\section{Introduction}
Flag varieties $G/B$ are among the most concrete varieties in  topology and algebraic geometry, and they have a cellular decomposition by Schubert cells, whose closure are called Schubert varieties. Consequently,  their equivariant singular cohomology are spanned by the Schubert classes. 

Associated to Schubert cells (or varieties), there are some others classes defined, for equivariant singular  cohomology $H^*_T$ and equivariant K-theory $K_T$, studied in \cite{AM15, AMSS17, RTV15, RV15, RTV17,  S15,  SZZ17}.  They are called Chern-Schwartz-MacPherson  (CSM) classes and Motivic Chern (mC) classes.  These classes coincide with the corresponding stable bases of Maulik-Okounkov for $H^*_T$ and $K_T$ of the Springer resolutions, respectively. Due to this fact, we always refer CSM classes (resp. mC classes) as cohomological stable basis (resp. K-theoretic stable basis). Moreover, they behave very similarly as the Schubert classes in corresponding theories. Roughly speaking, Schubert classes in $H^*_T$ and $K_T$ are permuted by Demazure operators (also called divided difference operators), and stable bases are permuted by the Demazure-Lusztig operators. All these operators generate various Hecke-type algebras. 

Structure constants of Schubert classes are  central objects in Schubert calculus and are closely related with representation theory and combinatorics.  In \cite{GK19}, the first author and Knutson obtain some formulas of the structure constants in $H^*_T$ and $K_T$. The authors used  geometric properties of  Bott-Samelson resolutions of Schubert varieties. They pull-back the Schubert classes to cohomology (or K-theory) of Bott-Samelson variety, apply cup product in this variety, then push-forward back to $G/B$. In \cite{S19}, Su generalized this method to the so-called Segre-Schwartz-MacPherson (SSM) classes, which are certain variant form of CSM classes (cohomolgical stable basis). 

We are interested in generalized cohomology theories, called the oriented cohomology theories, defined by Levine and Morel \cite{LM07}. They are contravariant functors defined on the category of smooth projective varieties over a field $k$ (usually of characteristic 0) to the category of commutative rings, such that for proper maps, there is a push-forward map on cohomology groups. Examples includes Chow rings (singular cohomology) and  K-theory. Chern classes are defined for each oriented cohomology theory $\mathbb{h}$, and there is an associated formal group law $F$ defined over $R=\mathbb{h}(\pt)$. 

For flag varieties, generalizing works of Kostant and Kumar \cite{KK86, KK90} on equivariant singular cohomology and equivariant K-theory of flag varieties, the ring $\mathbb{h}_T(G/B)$ has a nice algebraic model, constructed in \cite{HMSZ14} and studied in \cite{CZZ1, CZZ2, CZZ3} by Calm\`es, Zainoulline, and the second author. One can define the (formal) Demazure  operators $X_\al$ associated to each simple root $\al$. These operators generate a non-commutative algebra, called the formal affine Demazure algebra $\bfD_F$. 
 It is a free $\mathbb{h}_T(\pt)$-module with basis $X_{I_w}, w\in W$, where $X_{I_w}$ is (roughly speaking) a product of the operators $X_\al$,  with $I_w$ indicating a reduced word expression for $w$. 


The algebra $\bfD_F$ is also a co-commutative co-algebra, where the coproduct comes from the  twisted Leibniz rule of the operator $X_\al$. Taking the dual, one obtains a commutative ring $\bfD^*_F$, which is also a free $\mathbb{h}_T(\pt)$-module with the dual basis $\{X_{I_w}^*|w\in W\}$, and is isomorphic to $\mathbb{h}_T(G/B)$.  Indeed, for Chow group/singular cohomology/K-theory, $X_{I_w}^*$ coincides (up to various normalizations) to the Schubert classes.  


We notice that  the product structure on $\bfD_F^*$ actually comes from dualizing the coproduct structure of $\bfD_F$, which comes from the twisted Leibniz rule of $X_\al$. Therefore, the  structure constants of the basis $X_{I_w}^*$ actually comes from the twisted Leibniz rule of the product $X_{\be_1} X_{\be_2}\cdots X_{\be_k}$ for a reduced 
word $s_{\be_1}\cdots s_{\be_k}$
of $w\in W$. This is the main idea of the proof of  Theorem \ref{thm:Leibniz}, which implies the main result, Theorem \ref{thm:prod}. Specializing $\mathbb{h}$ to singular cohomology and K-theory, we recover the formulas of \cite{GK19}.

Moreover, as mentioned above, by replacing the Demazure operators by the Demazure-Lusztig operators, one obtains stable basis for $H^*_T$ and $K_T$. Both classes can be described similarly as above. That is,  the Demazure-Lusztig operators will generate the degenerate affine Hecke algebra (for $H^*_T$) and the affine Hecke algebra (for K-theory).  The dual of products of these operators are basically the cohomological/K-theoretic stable bases, so twisted Leibniz rules of them  will give the structure constants of stable bases. For instance, for cohomology, we recover the formula of Su \cite{S19} (see Remark \ref{rem:Su}). 


To work with the Demazure operators $X_\al$ and Demazure-Lusztig operators  $T_\al$ in the same time, we mainly work with a general element $Z_\al$ (see \S3) that can be specialized to $X_\al$ and $T_\al$. Our main results are Theorems \ref{thm:prod} and \ref{thm:cohstab}, which give the formula for structure constants of the basis determined by $Z_\al$ and apply it to the cohomological stable basis.

