha_12 = 
%(x1-x3)(x2-x4)-(x2-x3)(x1-x4) = x1x2-x2x3 -x1x4+x3x4 - (x2x1 -x1x3 -x2x4+x3x4_
%= -x2x3-x1x4+x1x3+x2x4 = (x1-x2) (x3-x4) == alpha3
\end{align*}




We now recall the AJS/Billey formula. The $T$-invariant inclusion $i$
of $T$-fixed points into $G/B$ results in a map in equivariant cohomology:
\begin{equation}\label{eq:fixedpointinclusion}
  i^*: H_T(G/B) 
  \longrightarrow \bigoplus_{w \in W} H_T^*(wB/B) \cong \bigoplus_{w \in W} H_T^*
\end{equation}
and $i$ is known to be an {\em injection.}  The inclusion
$i_w: wB/B\hookrightarrow G/B$ induces the projection to the $w$-term
in this sum, so we may write $i^* = \oplus_{w\in W}\ i^*_w$.


%The Weyl group $W := N(T)/T$ acts on $H_T^*$ by permuting these variables. %Let $H^*_T[W]$ denote 
%We denote by $H^*_T$ be the
%base ring of $T$-equivariant cohomology over $\ZZ$, which bears an
%action $w\cdot$ of the Weyl group $W := N(T)/T$.
%Let $H^*_T[W]$ denote its smash product with the group algebra, 
%i.e. 
%the free $H^*_T$-module with basis $W$ and multiplication 
%$wp := (w\cdot p) w$. 

\junk{
  Restricting \eqref{eq:structureconstants} to $u$,
  and using the upper triangularity once again,
  $$
  S_u|_u S_v|_u  = (S_u S_v)|_u = \sum_{w\geq u,v} c_{uv}^w S_w|_u 
  = \sum_{w = u} c_{uv}^w S_w|_u  = c_{uv}^u S_u|_u
  \qquad \therefore c_{uv}^u = S_v|_u. $$
} 

For any $v, w \in W$, the 
\defn{point restriction} $S_v|_w\in H_T^*$  is defined by $i_w^*(S_v)$, 
i.e. the image of $S_v$ under the map $i^*$ in \eqref{eq:fixedpointinclusion}, 
then projected to the $w$ summand. 
Since  \eqref{eq:fixedpointinclusion} is an inclusion,  
each Schubert class $S_v$ is described fully by the list
$\{i_w^*(S_v)\ :\ w\in W\}$ of these restrictions.
Note that $S_w|_u \neq 0$ implies $uB/B \in \overline{B_- wB}/B$, i.e. 
$u \geq w$ in \defn{Bruhat order}, and in fact the converse is also true.
This \defn{upper triangularity of the support} will be useful just below.

In the case $u=w$, the relation \eqref{eq:structureconstants} and this
upper triangularity imply that $c_{uv}^w = S_v|_w$. After choosing $Q$
a reduced word for $w$, the only choice of reduced word $P$ for $u$ is
$Q$ itself.  The formula thus simplifies to
$$ S_v|_w = 
  \sum_{R\subseteq Q \text{ reduced}\atop P = Q, \ \prod R = v}
  \prod_Q \left(    { \alpha_q}^{[q \in R]} \    r_q \right) \cdot 1, $$
which is just a restatement of the AJS/Billey formula.

After describing our geometric proof, we give an algebraic
interpretation of % this formula 
\cref{thm:main} as a coefficient of the product
of certain {\em Schubert structure operators}.  
Let  $H_T^*[\partial]$ denote the smash product of $H_T^*$ with $\ZZ[\partial]$, the algebra consisting of the free $H_T^*$-module 
$H_T^*\otimes_\ZZ \ZZ[\partial]$ with product given by, for $p, q\in H_T^*$,
$$
(p \otimes \partial_v) \cdot (q \otimes \partial_w) = p (\partial_v q) \otimes \partial_v\partial_w
$$
and extended linearly. This
smash product was first introduced by Kostant and Kumar in
\cite{Kostant-Kumar}.  Since $r_\alpha$ acts on $H_T^*(G/B)$
equivalently to $1-\alpha \partial_\alpha$, we will abuse notation and
denote by $r_\alpha\in H_T^*[\partial]$ the operator
$1-\alpha \partial_\alpha$.

\junk{red Is this only a simply laced case argument? The referee wants to know. AK: we haven't checked directly that our weirdo operators satisfy the
  $B_2,G_2$ braid relations, but it should likely follow from our main result
  proven the other way...?
  I'll look into modifying my code to check directly.}
Let
$$
  K^\alpha 
  := (\partial_\alpha r_\alpha \tensor 1 \tensor 1) 
  + (r_\alpha \tensor \partial_\alpha \tensor 1) 
  + (r_\alpha \tensor 1 \tensor \partial_\alpha) 
  + (\alpha r_\alpha \tensor \partial_\alpha \tensor \partial_\alpha) 
$$
in $H_T^*[\partial] \otimes \ZZ[\partial] \otimes \ZZ[\partial]$.
The \defn{Schubert structure operators} $K^\alpha$ braid and commute
appropriately (in the simply and doubly laced cases;
we conjecture but haven't checked the remaining $G_2$ case), and square to $0$. 
They act on
$H_T^*(G/B)\otimes H_T^*(G/B)\otimes H_T^*(G/B)$, resulting in another way 
(in \cref{sec:recursive}) to obtain the coefficients $c_{uv}^w$. 
It seems likely that further analysis of them would give a purely algebraic 
proof of \cref{thm:main}.  As an application of \cref{thm:main}, 
we derive two recursive formulas for structure constants.

\section{Ingredients of the proof}

Recall that the \defn{Bott-Samelson manifold} associated to a word
$Q = r_{\alpha_{i_1}}r_{\alpha_{i_2}}\cdots r_{\alpha_{i_\ell}}$ in
simple reflections is given by
$$
BS^Q = P_{\alpha_{i_1}} \times^B P_{\alpha_{i_2}} \times^B \cdots
\times^B P_{\alpha_{i_\ell}} /B
$$
where $P_{\alpha_{i_j}}$ is the minimal parabolic associated to the
simple reflection ${r_{i_j}}$ and the quotient results in an
equivalence of elements 
\junk{ in $P_{\alpha_{i_1}} \times^B P_{\alpha_{i_2}} \times^B \cdots \times^B
  P_{\alpha_{i_\ell}}$}
given by
$(g_1,g_2, \dots, g_\ell) \sim (g_1b_1,b_1^{-1}g_2b_2,\linebreak \dots,
b_{\ell-1}^{-1}g_\ell b_\ell).$ 
We denote the resulting equivalence classes with square
brackets, i.e. \linebreak $[g_1,g_2,\ldots,g_\ell] \in BS^Q$.

%that has been determined 
%and write $r_m$ to indicate a reflection in the position $m=t_b$ $L$, which we then denote $r_m$, though it is in fact the simple reflection $r_{i_{m}}$. We abuse notation in this fashion to avoid overly tedious sub-sub-indexing as well as concerns about ordering the elements in any subset of $Q$.


There is an action by $T$ on the left of $BS^Q$ with $2^{\#Q}$ fixed points;
more specifically the set of sequences
$(g_1,g_2,\dots, g_\ell) \in
P_{\alpha_{i_1}} \times P_{\alpha_{i_2}} \times \cdots
\times P_{\alpha_{i_\ell}}$ 
such that $\forall j,\  g_j \in \{1,s_j\}$
maps bijectively to the fixed point set $(BS^Q)^T$.
%Therefore fixed points of $T$ on $BS^Q$ are indexed by subsets $\{t_1,\dots, t_k\}\subset \{1,\dots, \ell\}$ consisting of those $j$ such that $g_j=s_j\cdot 1$.
In this way we index the fixed points by subsets $L\subseteq \{1,\ldots,\ell\}$,
but instead of writing ``$L$ is the $\{2,3\}$ subword of $(r_1,r_2,r_1)$''
we will write ``$L$ is the subword $-r_2 r_1$ of $(r_1,r_2,r_1)$'',
allowing e.g. distinction between the $r_1- -$ and $--r_1$ subwords.
In addition, 
the inclusion of the fixed points induces a map in equivariant cohomology
\begin{equation}\label{eq:inclusionBSfixedpoints}
H_T^*(BS^Q) \longrightarrow \bigoplus_{L\subseteq Q} H_T^*
\end{equation}
which is known to be an injection.