The paper is organized as follows: In \S2 we recall necessary notions from \cite{CZZ1, CZZ2, CZZ3}. We recall the definition of Demazure element, formal affine Demazure algebra, its dual, and relation with $\mathbb{h}_T(G/B)$. In \S3 we prove the twisted Leibniz rule for the element $Z_i$, which is used to derive the structure constants of the basis $Z_{I_w}^*$ in \S4. In \S5 we specialize our result to Demazure operators in singular cohomology and K-theory, and recover the formulas in \cite{GK19}. In \S6 we specialize our result to  Demazure-Lusztig operators in singular cohomology, which, as a by-product, recovers the formula in \cite{S19}. In \S7 we consider Demazure-Lusztig operators in K-theory and obtain a formula for K-theoretic stable basis. In \S8, for equivariant oriented cohomology, we generalize a result of Kostant-Kumar by relating our formula of structure constants with restriction formula of Schubert classes. 




\section{Preliminary} \label{sec:prelim}
We follow the notation in \cite{CZZ1,CZZ2, CZZ3}. Let $\Sigma\hookrightarrow \Lambda^\vee, \al\mapsto \al^\vee$ be a semi-simple root datum of rank $n$. That is, $\Sigma$ is the finite set of roots, and $\La$ is the lattice. Let $\{\al_1,...,\al_n\}$ be the set of simple roots, $\Sigma^+$ and $\Sigma^-$ be the set of positive and negative roots, respectively. Let $W$ be the Weyl group generated by the associated simple reflections $s_i:=s_{\al_i}$.  Note that $W$ acts on $\La$ since it preserves the root system. For each sequence $I=(i_1,...,i_k)$ with $i_j\in [n]$, denote  $\prod I=s_{i_1}\cdots s_{i_k}\in W$. If $\prod I$ is a reduced word expression for the resulting Weyl group element, we say that $I$ is a {\it reduced sequence}. Following  \cite[\S1]{GK19}, define the {\it Demazure product} $\wt \prod I=s_{i_1}\cdots s_{i_k}$ subject to the braid relation and $s_i^2=s_i, \forall i$. Note that if $I$ is a reduced sequence, then $\prod I=\wt \prod I$.  When $I$ is a reduced sequence for $w$, we may denote it by $I_w$ and abuse notation by calling it a reduced word for $w$. Finally, let $I^{\rev}$  denote the sequence obtained by reversing the sequence $I$. 

Let $F$ be a formal group law  over the coefficient ring $R$. Examples of formal group laws include the additive formal group law $F_a=x+y$ and the multiplicative formal group law $F_m=x+y-xy$. Suppose the root datum together with the formal group law satisfy the regularity condition of \cite[Lemma 2.7]{CZZ3}. This guarantees  that all the properties that we use from \cite{CZZ1, CZZ2, CZZ3}  hold. 

 Let $G$ be a split semi-simple  linear algebraic group with maximal torus $T$ and a Borel subgroup $B$. Let the associated root datum of $G$ be $\Sigma\hookrightarrow \La^\vee$, so $\La$ is the group of characters of $T$. 

Let $\mathbb{h}$ be an oriented cohomology theory of Levine and Morel. Roughly speaking, it is a contravariant functor from the category of smooth projective varieties to the category of commutative rings such that there is a push-forward map for any proper map. The Chern classes of vector bundles are defined. Associated to $\mathbb{h}$, there is a formal group law  is $F$ defined over $R=\mathbb{h}(\pt)$. That is, the first Chern class of line bundles over a smooth projective variety $X$ satisfies 
$$
c^\mathbb{h}_1(\calL_1\otimes \calL_2)=F(c_1^\mathbb{h}(\calL_1), c_1^\mathbb{h}(\calL_2)).
$$ For example, $F_a$ (resp. $F_m$) is associated to the Chow group (or singular cohomology) (resp. K-theory). 

Let $S$ be the formal group algebra defined in \cite{CPZ13}:
\[
S= R[[\La]]_F:=R[[x_\la|\la\in \La]]/J_F,
\]
where $J_F$ is the closure of the ideal generated by $x_0$ and $x_{\la+\mu}-F(x_\la, x_\mu),$ for all  $\la, \mu\in \La$. Indeed, if $\{t_1,...,t_n\}$ is a basis of $\La$, then $S$ is (non-canonically) isomorphic to $R[[x_{t_1},...,x_{t_n}]]$. According to \cite[\S3]{CZZ3}, $S\cong \mathbb{h}_T(\pt)$ with $x_\la$ corresponding to $c^\mathbb{h}_1(\calL_\la)$ where  $\calL_\la$ is the line bundle  associated to $\la\in \La$. Since $x_{-\la}$ is the formal inverse of $x_\la$, i.e.  $F(x_\la, x_{-\la})=0$ in $S$, we may write 
\[
x_{-\la}=-x_{\la}+\text{higher~degree~terms}\in S. 
\]
We will frequently need the special element of $S$ given by $\kappa_\la:=\frac{1}{x_\la}+\frac{1}{x_{-\la}}$. Note that the action of $W$ on $\La$ 
induces an action of $W$ on $S$. 

\begin{ex}\label{ex:FaFm} Two cases of the formal product appear widely in the literature. 
\begin{enumerate}
\item  If $F=F_a$ with $R=\bbZ$, then $\mathbb{h}$ is the singular cohomology/Chow groups, and  $S\cong \Sym_{\bbZ}(\La)^\wedge$  ( with $x_\la\mapsto \la$) is the completion of the polynomial ring at the augmentation ideal. In this case $x_{-\la}=-x_\la$ and $\kappa_\la=0$.
\item If $F=F_m$ with $R=\bbZ$, then $\mathbb{h}$ is K-theory, and $S\cong \bbZ[\La]^\wedge$ (with $x_\la\mapsto 1-e^{-\la}$) is the completion of the Laurent polynomial ring at the augmentation ideal. In this case $x_{-\la}=\frac{x_\la}{x_\la-1}$, and $\kappa_\la=1$. 
\end{enumerate}
\end{ex}


\subsection{The operator algebras $Q_W$ and $\bfD_F$} 
This paper is concerned with various divided difference operators acting on the cohomology of $G/B$. To create an algebraic framework for these operators, following \cite{CZZ1, CZZ2} we localize $S$ at $\{x_\alpha\}$ to create an algebra out of this localization and the Weyl group, as follows.
Define $Q:=S[\frac{1}{x_\al}|\al\in \Sigma]$, and 
$$Q_W:=Q\rtimes R[W],$$
as a left $Q$-module with basis $\{\de_w\}, w\in W$. 
We impose a product on $Q_W$ by
\[
{(}p\de_w {)}{(}p'\de_{w'}{)}=pw(p')\de_{ww'}, \quad \mbox{for all }p,p'\in Q,\mbox{ and } w,w'\in W,
\]
using the natural $W$ action on $Q$ induced from that on $\La$ and extending linearly. Note that $Q$ identified with $Q\delta_e$ is a subring of $Q_W$ under this product, where $e\in W$ denotes the identity element of $W$.
We routinely abuse notation and write $\de_\al$ for $\de_{s_\al}$, and use  $1=\de_e$ to denote the identity element of $Q_W$.
The ring $Q_W$ acts on $Q$ by 
\[p\de_w\cdot p'=pw(p'),\quad \mbox{for all } p, p'\in Q.\]

The action of $Q_W$ on $Q$ induces a coproduct structure on $Q_W$ as follows. Let $\eta=\sum\limits_{w\in W} q_w\de_w\in Q_W$. Then 
$$
\eta\cdot (pq) = \sum_w q_w w(pq) = \sum_w q_w w(p) w(q) = \sum_w q_w(\de_w\cdot p)(\de_w\cdot q).
$$
This action factors through the coproduct $\Delta: Q_W \rightarrow Q_W\otimes_Q Q_W$
\begin{equation}\label{eq:coprod}
\Delta(\eta) = \sum_w q_w\Delta(\de_w) = \sum_w q_w \de_w \otimes \de_w.
\end{equation}
In other words, the coproduct structure on $Q_W$ is actually induced from the $Q_W$-action on $Q$. 


For any simple root $\al$
we define the Demazure element $X_\al$ 
and the push-pull element $Y_\al$ in $Q_W$:
\begin{align*}
X_{\al}&:=\frac{1}{x_{\al}}(1-\de_{\al}), \mbox{ and}\\
Y_{\al}&:=\frac{1}{x_{-\al}}+\frac{1}{x_{\al}}\de_{\al}.
\end{align*}
We observe the relationship $Y_{\al}=\ka_{\al}-X_{\al}$. In particular, 
 if $F=F_a$ (resp. $F=F_m$), then $Y_\al=-X_\al$ (resp. $Y_\al=1-X_\al$). 

The way $Q_W$ acts on $Q$ implies that $X_\al$ acts similarly as the Demazure operator defined in \cite{Dem74} (and there denoted $D_\al$). In particular, $X_\al\cdot S \subset S$ and, for any $r\in R, X_\al\cdot r=0$ and $\de_\al\cdot r = s_\al(r)=0$.


Let $\bfD_F$ be the $R$-subalgebra of $Q_W$ 
$$
\bfD_F=\langle S, X_{\al_1},\dots, X_{\al_n}\rangle
$$
generated by $S$ and the elements $X_\al \in Q_W$ for simple roots $\al$, and call it the {\it formal affine Demazure algebra.} It is also generated by $S$ and $\{Y_\al:  \al ~\mbox{simple}\}$. 



Let  $w =s_{i_1}\cdots s_{i_k}$ be a reduced word decomposition and $I_w = (i_1,...,i_k)$ the corresponding sequence of reflections. Define
\begin{align*}
X_{I_w} &= X_{\al_{i_1}}\cdots X_{\al_{i_k}}\\
Y_{I_w} &= Y_{\al_{i_1}}\cdots Y_{\al_{i_k}}.
\end{align*}
In particular, $X_{(i)}=X_{\al_i}$ (and $Y_{(i)}=Y_{\al_i}$). We write $X_e:=1\in Q_W$ to indicate $X_I$ when $I$ is the empty sequence.

The Demazure and push-pull elements have following properties:
\begin{lem}\cite[Proposition 3.2]{Z15}\label{lem:1} Let $\al$ and $\be$ be simple roots. 
The following identities hold in $Q_W$:
\begin{enumerate}
\item \label{item:1}$X_\al^2=\ka_\al X_\al, \quad Y_\al^2=\ka_\al Y_\al$.
\item \label{item:2}$X_\al p=s_\al(p)X_\al+X_\al\cdot p, \quad p\in Q$. 
\item \label{item:3}If $(s_\al s_\be)^2=e$, then $X_{\al}X_\be=X_\be X_\al$. 
\item \label{item:4} If $(s_\al s_\be)^3=e$, then $X_\be X_\al X_\be-X_\al X_\be X_\al =\ka_{\al \be }X_\al -\ka_{\be \al }X_\be $, where 
\[
\ka_{\al \be }=\frac{1}{x_{\al +\be }x_\be }-\frac{1}{x_{\al +\be }x_{-\al }}-\frac{1}{x_\al x_\be }. 
\]
It is proved in \cite[Lemma 6.7]{HMSZ14} that $\ka_{\al \be }\in S$.
\item \label{item:5} Suppose $s_\al s_\be $ has order $m$ with $m=4$ or $6$,  and $I_w$ is a choice of reduced word for $w\in W$.  Then \[{\underbrace{X_\al X_\be X_\al \cdots}_m }-{\underbrace{X_\be X_\al X_\be \cdots}_m}=\sum_{v\in W}c_{I_v}X_{I_v},\] where $c_{I_v}=0$ if $v\not\leq\underbrace{s_\al s_\be s_\al \cdots}_m$. Moreover, $c_{I_v}=0$ if $\ell(v)=m-1$ or $v=e$. 
\end{enumerate}
\end{lem}
Lemma \ref{lem:1}.\eqref{item:4}-\eqref{item:5} imply that
 the operators $X_\al$ (and similarly $Y_\al$) do not satisfy braid relations for general $F$. For $F=F_a$ or $F=F_m$, they do; in these cases, the $\ka_{\al \be}$ and $c_{I_v}$ coefficients all vanish. In general, due to the failure of the braid relations, $X_{I_w}$ and $Y_{I_w}$ depend on the choice of $I_w$. 