\junk{ %% I object to this environment!
We provide some explanation of notation. For simplicity, we define $s_j = r_{i_j}$ and denote
$Q=s_1s_2\dots s_\ell$. Then each subword $L=s_{t_1}s_{t_2}\cdots s_{t_k}$  of $Q$ corresponds to a subset $\{t_1,\dots, t_k\}$ of $\{1,\dots, \ell\}$, which in turn indicates an ordered subset of possibly repeating simple reflections (those occurring as $s_{t_1},\cdots, s_{t_k}$), together with their positions within $Q$. We abuse notation and write $L\subseteq Q$ to indicate the ordered set of (possibly repeated) simple reflections occurring in positions $t_1,\dots, t_k$. Then for $t_b\in \{t_1,\dots, t_k\}$, and $s_{t_b}=r_m$, we simply write $m\in L\subseteq Q$ where the position has been suppressed.  Similarly, we write $\alpha_m$ for the simple root corresponding to the reflection.  While the same simple reflection $r_m$ may occur more than once in $L$, the corresponding position does not repeat. Furthermore, we indicate by $
\prod_{m\in L}$ the product that concatenates in the order of the subword $L$. For example, if $Q = r_1r_2r_1$ and $L = r_1 r_2$, then 
$$
\prod_{m\in L} \alpha_m r_m = \alpha_1r_1\alpha_2r_2.
$$
}


For any subword $L=s_{t_1}\cdots s_{t_k}$ of $Q$, there is a corresponding copy of $BS^L$
obtained as a submanifold of $BS^Q$ by
$$
BS^L = \left\{ [g_1,\cdots, g_\ell]\in BS^Q\  |\  g_j=1 \mbox{ if  $j\not\in L$} \right\}.
%$s_j$  does not occur in $L$} \right\}.
$$
The submanifolds $BS^L$ are $T$-invariant, and each $BS^L_\circ := BS^L \setminus \bigcup_{M \subsetneq L} BS^M$ contains a unique $T$-fixed point 
$[g_1, \dots, g_\ell]\in BS^L$, the one we also corresponded to $L$. 
%Thus we have a correspondence between
%$\{BS^L\}_{L\subseteq Q}$ and the set of $T$-fixed points on $BS^Q$.

%Let $I$ be a reduced word for the long word of the Weyl group. (Why??)
% Each invariant subvariety $BS^L$ has a corresponding Poincar\'e dual class in $H_T^*(BS^Q)$. 
The equivariant homology classes $\{[BS^L] \ :\ L\subseteq Q\}$ form a basis of
$H_*^T(BS^Q)$ as a (free) module over $H_T^*$. There exists a 
dual basis $\{T_J\}_{J\subseteq Q}$ of $H^*_T(BS^Q)$, 
again defined by the $H^*_T$-valued Alexander pairing $\langle,\rangle$;
we compute its point restrictions in \cref{lemma:Trestriction}.
% that multiplies cohomology classes and pushes forward to a point,
% i.e. $\int_{BS^Q} [BS^L] T_J =\delta_{LJ}$. 


Consider the natural map $\pi_R:\ BS^R \rightarrow G/B$ that multiplies
the terms, $[g_1,\ldots,g_\ell] \mapsto (\prod_i g_i) B/B$. 
The image is $B$-invariant, irreducible, and closed,
so necessarily some $X^w$ (but $w$ may not be $\prod R$).
However $\dim BS^R = \dim X^w$ if and only if $R$ is a reduced word, 
in which case the top homology class of $BS^R$ pushes forward to that of $X^w$. 
\junk{ Furthermore, for $I$ a reduced word for the longest word $w_0$, the map $\pi_I: BS^I\rightarrow G/B$ is surjective with $\pi_I(BS^Q\subset BS^I)=X^w$ whenever $w=\prod Q$. {\bf Who cares}}
The pushforward sends the homology class of $BS^R$ to that
of $X^w$ in $G/B$ whenever $R$ is a reduced word for $w$, and
otherwise sends it to $0$. These statements are true both for singular
homology and also, since the varieties involved are $T$-invariant,
for %Borel-Moore
equivariant homology \cite{Kumar-Nori,BrionH}.

We are interested in the transpose map in equivariant cohomology,
where we have the dual bases $\{T_J\}$, $\{S_w\}$ of $H^*_T(BS^Q)$, $H^*_T(G/B)$
respectively.
\junk{
  As with ordinary cohomology, there is a pairing between the
  equivariant homology $H^T_*(G/B)$  and the equivariant cohomology $H_T^*(G/B)$. The $H^*_T$-basis of equivariant homology classes associated with $\{X^w\}$ has a dual basis  $\{x_w\}\in H_T^*(G/B)$ given by the relationship
  $$
  \int_{X^w} x_v =\delta_{wv},
  $$
  where integration is shorthand notation for capping the equivariant homology element corresponding to the $T$-invariant variety $X^w$ with the equivariant cohomology class $x_v$, then pushing forward to a point \cite{Graham}.  A minor adaption of an argument in that work shows that $x_w = S_w$ is the opposite Schubert class (associated with the opposite Schubert variety $X_w$).
}
Since $(\pi_Q)_*([BS^R]) = [X^w]$ in equivariant homology,
%the equivariant %Borel-Moore 
%homology element associated to $BS^Q$ pushes forward to the homology element associated to $X^w$, 
the transpose statement is the lemma:
\begin{Lemma} 
%Let $[X_w]\in H_T^*(G/B)$ be the cohomology class associated with the variety $X_w$, and 
Let $\pi_Q: BS^Q \rightarrow G/B$ be the product map. Then
$$
\pi_Q^*(S_w) = \sum_{R\subseteq Q\text{ reduced} \atop  \prod R =w} T_R.
$$
\end{Lemma}
\begin{proof} 
  Let $[BS^L], [X^w]$ denote the equivariant homology classes, 
  and $\langle,\rangle_M$ denote the perfect $H^*_T$-valued pairing
  between $H^T_*(M)$ and $H_T^*(M)$ for $M$ a smooth compact
  oriented $T$-manifold.  Then
  \begin{eqnarray*}
    \langle \pi_Q^*(S_w), [BS^L]\rangle_{BS^Q} &=&
  \langle S_w, (\pi_Q)_*( [BS^L]) \rangle_{G/B} \\
   &=&  \begin{cases}
    \langle S_w, [X^v] \rangle 
    & \mbox{if $L$ is reduced, with product $v$}\\
    0 & \mbox{otherwise}
  \end{cases} \\
  &=&  \begin{cases}
    1
    & \mbox{if $L$ is reduced, with product $w$}\\
    0 & \mbox{otherwise.}
  \end{cases}
  \end{eqnarray*}
\junk{  This pairing is $1$ when $v=w$, and $0$ otherwise. 
  Therefore, $\langle \pi_I^*(S_w), [BS^L] \rangle=1$ if and only if $L$ is a
  reduced word for $w$. }
Since the $\{T_R\}$ are defined so that
  $\langle T_R, [BS^L] \rangle = \delta_{RL}$, we conclude that
  $\pi_Q^*(S_w) \linebreak = \sum_{R\subseteq Q\text{ reduced} \atop \prod R=w} T_R$.
\end{proof}

%This map describes the pullback of the Schubert classes since $S_w = S_w$. 
We pull back the equation
$S_u S_v = \sum_{x\in W} c_{uv}^x S_x$ along $\pi_Q:\ BS^Q\to G/B$ 
and simplify the right hand side of the equation:
\begin{equation}
  \label{eq:LHS}
  \pi_Q^*(S_u)\ \pi_Q^*(S_v) 
=  \sum_{x\in W} c_{uv}^x\ \pi_Q^*(S_x) 
= \sum_{x\in W} c_{uv}^x\sum_{R\subseteq Q\text{ reduced} \atop \prod R =x} T_R
= \sum_{R\subseteq Q\text{ reduced}}  c_{\ uv}^{\prod R}\  T_R.
\end{equation}
% (Soon $Q$ will be a reduced word for $w$, but we needn't assume this yet.)
By expanding the left hand side in a similar fashion, we obtain
$$
\pi_Q^*(S_u) \pi_Q^*(S_v) =  \sum_{R\subseteq Q \text{ reduced} \atop \prod R =u} T_R \sum_{S\subseteq Q\text{ reduced}\atop \prod S =v} T_S
= \sum_{R,S\subseteq Q\text{ reduced} \atop \prod R =u, \prod S = v} T_R T_S.
$$
Define $b_{RS}^J$ to be the structure constants for the multiplication
in $H_T^*(BS^Q)$ in the basis $\{T_J\}$, defined by the relationship
$$
 T_R T_S = \sum_{J\subset Q} b_{RS}^J T_J.
$$
Thus we have shown
\begin{equation}
  \label{eq:RHS}
\pi_Q^*(S_u) \pi_Q^*(S_v) = \sum_{R,S\subseteq Q\text{ reduced} \atop  \prod R =u, \prod S = v}  \sum_{J\subset Q} b_{RS}^J T_J.
\end{equation}
Now we take $Q$ to be reduced with product $w$ 
and look at the coefficient of $T_Q$ in 
(\ref{eq:LHS}) and (\ref{eq:RHS}):
\junk{

old:

We compare terms in (\ref{eq:LHS}) and (\ref{eq:RHS})
to obtain a relationship between the structure constants for the Schubert classes and those for the dual Bott-Samelson classes.  Let $Q$ be a reduced word expression for $w$. By comparing like terms (and using the independence of $\{T_J\}$),  we obtain 
}
\begin{equation}\label{eq:cs=sumofbs}
c_{uv}^w = \sum_{R,S\subset Q \text{ reduced} \atop  \prod R =u, \prod S = v} b_{RS}^Q.
\end{equation}

\begin{Theorem}\label{th:b-formula} Let the equivariant intersection
  numbers $b_{RS}^Q$ be defined as above. Then,
  $$
  b_{RS}^Q 
  = \prod_{q\in Q} \left( \alpha_q^{[q\in R, S]} \partial_q^{\left[q\notin R, S\right]}
    r_q \right)\cdot 1
  $$
  where the exponent $[q\in J]$ indicates inclusion of the factor only
  when $q\in J$. 
\end{Theorem}

\cref{thm:main} then follows directly from
\cref{th:b-formula} and \eqref{eq:cs=sumofbs}.

The proof of \cref{th:b-formula} is an inductive argument based on \cref{lemma:Trestriction} 
below; both proofs will appear elsewhere.

As with Schubert classes, we define the point restriction $T_J|_L$ to
be the restriction of $T_J \in H_T^*(BS^Q)$ under the map
\eqref{eq:inclusionBSfixedpoints} to the fixed point $L\subseteq
Q$. These restrictions can be computed explicitly:
%, which we prove in another paper:

\junk{{\color{red} RG: We have in fact established correct notation for $\prod_{m\in L} \alpha_{m}^{[m\in J]} r_m$ and it is
$\prod_{m\in L} \alpha_{i_m}^{[m\in J]} r_{i_m}$. Just sayin'.}}
\begin{Lemma}\label{lemma:Trestriction}
  The equivariant class $T_J \in H_T^*(BS^Q)$ 
  has the following restriction to a $T$-fixed point $L$:
  $$
  T_J|_L=\begin{cases}
    \displaystyle \left(\prod_{m\in L} \alpha_{m}^{[m\in J]} r_m \right)\cdot 1 &\mbox{if $J\subseteq L$}\\
    0 &\mbox{if $J \not\subseteq L$.}
  \end{cases}
  $$
  where the exponent $[m\in J]$ indicates inclusion of the factor only
  when $m\in J$. %, and the product is ordered by position in $L$, DUH 
  \end{Lemma}

\junk{

\begin{proof}
  We do this in reverse: {\em define} $T_J$ by its point restrictions above, and
  verify that 
  that $\langle [BS^K], T_J \rangle = \delta_{KJ}$ for all $K\subseteq Q$.
  First observe that 
  $$ H^*_T(BS^Q) \quad\stackrel{i^*}\into\quad H^*_T\left((BS^Q)^T \right)
  \quad\into\quad (frac\ H^*_T) \tensor_{H^*_T} H^*_T\left((BS^Q)^T \right)
  \iso \bigoplus_{R\subseteq Q} frac\ H^*_T $$
  identifies the first two as nested $H^*_T$-lattices in the third,
  a vector space over the fraction field $frac\ H^*_T$. 
  Our pairing $\langle [BS^K], \alpha \rangle$ can be defined for
  {\em any} $\alpha$ in this vector space, by the Atiyah-Bott / Berline-Vergne 
  equivariant localization formula
  $$ \langle [BS^K], \alpha \rangle 
  := \sum_{L \subseteq K} \frac{\alpha|_L}{e(T\ BS^K)|_L} 
  \qquad \in frac\ H^*_T $$
  where the denominator is the product of the weights
  {\color{red} 
  $\left\{ \left(\prod_{\ell < k, \ell\in L} r_{\alpha_\ell}\right)\cdot \alpha_k
    \ :\ k\in K\right\}$. RG: Do you like it now}
  in the tangent space $T_L BS^K$.
  Having defined $T_J$ by its restrictions, we know only that we have an
  element of $(frac\ H_T^*) \otimes H^*_T((BS^Q)^T)$, not necessarily of the sublattice $H^*_T(BS^Q)$.
  By the perfection of the pairing between $H^*_T(BS^Q)$ and $H_*^T(BS^Q)$,
  to check that $T_J \in H^*_T(BS^Q)$ it is necessary and sufficient
  that each $\langle [BS^K], T_J \rangle$ is in $H^*_T$.
  Of course, for the dual basis statement we will be confirming the
  stronger statement that $\langle [BS^K], T_J \rangle = \delta_{JK}$.

  ...

\end{proof}

  In particular, the support of $T_J$ is on the points $L \subseteq Q$ 
  that {\em contain} $J$, rather than those that are contained in $J$.

end junk}



\junk{
It follows that, for a fixed reduced word $Q$ for $w$, 
$$
c_{uv}^w =         
\sum_{R,S \subseteq Q\text{ reduced}\atop \prod R = u, \prod S = v}
\prod_Q \left( \alpha_q^{[q\in R, S]} \partial_q^{\left[q\notin R, S\right]} r_q \right)\cdot 1
$$
A quick comparison to $d_{uv}^w$ shows that $c_{uv}^w = d_{uv}^ww \cdot 1$. The statement of the theorem follows.
}

In the remainder we present these coefficients in terms of some
apparently natural families of operators, based on reflections and
divided difference operators.

\section{AJS/Billey operators}\label{sec:ajsb}

In the next two sections we interpret the AJS/Billey formula,
and \cref{thm:main}, in terms of certain operators;
our results are that these operators satisfy the various (nil-)Coxeter 
relations. We hope someday to run the arguments backward and use the relations
to give an algebraic proof of \cref{thm:main}.

Let $H^*_T[W]$ be the smash product of $H_T^*$ and the group algebra of $W$, i.e. the free $H^*_T$-module with basis $W$ and multiplication $wp := (w\cdot p) w$.
For each $w\in W$, we introduce an \defn{AJS/Billey operator} 
\begin{equation}
  \label{eq:Jdef}
  J_w := \sum_{v\leq w} (S_v|_w) w \ \tensor\ \partial_v 
  \quad \in H^*_T[W] \tensor_{\ZZ} \ZZ[\partial] 
\end{equation}

so in particular
$$ J_\alpha := J_{r_\alpha} = 
(r_\alpha\tensor 1)+(\alpha r_\alpha\tensor\partial_\alpha). $$
Note that these operators are homogeneous of degree $0$, where
the degrees of $\alpha,r_\alpha,\partial_\alpha$ are $+1,0,-1$ respectively.

\begin{Theorem}\label{thm:AJSBops}
  \begin{enumerate}
  \item If $Q$ is a reduced word for $w$, then $J_w = \prod_Q J_q$.
  \item If $\ell(w)+\ell(v) = \ell(wv)$, then $J_w J_v = J_{wv}$,
    and this fact is essentially equivalent to the AJS/Billey formula.
  \item $J_\alpha^2 = 1\tensor 1$, so in fact any word $Q$ for $w$ suffices
    in (1), and $J_w J_v = J_{wv}$ for all $w,v$.
  \end{enumerate}
\end{Theorem}

\begin{proof}
  \begin{enumerate}
  \item Let $Q$ be a reduced word for $w$. Then since $S_v|_{r_\alpha}$ is $0$ unless $v=1$ or $v=r_\alpha$,
    \begin{eqnarray*}
      \prod_Q J_q 
      &=& \prod_Q \sum_{v \leq r_q} (S_v|_{r_q}) r_q \tensor \partial_q 
      \quad=\quad \prod_Q 
          \left( (r_q \tensor 1) + (\alpha_q r_q \tensor \partial_q) \right) \\
      &=& \sum_{R\subseteq Q} \left(\prod_Q \alpha_q^{[q\in R]} r_q\right) 
          \tensor \prod_R \partial_r 
      \quad=\quad 
%    &=& 
          \sum_v \sum_{R\subseteq Q\text{ reduced}\atop \prod R=v} 
                    \left(\prod_Q \alpha_q^{[q\in R]} r_q\right) 
          \tensor \partial_v
     \end{eqnarray*}         
as $\prod_R \partial_r=0$ unless $R$ is reduced. The AJS/Billey formula 
%(Billey, 1999)
 states that
$$
\sum_{R\subseteq Q\text{ reduced}\atop \prod R=v} 
                    \prod_Q \alpha_q^{[q\in R]} r_q = S_v|_ww,
$$
from which it follows that %\quad\text{ and therefore } \quad
$$
\prod_Q J_q =  \sum_{v\leq w} (S_v|_w) w \tensor \partial_v=J_w.
$$