For the purposes of this paper, we fix a reduced sequence $I_w$ of $w$ for each $w\in W$. While the specific coefficients and calculations regarding $X_{I_w}$ and $Y_{I_w}$  depend on this choice, statements regarding bases and ring phenomena do not.

Let $\bfD_F$ be the $R$-subalgebra of $Q_W$ generated by $S$ and $X_i, i=1,...,n$, and call it the formal affine Demazure algebra. It is also generated by $S$ and $Y_i, i=1,...,n$. 
From \cite[Proposition 7.7]{CZZ1}, we know the algebra $\bfD_F$ is a free left $S$-module with basis $\{X_{I_w}\}_{w\in W}$, and similarly it is a free left $S$-module with basis $\{Y_{I_w}\}_{w\in W}$.


By construction, $\{\de_v:\ v\in W\}$ form a basis of $Q_W$ as a module over $Q$. In \cite{CZZ1}, and extended in \cite{CZZ2}, the second author proves that $\{X_{I_v}: \ v\in W\}$  and  $\{Y_{I_v}: \ v\in W\}$  also form bases of $Q_W$ as a module over $Q$, and that the change of basis matrix from $\{X_{I_v}\}$ (or from $\{Y_{I_v}\}$) to $\{\de_v\}$ consists of elements of $S$.  In particular, $\{\delta_v\}$ are elements of $\bfD_F$. The lower-triangularity of the change of bases matrices is expressed in the following lemma.


\begin{lem}\cite[Lemma 3.2, Lemma 3.3]{CZZ2} 
For each $v\in W$, choose a reduced decomposition of $v$ and let $I_v$ be its corresponding sequence. There exist elements $a^X_{I_w,v}\in Q$ for $v\in W$,  and $b^X_{w,I_v}\in S$ 
 such that
$$
X_{I_w}=\sum_{v\le w}a^X_{I_w,v}\ \de_v, \mbox{ and}\quad \de_{w}=\sum_{v\le w}b^X_{w,I_v}X_{I_v}.
$$
Similarly, there exist $a^Y_{I_w,v}\in Q$ and $b^Y_{w,I_v}\in S$ such that
$$
Y_{I_w}=\sum_{v\le w}a^Y_{I_w,v}\ \de_v, \mbox{ and}\quad \de_{w}=\sum_{v\le w}b^Y_{w,I_v}Y_{I_v}.
$$
\end{lem}



\begin{ex}\label{ex:changebasis}
Consider the root datum $A_2$, with $W=\{e, s_1, s_2, s_1s_2, s_2s_1, w_0\}$, where $w_0$ is the longest element and $s_i$ is the reflection corresponding to $\alpha_i$ for $i=1,2$. We fix the reduced sequence $I_{w_0} = (1,2,1)$ so that $w_0=s_1s_2s_1$. 
For simplicity,  let $\al_{13}=\al_1+\al_2$. By a direct computation,
\begin{align*}
&\de_{e\phantom{1}} =X_e &\qquad &\de_{s_1s_2}=1-x_1X_{(1)}-x_{\al_{13}}X_{(2)}+x_{\al_1}x_{\al_{13}}X_{(1,2)},\\
&\de_{1} =1-x_{\al_1}X_{(1)},&\qquad &\de_{s_2s_1}=1-x_{\al_2}X_{(2)}-x_{\al_{13}}X_{(1)}+x_2x_{\al_{13}}X_{(2,1)},\\ 
&\de_{2}  =1-x_{\al_2}X_{(2)} ,&\qquad  &\de_{w_0\phantom{w}}=1-x_{\al_{13}}X_{(2)}-(x_{\al_1}+x_{\al_2}-\kappa_{\al_1} x_{\al_1}x_{\al_2})X_{(1)} \\
\phantom{\de_2}, &\phantom{=1-x_2X_2 }&\quad &\phantom{\de_{121}}+x_{\al_1}x_{\al_{13}}X_{(1,2)}+x_{\al_2}x_{\al_{13}}X_{(2,1)}-x_{\al_1}x_{\al_2}x_{\al_{13}}X_{I_{w_0}}.
\end{align*}
Notice that nonzero coefficients $b^X_{w, I_v}$ are elements of $S$ with $v\leq w$. 
\end{ex}



\subsection{The dual operator algebras}

The dual $Q$-module
$$
Q_W^*=\Hom_Q(Q_W, Q)\cong \Hom(W, Q),
$$ 
contains a natural basis $\{f_w\}_{w\in W}$ dual to $\{\delta_w\}_{w\in W}$, defined by 
$$
\langle f_w, \de_v\rangle=\begin{cases}
					1,&\mbox{if $w=v$};\\
					0,& \mbox{else}.
					\end{cases}
$$
The module $Q_W^*$ forms a ring with product $f_w f_v = 1$ if an only if $w=v$, and $0$ otherwise, extended linearly to all elements of $Q_W^*$, 
and unity $\mathbf{1}=\sum_{w\in W}f_w$.  This product structure is equivalent to the one induced from the coproduct structure (see \S\ref{sec:prod} below).