  \item From (1) the equality $J_w J_v = J_{wv}$  follows
   by concatenating words for $w$ and $v$. \\
    Conversely, the equality implies $J_w = \prod_Q J_q$ when $Q$ is a reduced word for $w$,
    which in turn implies the AJS/Billey formula by the calculation above.
  \item 
    \begin{eqnarray*}
      J_\alpha^2 
  &=& \left( (r_\alpha\tensor 1)+(\alpha r_\alpha\tensor\partial_\alpha)\right)^2
  = \left( (r_\alpha\tensor 1)+(\alpha r_\alpha\tensor\partial_\alpha)\right)
      \left( (r_\alpha\tensor 1)+(\alpha r_\alpha\tensor\partial_\alpha)\right)\\
  &=& (1\tensor 1) 
      + (r_\alpha \alpha r_\alpha \tensor \partial_\alpha)
      + (\alpha \tensor \partial_\alpha)
      + ( \alpha r_\alpha  \alpha r_\alpha \tensor \partial_\alpha^2) = 1\tensor 1
    \end{eqnarray*}
  \end{enumerate}
\end{proof}


Let $(G/B)_\Delta$ denote the diagonal copy of $G/B$ in $(G/B)^2$,
which is invariant under the diagonal $T$-action on $(G/B)^2$. 
The corresponding Poincar\'e dual class
$D^{w_0} \in H^*_T((G/B)^2)$ 
 % $[(G/B)_\Delta]$ 
of this submanifold can be described explicitly in
terms of the Poincar\'e duals \linebreak $S^v \in H^*_T(G/B)$ to the $X^v$.
Under the isomorphism
$$
H^*_T((G/B)^2) \cong H^*_T(G/B)\otimes_{H_T^*} H^*_T(G/B)
$$
we have from \cite{Brion} the factorization of the diagonal
%Let $[(G/B)_\Delta]$ denote the class of the diagonal in $(G/B)^2$, mainly 
\begin{equation}
  \label{eq:factorization}
%[(G/B)_\Delta]\in H^*_T((G/B)^2) &\cong H^*_T(G/B)\otimes_{H_T^*} H^*_T(G/B)\\
D^{w_0} %[(G/B)_\Delta] %&=  \sum_v [X_v]\otimes [X^v] 
 = \sum_v S_v \tensor S^v
\quad = \sum_v S_v \tensor (\partial_v \cdot S^{1})
\end{equation}
Consider its restriction along 
$i_w\times Id:\ \{wB/B\} \times G/B \to (G/B)^2$\, :
$$
 D^{w_0}  %= \sum_v S_v \tensor S^v
= \sum_v S_v \tensor (\partial_v \cdot S^{1})
%\stackrel{(i_w\times Id)^*}\longmapsto 
\xmapsto{ (i_w\times Id)^* }
\sum_v (S_v|_w) \tensor (\partial_v \cdot S^1)
= J_w \cdot (S_1 \tensor S^1). 
$$
While we won't directly use this suggestive calculation of the $S_v|_w$,
it will inform a similar operator-theoretic calculation of the 
$c_{uv}^w$ in the next section.
Towards that end we rephrase the equation above 
using the equivariant Euler class $e(T\, G/B)$ of the tangent bundle:
\begin{equation}
  \label{eq:J}
 \left( e(T\, G/B) \tensor 1\right)\ D^{w_0} 
= \sum_{w\in W} (i_w \times Id)_* \big(J_w \cdot (S_1 \tensor S^1) \big)
\end{equation}


\junk{

\subsection{Expanding in $(\partial_u)$ -- will we need this?}

In the next section we'll need a generalization in which the first
tensor factor lives in $H^*_T[\partial]$, not just the subring $H^*_T[W]$. 
To compute the expansion $w = \sum_u m_w^u \partial_u$ of a basis
element from the subring in the basis of the whole, 
we first apply both sides to $S_{v}$
$$  w \cdot S_v  = \sum_u m_w^u \partial_u \cdot S_{v}
      = \sum_{u\atop \ell(uv) = \ell(v)-\ell(u)} m_w^u S_{uv}
$$
then restrict to the basepoint. The LHS becomes
$$ (w\cdot S_v)|_1 = w\cdot(S_v|_{w^{-1}}) $$
whereas the RHS becomes
$$ \left(\sum_{u\atop \ell(uv) = \ell(v)-\ell(u)} m_w^u S_{uv}\right)\Bigg|_1
= \sum_{u\atop \ell(uv) = \ell(v)-\ell(u)} m_w^u (S_{uv}|_1)
= m_w^{v^{-1}} $$
Hence
$w = \sum_u \left(w\cdot (S_{u^{-1}}|_{w^{-1}})\right) \partial_u$,
and $J_w = \sum_{u,v} (S_v|_w) 
\left(w\cdot (S_{u^{-1}}|_{w^{-1}})\right) \partial_u \ \tensor\ \partial_v$.

}

\section{Schubert structure operators}

Analogously to $J_\alpha \in H^*_T[W] \tensor \ZZ[\partial]$, 
we introduce in $H^*_T[\partial] \tensor \ZZ[\partial] \tensor \ZZ[\partial]$ 
elements 
$$
  K^\alpha 
  := (\partial_\alpha r_\alpha \tensor 1 \tensor 1) 
  + (r_\alpha \tensor \partial_\alpha \tensor 1) 
  + (r_\alpha \tensor 1 \tensor \partial_\alpha) 
  + (\alpha r_\alpha \tensor \partial_\alpha \tensor \partial_\alpha),
$$
{ where $r_\alpha = (1-\alpha\partial_\alpha) \in H^*_T[\partial] $.}
These are homogeneous of degree $-1$.
Note that $r_\alpha \partial_\alpha = \partial_\alpha = -\partial_\alpha r_\alpha$.