The ring $Q_W$ acts on $Q_W^*$ by 
\[
\langle z\bullet f, z'\rangle =\langle f, z' z\rangle , \quad \mbox{for all }z, z'\in Q_W,\  f\in Q_W^*.
\]
In the bases $\{\de_w\}$ of $Q_W$ and $\{f_w\}$ of $Q_W^*$, the action has explicit formulation
\begin{equation}p\de_w\bullet (qf_v)=qvw^{-1}(p)f_{vw^{-1}}, \mbox{ for all }p,q\in Q. 
\end{equation}
Denote 
$$
\quad \pt_w=(\prod_{\al<0}x_\al)\bullet f_w=w\left(\prod_{\al<0}x_\al\right)f_w\in Q_W^*. 
$$
 
Let  $\bfD_F^*:=\Hom_S(\bfD_F, S)\subset Q_W^*$ be the dual $S$-module to $\bfD_F$.  It is proved in \cite[Lemma 10.3]{CZZ2} that $\pt_w\in \bfD_F^*$. 
Let
\begin{align*}
\zeta_{I_w}^X &=X_{I_w^{\rev}}\bullet \pt_e\\
\zeta_{I_w}^Y& =Y_{I_w^{\rev}}\bullet \pt_e.
\end{align*} 
Then $\{\zeta_{I_w}^X\}$ forms a basis of $D_F^*$ over $S$, as does $\{\zeta_{I_w}^Y\}$.

Finally, Let $\{X_{I_w}^*\},$ (respectively $\{Y_{I_w}^*\}$) be the bases dual to $\{X_{I_w}\}$ (resp. $\{Y_{I_w}\}$) in $\bfD^*_F$, which are also $Q$-basis of $Q_W^*$. 


\begin{ex}\label{ex:dualbasischange}
Consider the root datum $A_2$, with $W=\{e, s_1, s_2, s_1s_2, s_2s_1, w_0\}$. We fix the reduced sequence $w_0=s_1s_2s_1$. The classes $X_{I_v}^*$ for each $v\in W$ are determined by duality.  Under the dual pairing, 
\begin{align*}
\langle X_{I_v}^*, \de_w \rangle &= \langle X_{I_v}^*,\sum_{u\in W} b_{w,I_u}^X X_{I_u} \rangle 
= b_{w, I_v}^X.
\end{align*}
Set
$
X_{I_v}^* = \sum\limits_{u\in W} m_{I_v,u} f_u,
$
which implies
\begin{align*}
\langle X_{I_v}^*, \de_w \rangle &=\langle \sum\limits_{u\in W} m_{I_v,u} f_u, \de_w\rangle = m_{I_v,w}, 
\end{align*}
and thus $X_{I_v}^*=   \sum\limits_{w\in W} b_{w, I_v}^X f_w$. The calculations from Example~\ref{ex:changebasis} imply
\begin{align*}
&X_e^*=\mathbf{1}=\sum_{w\in W}f_w ,&\quad &X_{(1,2)}^*=x_{\al_1}x_{\al_{13}}(f_{s_1s_2}+f_{w_0}) ,\\
&X_{(1)}^*=-x_{\al_1}(f_{s_1}+f_{s_1s_2})-x_{\al_{13}}f_{s_2s_1}-yf_{w_0} ,&\quad &X_{(2,1)}^*=x_{\al_2}x_{\al_{13}}(f_{s_2s_1}+f_{w_0}),\\
&X_{(2)}^*=-x_{\al_2}(f_{s_2}+f_{s_2s_1})-x_{\al_{13}}(f_{s_1s_2}+f_{w_0}),&\quad &X^*_{I_{w_0}}=-x_{\al_1}x_{\al_2}x_{{\al_{13}}}f_{w_0}.\\
\end{align*}
where $y=x_{\al_1}+x_{\al_2}-\kappa_{\al_1} x_{\al_1}x_{\al_2}$. In case $F=F_a$ or $F_m$, we have $y=x_{\al_{13}}$.
\end{ex}
The following proposition explains the relationship between the algebraic construction above and equivariant oriented cohomology of $G/B$. 


For each reduced sequence $I_w$, let $\calX_{I_w}\to G/B$ denote the Bott-Samelson resolution. The push-forward in $\mathbb{h}_T$ of the fundamental class along this resolution is called the {\it Bott-Samelson class of $I_w$}, which we denote by $\eta_{I_w}$.
Define a map
$$
\Phi: \bfD^*_{F}\quad \longrightarrow \quad \mathbb{h}_T(G/B)
$$
given by $\Phi(\zeta_{I_w}^Y) = \eta_{I_w}$ and $\Phi({\mathbf{1}}) = [G/B]$, the fundamental class of $G/B$, and extended as a module over $S$.




\begin{prop}\label{prop:Schubert} 
\begin{enumerate} The isomorphism $\Phi$ satisfies the following properties:
\item \cite[Theorem 8.2, Lemma 8.8]{CZZ3} The map $\Phi$ is a canonical isomorphism.
\item \cite[Theorem 14.7]{CZZ2} 
 The basis $\{\Phi(X_{I_w}^*):\ w\in W\}$ (resp. $\{\Phi(Y_{I_w}^*)):\ w\in W\}$) is dual to $\zeta^X_{I_w}$ (resp. $\zeta^Y_{I_w}$) via the nondegenerate Poincar\'e dual pairing on $\mathbb{h}_T(G/B)$ given by multiplying and pushing forward to a point.
 
\item \cite[Corollary 6.4]{CZZ3}
Let $i_w: wB \hookrightarrow G/B$ be the inclusion of the $T$-fixed point corresponding to $w\in W$, and $(i_w)_*: \mathbb{h}_T(wB)\rightarrow \mathbb{h}_T(G/B)$ the pushforward map. Then  $\Phi(\pt_w) = (i_w)_*(1)$.

\item There is a commutative diagram
\[
\xymatrix{\bfD_F^*\ar@{^(->}[rr]\ar[d]^{ \cong} &  & Q_W^*\ar[d]^{ \cong}\\
        \mathbb{h}_T(G/B)\ar[rr]^{\bigoplus\limits_{w\in W} i_w^*} &  & Q\otimes_{S}\mathbb{h}_T(W),}
\]
where the top horizontal map is the embedding of the $S$-module into the $Q$-module $Q_W^*$. 
\end{enumerate}
\end{prop}
\begin{ex}\label{ex:Schubert} Let $X(w)=\overline{BwB/B}$ be the Schubert variety and $Y(w)=\overline{B^-wB/B}$ be the opposite Schubert variety. If $F=F_a$ or $F=F_m$, we write $w$ for $I_w$ since $X_{I_w}$ and $Y_{I_w}$ are independent of the reduced sequence.
\begin{enumerate}
\item If $F=F_a$,
then $\zeta^Y_w=[X(w)]$, and $\zeta^X_w=(-1)^{\ell(w)}[X(w)]$, where each homology classes is identified with its  Poincar\'e dual cohomology class.
Then $Y_{w}^* = [Y(w)]$ and similarly $X_{w}^* = (-1)^{\ell(w)}[Y(w)]$.
\item If $F=F_m$, then $\zeta^Y_w=[\calO_{X(w)}]$ is the class of the structure sheaf of $X(w)$, $Y_w^*=[\calO_{Y(w)}(-\partial Y(w))]$, $\zeta_w^X=(-1)^{\ell(w)}[\calO_{X(w)}(-\partial X(w))]$, and $X^*_w=(-1)^{\ell(w)}[\calO_{Y(w)}]$. 
\end{enumerate}
\end{ex}







\section{Generalized Demazure operators and the generalized  Leibniz rule}

In this section, we generalize the operators $X_{I_v}$ and $Y_{I_v}$ to a more general class of elements of $Q_W$, and prove the generalized  Leibniz rule for $\bfD_F$ acting on $Q$. We use this result to compute the coproduct structure in $Q_W$, and then the product structure in $Q_W^*$.

Let $\{a_\al, b_\al \in Q:\ \al\in \Sigma\}$ be a set of elements with the property that, for all $w\in W$, 
$$
w(a_\al) = a_{w(\al)}, \quad\quad w(b_\al) = b_{w(\al)}, \text{ and }b_\al\text{ are all invertible in }Q. 
$$
For any simple root $\al$, define operators $Z_\al\in Q_W$ by
$$
Z_\al=a_\al+b_\al\de_\al.
$$
Clearly $X_\al$ and $Y_\al$ result from special choices of $a_\al$ and $b_\al$ in $Z_\al$. For any sequence $I=(i_1,...,i_k)$ of simple roots, 
define $Z_I\in Q_W$ by
$$
Z_I=Z_{\al_{i_1}}Z_{\al_{i_2}}\cdots Z_{\al_{i_k}}.
$$
We call $Z_I$ {\it generalized Demazure operators.}

As before, we choose a reduced word expression $I_v$ for each $v\in W$.

\begin{lem} The set of generalized Demazure operators $\{Z_{I_v}\}$ forms a basis of $Q_W$ as a module over $Q$.
\end{lem}
\begin{proof}
This  follows from the fact that $b_\al\in Q$ is invertible for all simple roots $\al$ (hence, for all roots $\al$).
\end{proof}



\begin{lem}\label{lem:Zcoeff}
For any sequence $J$, define coefficients $c_{J, I_w}\in Q$ by
\begin{equation}\label{eq:Zcoeff}
Z_J=\sum_{w\in W}c_{J, I_w}Z_{I_w},
\end{equation}
Then $c_{J,I_w}=0$ unless $w\le \wt \prod J$. 
\end{lem}

\begin{proof} 
Clearly $Z_\al = a_\al+b_\al\delta_i$ has support on $\{w:\ w\leq s_\al\}$.  An immediate observation of the product in $Q_W$ shows inductively that $Z_J$ may be expressed as a $Q$-linear combination of $\de_v$ for $v\leq \wt\prod J$.  

For any $v\in W$ and reduced sequence $I_v=(i_1,\dots, I_k)$, let $\ga_j = \al_{i_j}$ for $j=1,\dots, k$. The coefficient of $\de_v$ in $Z_{I_v}$ is 
$$
b_{\ga_1} s_{\ga_1}(b_{\ga_2}) s_{\ga_1}s_{\ga_2}(b_{\ga_3})\dots s_{\ga_1}\dots s_{\ga_{k-1}} (b_{\ga_k}).
$$
 In particular, since $b_{\ga_j}$ is invertible, so is $w(b_{\ga_j})$ for any Weyl group element $w$, and thus the coefficient of $\de_v$ in $Z_{I_v}$ is nonzero.