\begin{Lemma}
  $(K^\alpha)^2 = 0$.
\end{Lemma}

\begin{proof}
  At the end we use the equality of operators $\partial_\alpha \alpha + \alpha \partial_\alpha = 2$,
  derivable from the twisted Leibniz identity
  $\partial_\alpha\cdot (xy) = (\partial_\alpha \cdot x)y + (r_\alpha\cdot x)(\partial_\alpha\cdot y)$.
  {\small
  \begin{eqnarray*}
    (K^\alpha)^2 
    &=& (\partial_\alpha r_\alpha \tensor 1 \tensor 1) 
        \left( (\partial_\alpha r_\alpha \tensor 1 \tensor 1) 
  + (r_\alpha \tensor \partial_\alpha \tensor 1) 
  + (r_\alpha \tensor 1 \tensor \partial_\alpha) 
  + (\alpha r_\alpha \tensor \partial_\alpha \tensor \partial_\alpha) \right) \\
  &+& (r_\alpha \tensor \partial_\alpha \tensor 1) 
        \left( (\partial_\alpha r_\alpha \tensor 1 \tensor 1) 
  + (r_\alpha \tensor \partial_\alpha \tensor 1) 
  + (r_\alpha \tensor 1 \tensor \partial_\alpha) 
  + (\alpha r_\alpha \tensor \partial_\alpha \tensor \partial_\alpha) \right) \\
  &+& (r_\alpha \tensor 1 \tensor \partial_\alpha) 
        \left( (\partial_\alpha r_\alpha \tensor 1 \tensor 1) 
  + (r_\alpha \tensor \partial_\alpha \tensor 1) 
  + (r_\alpha \tensor 1 \tensor \partial_\alpha) 
  + (\alpha r_\alpha \tensor \partial_\alpha \tensor \partial_\alpha) \right) \\
  &+& (\alpha r_\alpha \tensor \partial_\alpha \tensor \partial_\alpha) 
        \left( (\partial_\alpha r_\alpha \tensor 1 \tensor 1) 
  + (r_\alpha \tensor \partial_\alpha \tensor 1) 
  + (r_\alpha \tensor 1 \tensor \partial_\alpha) 
  + (\alpha r_\alpha \tensor \partial_\alpha \tensor \partial_\alpha) \right)
\\
  &=&(\partial_\alpha r_\alpha \partial_\alpha r_\alpha \tensor 1 \tensor 1) 
  + (\partial_\alpha r_\alpha r_\alpha \tensor \partial_\alpha \tensor 1) 
  + (\partial_\alpha r_\alpha r_\alpha \tensor 1 \tensor \partial_\alpha) 
  + (\partial_\alpha r_\alpha \alpha r_\alpha \tensor \partial_\alpha \tensor \partial_\alpha) \\
  &+& (r_\alpha \partial_\alpha r_\alpha \tensor \partial_\alpha \tensor 1) 
  + (r_\alpha r_\alpha \tensor \partial_\alpha \partial_\alpha \tensor 1) 
  + (r_\alpha r_\alpha \tensor \partial_\alpha \tensor \partial_\alpha) 
  + (r_\alpha \alpha r_\alpha \tensor \partial_\alpha \partial_\alpha \tensor \partial_\alpha) \\
  &+& (r_\alpha \partial_\alpha r_\alpha \tensor 1 \tensor \partial_\alpha) 
  + (r_\alpha r_\alpha \tensor \partial_\alpha \tensor \partial_\alpha) 
  + (r_\alpha r_\alpha \tensor 1 \tensor \partial_\alpha \partial_\alpha) 
  + (r_\alpha \alpha r_\alpha \tensor \partial_\alpha \tensor \partial_\alpha \partial_\alpha) \\
  &+& (\alpha r_\alpha \partial_\alpha r_\alpha \tensor \partial_\alpha \tensor \partial_\alpha) 
  + (\alpha r_\alpha r_\alpha \tensor \partial_\alpha \partial_\alpha \tensor \partial_\alpha) 
  + (\alpha r_\alpha r_\alpha \tensor \partial_\alpha \tensor \partial_\alpha \partial_\alpha) 
  + (\alpha r_\alpha \alpha r_\alpha \tensor \partial_\alpha \partial_\alpha \tensor \partial_\alpha \partial_\alpha)
\\
  &=&0 %(\partial_\alpha r_\alpha \partial_\alpha r_\alpha \tensor 1 \tensor 1) 
  + (\partial_\alpha  \tensor \partial_\alpha \tensor 1) 
  + (\partial_\alpha  \tensor 1 \tensor \partial_\alpha) 
  - (\partial_\alpha \alpha \tensor \partial_\alpha \tensor \partial_\alpha) 
  - ( \partial_\alpha \tensor \partial_\alpha \tensor 1) 
+ 0 %  + (1 \tensor \partial_\alpha \partial_\alpha \tensor 1) 
  + (1 \tensor \partial_\alpha \tensor \partial_\alpha) 
+ 0 %  + (r_\alpha \alpha r_\alpha \tensor \partial_\alpha \partial_\alpha \tensor \partial_\alpha) 
\\
  &-& ( \partial_\alpha \tensor 1 \tensor \partial_\alpha) 
  + (1 \tensor \partial_\alpha \tensor \partial_\alpha) 
+ 0%  + (1 \tensor 1 \tensor \partial_\alpha \partial_\alpha) 
+ 0%  + (r_\alpha \alpha r_\alpha \tensor \partial_\alpha \tensor \partial_\alpha \partial_\alpha) 
  - (\alpha  \partial_\alpha \tensor \partial_\alpha \tensor \partial_\alpha) 
+0% + (\alpha  \tensor \partial_\alpha \partial_\alpha \tensor \partial_\alpha) 
+0% + (\alpha  \tensor \partial_\alpha \tensor \partial_\alpha \partial_\alpha) 
+0% + (\alpha r_\alpha \alpha r_\alpha \tensor \partial_\alpha \partial_\alpha \tensor \partial_\alpha \partial_\alpha)
\\
  &=&
  - (\partial_\alpha \alpha \tensor \partial_\alpha \tensor \partial_\alpha) 
  + (1 \tensor \partial_\alpha \tensor \partial_\alpha) 
  + (1 \tensor \partial_\alpha \tensor \partial_\alpha) 
  - (\alpha  \partial_\alpha \tensor \partial_\alpha \tensor \partial_\alpha) 
\\
    &=& (2- \alpha  \partial_\alpha - \partial_\alpha \alpha)
        \tensor \partial_\alpha \tensor \partial_\alpha = 0.
  \end{eqnarray*}
  }
%  Now since $\partial_\alpha(xy) = (\partial_\alpha x)y + (r_\alpha\cdot x)y$,
%  we have $\partial_\alpha(\alpha y) = 2y - \alpha \partial_\alpha y$,
%  or $\partial_\alpha \alpha + \alpha \partial_\alpha = 2$.
%\vskip -.3in
\end{proof}

\begin{Theorem}\label{thm:K^wdef}
  The operators $K^\alpha$ obey the commutation and (simply- or doubly-laced)
  braid relations, and as such, we can define $K^w := \prod_Q K^q$ 
  (for $W$ simply- or doubly-laced) using any reduced word $Q$ for $w$.
\end{Theorem}

\begin{proof}
  The commutation operations are obvious. For braiding, we compute
  $K^\alpha K^\beta K^\alpha$ for the simple roots in $SL_3$.
  \begin{eqnarray*}
    K^\alpha K^\beta K^\alpha
    &=&
\big( -(\partial_\alpha \tensor 1 \tensor 1) 
  + (r_\alpha \tensor \partial_\alpha \tensor 1) 
  + (r_\alpha \tensor 1 \tensor \partial_\alpha) 
  + (\alpha r_\alpha \tensor \partial_\alpha \tensor \partial_\alpha) \big)\\
    &&
\big(- (\partial_\beta \tensor 1 \tensor 1) 
  + (r_\beta \tensor \partial_\beta \tensor 1) 
  + (r_\beta \tensor 1 \tensor \partial_\beta) 
  + (\beta r_\beta \tensor \partial_\beta \tensor \partial_\beta) \big) \\
    &&
\big( -(\partial_\alpha \tensor 1 \tensor 1) 
  + (r_\alpha \tensor \partial_\alpha \tensor 1) 
  + (r_\alpha \tensor 1 \tensor \partial_\alpha) 
  + (\alpha r_\alpha \tensor \partial_\alpha \tensor \partial_\alpha) \big)
\end{eqnarray*}
We group the $4^3$ terms ($15$ of which vanish by 
$\partial_\alpha^2 = 0$) according to their second and
third tensor factors
\junk{
, written on left and top of this array:
$$
\begin{blockarray}{rcccccc}
  &  1 &  \partial_\alpha &  \partial_\beta &  \partial_\alpha\partial_\beta &
  \partial_\beta\partial_\alpha &  \partial_\alpha\partial_\beta\partial_\alpha \\ \\
    \begin{block}{r[cccccc]}             
1
 & \begin{matrix} -\partial_\alpha \partial_\beta \partial_\alpha  \end{matrix} 
 & \begin{matrix} \partial_\alpha \partial_\beta r_\alpha  \\
+r_\alpha \partial_\beta \partial_\alpha  \end{matrix} 
 & \begin{matrix} \partial_\alpha r_\beta \partial_\alpha  \end{matrix} 
 & \begin{matrix} -r_\alpha r_\beta \partial_\alpha  \end{matrix} 
 & \begin{matrix} -\partial_\alpha r_\beta r_\alpha  \end{matrix} 
 & \begin{matrix} r_\alpha r_\beta r_\alpha  \end{matrix} 
\\ \\ 
\partial_\alpha 
 & \begin{matrix} \partial_\alpha \partial_\beta r_\alpha  \\
+r_\alpha \partial_\beta \partial_\alpha  \end{matrix} 
 & \begin{matrix} \partial_\alpha \partial_\beta \alpha r_\alpha  \\
-2r_\alpha \partial_\beta r_\alpha  \\
+\alpha r_\alpha \partial_\beta \partial_\alpha  \end{matrix} 
 & \begin{matrix} -\partial_\alpha r_\beta r_\alpha  \\
-r_\alpha r_\beta \partial_\alpha  \end{matrix} 
 & \begin{matrix} r_\alpha r_\beta r_\alpha  \\
-\alpha r_\alpha r_\beta \partial_\alpha  \end{matrix} 
 & \begin{matrix} -\partial_\alpha r_\beta \alpha r_\alpha  \\
+r_\alpha r_\beta r_\alpha  \end{matrix} 
 & \begin{matrix} r_\alpha r_\beta \alpha r_\alpha  \\
+\alpha r_\alpha r_\beta r_\alpha  \end{matrix} 
\\ \\ 
\partial_\beta 
 & \begin{matrix} \partial_\alpha r_\beta \partial_\alpha  \end{matrix} 
 & \begin{matrix} -\partial_\alpha r_\beta r_\alpha  \\
-r_\alpha r_\beta \partial_\alpha  \end{matrix} 
 & \begin{matrix} \partial_\alpha \beta r_\beta \partial_\alpha  \end{matrix} 
 & \begin{matrix} -r_\alpha \beta r_\beta \partial_\alpha  \end{matrix} 
 & \begin{matrix} -\partial_\alpha \beta r_\beta r_\alpha  \end{matrix} 
 & \begin{matrix} r_\alpha \beta r_\beta r_\alpha  \end{matrix} 
\\ \\ 
\partial_\alpha \partial_\beta 
 & \begin{matrix} -r_\alpha r_\beta \partial_\alpha  \end{matrix} 
 & \begin{matrix} r_\alpha r_\beta r_\alpha  \\
-\alpha r_\alpha r_\beta \partial_\alpha  \end{matrix} 
 & \begin{matrix} -r_\alpha \beta r_\beta \partial_\alpha  \end{matrix} 
 & \begin{matrix} -\alpha r_\alpha \beta r_\beta \partial_\alpha  \end{matrix} 
 & \begin{matrix} r_\alpha \beta r_\beta r_\alpha  \end{matrix} 
 & \begin{matrix} \alpha r_\alpha \beta r_\beta r_\alpha  \end{matrix} 
\\ \\
\partial_\beta \partial_\alpha 
 & \begin{matrix} -\partial_\alpha r_\beta r_\alpha  \end{matrix} 
 & \begin{matrix} -\partial_\alpha r_\beta \alpha r_\alpha  \\
+r_\alpha r_\beta r_\alpha  \end{matrix} 
 & \begin{matrix} -\partial_\alpha \beta r_\beta r_\alpha  \end{matrix} 
 & \begin{matrix} r_\alpha \beta r_\beta r_\alpha  \end{matrix} 
 & \begin{matrix} -\partial_\alpha \beta r_\beta \alpha r_\alpha  \end{matrix} 
 & \begin{matrix} r_\alpha \beta r_\beta \alpha r_\alpha  \end{matrix} 
\\ \\
\partial_\alpha \partial_\beta \partial_\alpha 
 & \begin{matrix} r_\alpha r_\beta r_\alpha  \end{matrix} 
 & \begin{matrix} r_\alpha r_\beta \alpha r_\alpha  \\
+\alpha r_\alpha r_\beta r_\alpha  \end{matrix} 
 & \begin{matrix} r_\alpha \beta r_\beta r_\alpha  \end{matrix} 
 & \begin{matrix} \alpha r_\alpha \beta r_\beta r_\alpha  \end{matrix} 
 & \begin{matrix} r_\alpha \beta r_\beta \alpha r_\alpha  \end{matrix} 
 & \begin{matrix} \alpha r_\alpha \beta r_\beta \alpha r_\alpha  \end{matrix} 
\\ 
   \end{block} 
  \end{blockarray}
$$
This is supposed to be entrywise equal to the following,
coming from $K^\beta K^\alpha K^\beta$:

$$
\begin{blockarray}{rcccccc}
  &  1 &  \partial_\alpha &  \partial_\beta &  \partial_\alpha\partial_\beta &
  \partial_\beta\partial_\alpha &  \partial_\beta\partial_\alpha\partial_\beta \\ \\
    \begin{block}{r[cccccc]}             
1
 & \begin{matrix} -\partial_\beta \partial_\alpha \partial_\beta  \end{matrix} 
 & \begin{matrix} \partial_\beta r_\alpha \partial_\beta  \end{matrix} 
 & \begin{matrix} \partial_\beta \partial_\alpha r_\beta  \\
+r_\beta \partial_\alpha \partial_\beta  \end{matrix} 
 & \begin{matrix} -\partial_\beta r_\alpha r_\beta  \end{matrix} 
 & \begin{matrix} -r_\beta r_\alpha \partial_\beta  \end{matrix} 
 & \begin{matrix} r_\beta r_\alpha r_\beta  \end{matrix} 
\\  \\
\partial_\alpha 
 & \begin{matrix} \partial_\beta r_\alpha \partial_\beta  \end{matrix} 
 & \begin{matrix} \partial_\beta \alpha r_\alpha \partial_\beta  \end{matrix} 
 & \begin{matrix} -\partial_\beta r_\alpha r_\beta  \\
-r_\beta r_\alpha \partial_\beta  \end{matrix} 
 & \begin{matrix} -\partial_\beta \alpha r_\alpha r_\beta  \end{matrix} 
 & \begin{matrix} -r_\beta \alpha r_\alpha \partial_\beta  \end{matrix} 
 & \begin{matrix} r_\beta \alpha r_\alpha r_\beta  \end{matrix} 
\\  \\
\partial_\beta 
 & \begin{matrix} \partial_\beta \partial_\alpha r_\beta  \\
+r_\beta \partial_\alpha \partial_\beta  \end{matrix} 
 & \begin{matrix} -\partial_\beta r_\alpha r_\beta  \\
-r_\beta r_\alpha \partial_\beta  \end{matrix} 
 & \begin{matrix} \partial_\beta \partial_\alpha \beta r_\beta  \\
-2r_\beta \partial_\alpha r_\beta  \\
\beta r_\beta \partial_\alpha \partial_\beta  \end{matrix} 
 & \begin{matrix} -\partial_\beta r_\alpha \beta r_\beta  \\
+r_\beta r_\alpha r_\beta  \end{matrix} 
 & \begin{matrix} r_\beta r_\alpha r_\beta  \\
-\beta r_\beta r_\alpha \partial_\beta  \end{matrix} 
 & \begin{matrix} r_\beta r_\alpha \beta r_\beta  \\
\beta r_\beta r_\alpha r_\beta  \end{matrix} 
\\  \\
\partial_\alpha \partial_\beta 
 & \begin{matrix} -\partial_\beta r_\alpha r_\beta  \end{matrix} 
 & \begin{matrix} -\partial_\beta \alpha r_\alpha r_\beta  \end{matrix} 
 & \begin{matrix} -\partial_\beta r_\alpha \beta r_\beta  \\
+r_\beta r_\alpha r_\beta  \end{matrix} 
 & \begin{matrix} -\partial_\beta \alpha r_\alpha \beta r_\beta  \end{matrix} 
 & \begin{matrix} r_\beta \alpha r_\alpha r_\beta  \end{matrix} 
 & \begin{matrix} r_\beta \alpha r_\alpha \beta r_\beta  \end{matrix} 
\\  \\
\partial_\beta \partial_\alpha 
 & \begin{matrix} -r_\beta r_\alpha \partial_\beta  \end{matrix} 
 & \begin{matrix} -r_\beta \alpha r_\alpha \partial_\beta  \end{matrix} 
 & \begin{matrix} r_\beta r_\alpha r_\beta  \\
-\beta r_\beta r_\alpha \partial_\beta  \end{matrix} 
 & \begin{matrix} r_\beta \alpha r_\alpha r_\beta  \end{matrix} 
 & \begin{matrix} -\beta r_\beta \alpha r_\alpha \partial_\beta  \end{matrix} 
 & \begin{matrix} \beta r_\beta \alpha r_\alpha r_\beta  \end{matrix} 
\\  \\
\partial_\beta \partial_\alpha \partial_\beta 
 & \begin{matrix} r_\beta r_\alpha r_\beta  \end{matrix} 
 & \begin{matrix} r_\beta \alpha r_\alpha r_\beta  \end{matrix} 
 & \begin{matrix} r_\beta r_\alpha \beta r_\beta  \\
+\beta r_\beta r_\alpha r_\beta  \end{matrix} 
 & \begin{matrix} r_\beta \alpha r_\alpha \beta r_\beta  \end{matrix} 
 & \begin{matrix} \beta r_\beta \alpha r_\alpha r_\beta  \end{matrix} 
 & \begin{matrix} \beta r_\beta \alpha r_\alpha \beta r_\beta  \end{matrix} 
\\ 
   \end{block} 
  \end{blockarray}
$$

}
Using the relations
$$ \partial_\alpha \alpha = 2 - \alpha \partial_\alpha
\qquad
\partial_\beta \beta = 2 - \beta \partial_\beta
\qquad
\partial_\alpha \beta = -1 + \alpha \partial_\alpha + \beta \partial_\alpha
\qquad
 \partial_\beta \alpha = -1 + \alpha \partial_\beta + \beta \partial_\beta
$$
we can write each matrix entry uniquely as $\sum_w h_w \partial_w$,
$h_w \in H^*_T$, to compare the two operators. We left the resulting
comparison of $>1000$ terms to a computer. The corresponding $B_2$ 
calculation involved closer to $140,000$ terms. % in intermediate calculation.
% (which is also how the two matrices
%above were produced).
\end{proof}

We are confident that the $K^\alpha$ satisfy the $G_2$
braid relation but have not done the computation (having run out of
memory at $3M+$ terms). 