Let $A = \{w\in W:  c_{J,I_w}\neq 0 \mbox{ and } w\not \leq \wt \prod J\}$, and assume $A$ is nonempty. Pick $v\in A$ to be a maximal element of $A$ in the Bruhat order. By support considerations, the only terms contributing to the coefficient of $\de_v$ in \eqref{eq:Zcoeff} is $c_{J,I_v}Z_{I_v}$. Since the coefficient of $\de_v$ in $Z_{I_v}$ is a unit, we conclude $c_{J,I_v}=0$, contrary to assumption.
\end{proof}


We have the following lemma describing the action of $Z_\al$ on a product.
\begin{lem}\label{le:Zi1} For a simple root $\al$, and $p,q\in Q$, we have 
\begin{equation*
Z_\al\cdot (pq)=\frac{a_\al(a_\al+b_\al)}{b_\al}pq-\frac{a_\al}{b_\al}\left[\left(Z_\al\cdot p\right)q+p(Z_\al\cdot q)\right]+\frac{1}{b_\al}(Z_\al\cdot p)(Z_\al\cdot q). 
\end{equation*}
\end{lem}
\begin{proof}One just has to plug in $Z_\al=a_\al+b_\al\de_{s_\al}$ and use the definition of the action $\de_{s_\al}\cdot p=s_\al(p)$. A comparison of both sides yields the identity.
\end{proof}



The coefficients occurring in Lemma~\ref{le:Zi1} may be generalized to the case of the action of $Z_I$ on a product $pq$. 
\begin{defn}\label{def:LeibnizCoeff} For each simple root $\al$, let $Z_\al = a_\al + b_\al \de_\al$ with $a_\al, b_\al\in Q$ and $b_\al$  invertible. 
Let $I = (i_1,\dots, i_k)$ be a sequence of indices of simple roots, with $\ga_j := \al_{i_j}$ corresponding to the $j$th entry of $I$. 
For $E, F\subset \{1,\dots k\}$,
define the {\bf Leibniz coefficients} $\bfz_{E,F}^I\in Q$ by
\begin{equation}\label{eq:Leibniz}
\bfz^I_{E,F}=(B^Z_1 B^Z_2\cdots  B^Z_k)\cdot 1,
\end{equation}
 where the operators $B^Z_j\in Q_W$ are given by 
\begin{equation}\label{eq:BZ}
B_j^Z=
\begin{cases}
\frac{1}{b_{\ga_j}}\de_{\ga_j}, &\text{ if }j\in E\cap F,\\
\vspace{-.1in}\\
-\frac{a_{\ga_j}}{b_{\ga_j}}\de_{\ga_j}, &\text{ if }j\in  E \text{ or }F, \text{ but not both},\\
\vspace{-.1in}\\
a_{\ga_j}+\frac{a_{\ga_j}^2}{b_{\ga_j}}\de_{\ga_j}, &\text{ if }j\not\in E\cup F.\\
\end{cases}
\end{equation}
\end{defn}

\begin{ex}\label{ex:LeibnizCoeffXY} Let $\ga_j = \al_{i_j}$ indicate the $j$th root listed in the sequence $I$.
If $Z=X$, then using the specific choice of coefficients for the Demazure operator yields
$$
B_j^X=
\begin{cases}
-x_{\ga_j}\de_{\ga_j}, &\text{ if }j\in E\cap F,\\
\vspace{-.1in}\\
\de_{\ga_j}, &\text{ if }j\in  E \text{ or }F, \text{ but not both},\\
\vspace{-.1in}\\
X_{\ga_j}, &\text{ if }j\not\in E\cup F.\\
\end{cases}
$$
and similarly letting $Z=Y$ be the push-pull operators,
$$
B_j^Y=
\begin{cases}
x_{\ga_j}\de_{\ga_j}, &\text{ if }j\in E\cap F,\\
\vspace{-.1in}\\
\frac{x_{\ga_j}}{x_{-\ga_j}}\de_{\ga_j}, &\text{ if }j\in  E \text{ or }F, \text{ but not both},\\
\vspace{-.1in}\\
\frac{1}{x_{-\ga_j}} +\frac{x_{\ga_j}}{(x_{-\ga_j})^2}\de_{\ga_j} , &\text{ if }j\not\in E\cup F.\\
\end{cases}
$$
\end{ex}

Now we prove the main technical result of this paper, generalizing \cite[Lemma 4.8]{CZZ1}. 

\begin{thm}[Generalized Leibniz Rule]\label{thm:Leibniz} 
Let $Z_I$ be a generalized Demazure operator for $I=(i_1,...,i_k)$, and let $\ga_j = \al_{i_j}$ denote the $j$th simple root in the list. Then for any $p,q\in Q$, 
\[
Z_I\cdot (pq)=\sum_{E,F\subset [k]}\bfz^I_{E,F}(Z_{E}\cdot p)( Z_{F}\cdot q), 
\]
where $\bfz^I_{E,F}$ are the Leibniz coefficients defined in \eqref{eq:Leibniz}
\end{thm}
\begin{proof}
For any simple root $\al$, observe the following two identities:
\begin{align}
&a_{\al}(1-\de_{\al})+\frac{a_{\al}(a_{\al}+b_{\al})}{b_{\al}}\de_{\al}=a_{\al}+\frac{a_{\al}^2}{b_{\al}}\de_{\al}=\frac{a_{\al}}{b_{\al}}Z_\al\de_\al ,\label{eq:case3}
\\
&Z_\al\cdot (pq)=a_\al(p-s_\al(p))q+s_\al(p)(Z_\al\cdot q). \label{eq:Zi2}
\end{align}
We prove the theorem by induction on $k$. If $k=1$, the theorem holds by Lemma~\ref{le:Zi1}.