As a result of \cref{thm:K^wdef}, we may define operators $d_{uv}^w\in H_T^*[\partial]$ by 
$$
K^w := \sum_{u,v} d_{uv}^ww \otimes \partial_u\otimes \partial_v.
$$
The successive application of $K^\alpha$ for each reflection $r_\alpha$ in a reduced word for $w$ then results in the statement that
$$
 d_{uv}^ww  = \prod_Q \left( \alpha_q^{[q\in R, S]} \partial_q^{\left[q\notin R, S\right]}
    r_q \right)
$$
As these operators applied to $1$ are the terms appearing in \cref{thm:main}, we deduce that 
$$
K^w(S_1\otimes S^1 \otimes S^1) = \sum_{u, v} c_{uv}^w \otimes S^u \otimes S^v
$$
%{\color{red} we can presumably fix this statement by replacing $S^u \otimes S^v$ with $S^{u^{-1}} \otimes S^{v^{-1}} $.}
which we now manipulate to get a $K^\alpha$ analogue of (\ref{eq:J}).

Let $D_{12} \in H^*_T\left( (G/B)^3 \right)$ denote the Poincar\'e dual of
the partial diagonal 
$\{(F_1,F_2,F_3) \in (G/B)^3\ :\ F_1 = F_2\}$,
and $D_{13}$ denote that of $\{(F_1,F_2,F_3) \in (G/B)^3\ :\ F_1 = F_3\}$ 
likewise. Then $D_{123} := D_{12} \cap D_{13}$ is the class of the full diagonal.
By two applications of (\ref{eq:factorization}), we get
\begin{eqnarray*}
  D_{123} &=& D_{12} \cap D_{23} 
 = \left(\sum_u (S_u \tensor S^u \tensor 1)\right)
\left(\sum_v ( S_v \tensor 1 \tensor S^v)\right) 
 = \sum_{u,v} S_u S_v \tensor S^u \tensor S^v
\\ &=& \sum_{u,v} \left( \sum_w c_{uv}^w S_w \right) \tensor S^u \tensor S^v 
 =  \sum_w (S_w \tensor 1\tensor 1) 
       \sum_{u,v} \left( c_{uv}^w \tensor S^u \tensor S^v \right)
\end{eqnarray*}
Combined with the above equation, we get 
\begin{equation}
  \label{eq:K}
    D_{123} = \sum_w (S_w \tensor 1\tensor 1)\ K^w(S_1\otimes S^1 \otimes S^1),
\end{equation}
a distinct echo of  (\ref{eq:J}).

{\em Question.} What is a closed form for $K^w$, analogous to
that of $J^w$ in (\ref{eq:Jdef})? 

\section{Recursive formulas for structure constants}
\label{sec:recursive}

\begin{Corollary}
%Inductive statement about structure constants using $K^w$.
Fix a reflection $r_\alpha$, and
let $\overline{s}$ denote $r_\alpha s$ for $s\in W$. If $\overline{w}<w$, then
%$$
%c_{uv}^w=\partial_\alpha r_\alpha d_{uv}^{\overline{w}}\cdot 1 +[\overline{u} < u] c_{\overline{u}, v}^{\overline{w}} +   [\overline{v} < v] c_{u, \overline{v}}^{\overline{w}} +  [\overline{u} < u] [\overline{v} < v]   \alpha c_{\overline{u},\overline{v}}^{\overline{w}}
%$$
$$
c_{uv}^w
\quad=\quad
(\partial_\alpha r_\alpha) \cdot c_{uv}^{\overline{w}} 
\quad+\quad
[\overline{u} < u] c_{\overline{u}, v}^{\overline{w}} 
\quad+\quad   
[\overline{v} < v] c_{u, \overline{v}}^{\overline{w}} 
\quad+\quad  
[\overline{u} < u] [\overline{v} < v] \alpha c_{\overline{u},\overline{v}}^{\overline{w}}
$$
where $[\overline{s} < s]$ indicates $1$ if $\overline{s} < s$, and
$0$ otherwise (i.e. $\overline{s} > s$).

Similarly, let $\underline{s}$ denote $sr_\alpha$. If $\underline{w}<w$, then
$$
c_{uv}^w 
\quad=\quad
[\underline{u}<u] \left({c}_{\underline{u},v}^{\underline{w}}\right) 
\quad+\quad
[\underline{v}<v]\left({c}_{u,\underline{v}}^{\underline{w}}\right) 
\quad+\quad
[\underline{u}<u] [\underline{v}<v] \left(d_{\underline{u},\underline{v}}^{\underline{w}} \cdot \alpha\right)
$$
\end{Corollary}


\begin{proof} 
Suppose $w=r_\alpha r_{\alpha_1}\cdots r_{\alpha_k}$ is a reduced word expression for $w$. Then $K^w =K^\alpha K^{\overline{w}}$, where $\overline{w} = r_\alpha w$. In particular
{\small
\begin{align*}
\sum_{u,v} c_{uv}^w \otimes { S^{u} \otimes S^{v} } &= K^w (S_1\otimes S^1 \otimes S^1)
= \left(K^\alpha \sum_{s, t} d_{st}^{\overline{w}}\overline{w}\otimes \partial_s \otimes \partial_t\right) (S_1\otimes S^1 \otimes S^1)\\
& = \sum_{s, t}\big( \partial_\alpha r_\alpha d_{st}^{\overline{w}}\overline{w} \otimes \partial_s \otimes \partial_t +r_\alpha d_{st}^{\overline{w}}\overline{w} \otimes \partial_\alpha \partial_s \otimes \partial_t
+ r_\alpha d_{st}^{\overline{w}}\overline{w} \otimes \partial_s \otimes  \partial_\alpha \partial_t \\
&  \hspace{.3in} +  \alpha r_\alpha d_{st}^{\overline{w}}\overline{w} \otimes \partial_\alpha \partial_s \otimes  \partial_\alpha \partial_t\big) (S_1\otimes S^1 \otimes S^1)\\
\end{align*}
The term $c_{uv}^w \otimes S^u\otimes S^v$ on the left is obtained as the image of $S_1\otimes S^1 \otimes S^1$ under those tensors with terms $\partial_u\otimes \partial_v$ in the second and third positions. Note that $\partial_\alpha\partial_s =\partial_{s'}$ exactly when $r_\alpha s =s'$ {\em and} $\ell(s') = \ell(s)+1$. If $r_\alpha s =s'$ but $\ell(s') \neq \ell(s)+1,$ then $\partial_\alpha\partial_s =0$.
Let $\overline{v}=r_\alpha v$ and $\overline{u}=r_\alpha u$. By matching the terms,
\begin{align*}
c_{uv}^w \otimes  { S^{u} \otimes S^{v} }  &= \big(\partial_\alpha r_\alpha d_{uv}^{\overline{w}}\overline{w} \otimes \partial_u \otimes \partial_v 
+r_\alpha d_{\overline{u},v}^{\overline{w}}\overline{w} \otimes \partial_\alpha \partial_{\overline{u}} \otimes \partial_v
+ r_\alpha d_{u,\overline{v}}^{\overline{w}}\overline{w} \otimes \partial_u \otimes  \partial_\alpha \partial_{\overline{v}} \\
&  \hspace{.3in} +  \alpha \partial_\alpha d_{\overline{u},\overline{v}}^{\overline{w}}\overline{w} \otimes \partial_\alpha \partial_{\overline{u}} \otimes  \partial_\alpha \partial_{\overline{v}} \big) (S_1\otimes S^1 \otimes S^1)\\
&= \big(\partial_\alpha r_\alpha d_{uv}^{\overline{w}}\overline{w} \otimes \partial_u \otimes \partial_v 
+ [\overline{u} < u] \ r_\alpha d_{\overline{u},v}^{\overline{w}}\overline{w} \otimes \partial_u \otimes \partial_v
+ [\overline{v} < v]\  r_\alpha d_{u,\overline{v}}^{\overline{w}}\overline{w} \otimes \partial_u \otimes  \partial_v \\
& \hspace{.3in}  +  [\overline{u} < u] [\overline{v} < v]\  \alpha r_\alpha d_{\overline{u},\overline{v}}^{\overline{w}}\overline{w} \otimes \partial_u \otimes  \partial_v \big) (S_1\otimes S^1 \otimes S^1).
\end{align*}
}
We evaluate the expression on the right and isolate the first tensor to obtain
{\small
\begin{align*}
c_{uv}^w &= (\partial_\alpha r_\alpha d_{uv}^{\overline{w}}\overline{w})\cdot 1 + [\overline{u} < u]\  (r_\alpha d_{\overline{u},v}^{\overline{w}}\overline{w} )\cdot 1+  [\overline{v} < v]\  (r_\alpha d_{u,\overline{v}}^{\overline{w}}\overline{w} )\cdot 1 +  [\overline{u} < u] [\overline{v} < v] \ (\alpha r_\alpha d_{\overline{u},\overline{v}}^{\overline{w}}\overline{w}) \cdot 1\\
& = (\partial_\alpha r_\alpha d_{uv}^{\overline{w}})\cdot 1 +[\overline{u} < u] c_{\overline{u}, v}^{\overline{w}} +   [\overline{v} < v] c_{u, \overline{v}}^{\overline{w}} +  [\overline{u} < u] [\overline{v} < v]   \alpha c_{\overline{u},\overline{v}}^{\overline{w}}.
\end{align*}
}
A similar proof holds for the second recursion.
\end{proof}

{
We finish with an example illustrating the use of the first recursive formula.
\begin{Example} We compute $c_{u,v}^w$ in the $S_3$ case, with
  $u=[312]$, $v=[132]$ and $w=w_0=[321]$ in 1-line notation. First we
  use $\overline{w}=r_1w$. Then $\overline{u}=r_1u \not< u$ and
  $\overline{v}=r_1v\not< v$. The three latter terms in the sum of the
  first recursion relationship drop out and we obtain
$$
c_{[312],[132]}^{[321]} = 
c_{uv}^w = \partial_1 r_1 \cdot c_{uv}^{\overline{w}}
 = \partial_1 r_1 \cdot c_{[312],[132]}^{[312]}
$$
We set about to compute $c_{uv}^{\overline{w}}$. Note that $r_2r_1$ is a reduced word for $\overline{w}$. There is only one subword for $u$, mainly $r_2r_1$, and one subword for $v$, mainly $r_2-$. Therefore $c_{uv}^{\overline{w}} = \alpha_2 r_2 r_1\cdot 1$ and we obtain
$$
c_{uv}^w = \partial_1 r_1 \alpha_2 r_2 r_1\cdot 1 = \partial_1 (r_1(\alpha_2))=\partial_1 (\alpha_1+\alpha_2) = 1.
$$
As a check on this result, we consider the recursion with $r_2$ 
instead of $r_1$, so $\overline{w} = r_2w=[231]$. 
Then $\overline{u}=r_2u =[213]< u$ and $\overline{v}=r_2v=1\leq v$. 
In principle all four terms are nonzero:
$$
c_{uv}^w=\partial_2 r_2 \cdot c_{uv}^{\overline{w}} +c_{\overline{u}, v}^{\overline{w}} +   c_{u, \overline{v}}^{\overline{w}} +   \alpha c_{\overline{u},\overline{v}}^{\overline{w}}.
$$
However $u \not \leq \overline{w}$, so the first and third terms
$c_{uv}^{\overline{w}}$ and $c_{u, \overline{v}}^{\overline{w}}$ vanish. 
The last term $c_{\overline{u},\overline{v}}^{\overline{w}} = c_{[213],1}^{[231]}=0$ because $S_{[213]}S_{1} = S_{[213]}$. Thus $c_{uv}^w = c_{\overline{u}, v}^{\overline{w}} = c_{[213],[132]}^{[231]}$ is the only remaining nonzero term. This smaller structure constant is easily seen to be 1, for instance by another application of same inductive formula with $r_1[231]=[132]<[231]$.  Note that $r_1[132]\not<[132]$ which forces two terms in the recursive sum to be $0$. We obtain
$$
c_{[213],[132]}^{[231]}
\quad=\quad \partial_1 r_1 \cdot c_{[213],[132]}^{[132]} \quad+\quad c_{1,[132]}^{[132]}
\quad=\quad 0\quad+\quad 1
$$
where the last two equalities follow from $[213]\not\leq [132]$ and 
 $S_1S_{[132]}=S_{[132]}$.
\end{Example}
}


\printbibliography

\end{document}

\junk{

\begin{thebibliography}{100}
\newcommand\arxiv[1]{%use this to give preprint # in refs
%\special{html:<a href="http://arxiv.org/abs/#1">}{\tt arXiv:#1}\special{html:</a>}}
\href{http://arxiv.org/abs/#1}{\tt arXiv:#1}}



\bibitem  Andersen, H.H., Jantzen, J.C., Soergel, W., Representations of quantum
groups at a pth root of unity and of semisimple groups in characteristic
$p$.: independence of $p$. {\em Ast\'erisque,} (220):321, 1994.
%\bibitem[AGM]{AGM} Anderson, Griffeth, Miller, {\em Transersality..}

%\bibitem[AB]{AB:equivariantcohomology} M. Atiyah and R. Bott, Equivariant Cohomology and Moment Maps.

\bibitem[Bi]{Bi} S. Billy, Kostant polynomials and the cohomology ring for G/B. {\em Duke Math. J.}
Volume 96, Number 1 (1999), 205-224.

%\bibitem[Br] M. Brion, Positivity in the Grothendieck group of complex flag varieties, J. Algebra 258 (2002), 137-159.
 
%\bibitem[Br2]{BriLectures} M. Brion, Lectures on the Geometry of Flag Varieties. Topics in cohomological studies of algebraic varieties, 33--85, Trends Math., Birkhauser, Basel, 2005.

\bibitem[BL]{BL} Brion-Lakshmibai

%\bibitem[Dem]{Dem} M. Demazure, {\em D\'esingularisation des vari\'et'es de Schubert g\'en\'eralis\'ees,} Ann. Sci. \'Ecole Norm. Sup. (4)\bf{7} (1974), 53-88.
%\bibitem[De]{De} Deodhar

\bibitem[Gr]{Gr} W. Graham, Positivity in equivariant Schubert calculus. {\em Duke Math. J.}
Volume 109, Number 3 (2001), 599-614.


%\bibitem[GK]{GK} Grossberg-Karshon

%\bibitem[HS]{HS} Harada-Sjamaar

%\bibitem[Kn]{Knutson:dctriviality}

%\bibitem[KM]{KM} A. Knutson, E. Miller

\bibitem[Wi1]{Willems:EquivariantCohomology} M. Willems, Cohomologie \'equivariante des tours de Bott e calcul de Schubert \'equivariant. {\em  J. Inst. Math. Jussieu}  {\bf 5}  (2006),  no. 1, 125--159.

\bibitem[Wi2]{WIllems:EquK-theoryBSFlag} M. Willems, Cohomologie et $K$-th\'eorie 
\'equivariantes des vari\'et\'es de Bott-Samelson et des vari\'et\'es de drapeaux.

%\bibitem[Wi]{Willems:Chevalley} M. Willems, A Chevalley Formula in Equivariant $K$-theory



\end{thebibliography}
}

\end{document}

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: t
%%% End: 




\begin{Theorem}
$K^w \cdot (S_1 \tensor S^1 \tensor S^1) = \sum_{u,v} c_{uv}^w \tensor S^u \tensor S^v.$
\end{Theorem}
%&=& \sum_{u,v} c_{uv}^w \tensor S^u \tensor S^v \qquad 
%\text{this is our intended theorem} \\
\begin{proof}
{\color{blue} 

Analogously to our calculation 
$J_w\cdot (S_1 \tensor S^1) = (i_w \times Id)^*[(G/B)_\Delta]$ 
from \cref{sec:ajsb}, we have
\begin{eqnarray*}
K^w \cdot (S_1 \tensor S^1 \tensor S^1)
&= \sum_{u,v} (d_{uv}^w w\cdot 1) \tensor \partial_u S^1 \tensor \partial_v S^1 \\
&= \sum_{u,v} (d_{uv}^w w\cdot 1) \tensor S^u \tensor S^v \\
\end{eqnarray*}
It is sufficient to show that $d_{uv}^ww\cdot 1=c_{uv}^w$.