\junk{Now assume it holds for all $I$ with $\ell(I)<k$, and let $I=(i_1,...,i_{k})$. Denote $J=(i_2,...,i_{k})$. We have
\begin{align*}
&Z_I\cdot(pq)\\
&=(Z_{i_1} Z_J)\cdot (pq)\\
&=Z_{i_1}\cdot [\sum_{E_1,E_2\subset [k-1]}\bfz^{J}_{E_1,E_2}(Z_{J|_{E_1}}\cdot p) (Z_{J|_{E_2}}\cdot q)]\\
&\overset{\eqref{eq:Zi2}}=\sum_{E_1,E_2\subset [k-1]}a_{i_1}[\bfz^{J}_{E_1,E_2}-s_{i_1}(\bfz^{J}_{E_1,E_2})](Z_{J|_{E_1}}\cdot p)(Z_{J|_{E_2}}\cdot q)\\
&+\sum_{E_1,E_2\subset [k-1]}s_{i_1}(\bfz^{J}_{E_1,E_2})Z_{i_1}\cdot [(Z_{J|_{E_1}}\cdot p)(Z_{J|_{E_2}}\cdot q)]\\
&\overset{Lemma \ref{le:Zi1}}=\sum_{E_1,E_2\subset [k-1]}a_{i_1}[\bfz^{J}_{E_1,E_2}-s_i(\bfz^{J}_{E_1,E_2})](Z_{J|_{E_1}}\cdot p)(Z_{J|_{E_2}}\cdot q)\\
&+\sum_{E_1,E_2\subset [k-1]}s_{i_1}(\bfz^{J}_{E_1,E_2})\frac{a_{i_1}(a_{i_1}+b_{i_1})}{b_{i_1}}(Z_{J|_{E_1}}\cdot p)(Z_{J|_{E_2}}\cdot q)\\
&-\sum_{E_1,E_2\subset [k-1]}s_{i_1}(\bfz^{J}_{E_1,E_2})\frac{a_{i_1}}{b_{i_1}}\left[[Z_i\cdot (Z_{J|_{E_1}}\cdot p)](Z_{J|_{E_2}}\cdot q)+(Z_{J|_{E_1}}\cdot p)[Z_{i_1}\cdot (Z_{J|_{E_2}}\cdot q)]\right]\\
&+\sum_{E_1,E_2\subset [k-1]}s_{i_1}(\bfz^{J}_{E_1,E_2})\frac{1}{b_{i_1}}(Z_{i_1}\cdot Z_{J|_{E_1}}\cdot p)(Z_{i_1}\cdot Z_{J|_{E_2}}\cdot q)\\
&\overset{\eqref{eq:case3}}=\sum_{E_1,E_2\subset [k-1]}[\frac{a_{i_1}}{b_{i_1}}Z_{i_1}\de_{i_1}\cdot \bfz^{J}_{E_1,E_2}](Z_{J|_{E_1}}\cdot p)(Z_{J|_{E_2}}\cdot q)\\
&-\sum_{E_1,E_2\subset [k-1]}(\frac{a_{i_1}}{b_{i_1}}\de_{i_1}\cdot \bfz^{J}_{E_1,E_2})\left[[Z_{i_1}\cdot (Z_{J|_{E_1}}\cdot p)](Z_{J|_{E_2}}\cdot q)+(Z_{J|_{E_1}}\cdot p)[Z_{i_1}\cdot (Z_{J|_{E_2}}\cdot q)]\right]\\
&+\sum_{\tiny E_1,E_2\subset [k-1]}(\frac{1}{b_{i_1}}\de_{i_1}\cdot \bfz^{J}_{E_1,E_2})(Z_{i_1}\cdot Z_{J|_{E_1}}\cdot p)(Z_{i_1}\cdot Z_{J|_{E_2}}\cdot q).
\end{align*}
Comparing the coefficients with $B^Z_{1}(\bfz^{J}_{E_1,E_2})$ from  \eqref{eq:BZ}, we see that they coincide. The proof then follows by induction.
}

Now assume it holds for all $I$ with $\ell(I)<k$, and let $I=(i_1,...,i_{k})$. Let $J=(i_2,...,i_{k})$ and let $\al=\al_{i_1}$.
We have
\begin{align*}
Z_I\cdot(pq)&=(Z_\al Z_J)\cdot (pq) = Z_\al\cdot ( Z_J\cdot (pq))\\
&=Z_\al\cdot \left[\sum_{E,F\subset \{2,\dots, k\}}\bfz^{J}_{E,F}(Z_{E}\cdot p) (Z_{F}\cdot q)\right]\\
&=\sum_{E,F\subset  \{2,\dots, k\}}a_\al\left[\bfz^{J}_{E,F}-s(\bfz^{J}_{E,F})\right](Z_{E}\cdot p)(Z_{F}\cdot q)\\
&\phantom{"="}+\sum_{E,F\subset  \{2,\dots, k\}}s_\al(\bfz^{J}_{E,F})Z\cdot \left[(Z_{E}\cdot p)(Z_{F}\cdot q)\right] \mbox{ by Equation \eqref{eq:Zi2}}\\
&=\sum_{E,F\subset  \{2,\dots, k\}}a_\al\left[\bfz^{J}_{E,F}-s_\al(\bfz^{J}_{E,F})\right](Z_{E}\cdot p)(Z_{F}\cdot q)\\
&\phantom{"="}+\sum_{E,F\subset  \{2,\dots, k\}}s_\al(\bfz^{J}_{E,F})\frac{a_\al(a_\al+b_\al)}{b_\al}(Z_{E}\cdot p)(Z_{F}\cdot q)\\
&\phantom{"="}-\sum_{E,F\subset  \{2,\dots, k\}}s_\al(\bfz^{J}_{E,F})\frac{a_\al}{b_\al}\left[(Z_\al Z_{E}\cdot p)](Z_{F}\cdot q)+(Z_{E}\cdot p)(Z_\al Z_{F}\cdot q)\right]\\
&\phantom{"="}+\sum_{E,F\subset  \{2,\dots, k\}}s_\al(\bfz^{J}_{E,F})\frac{1}{b_\al}(Z_\al Z_{E}\cdot p)(Z_\al Z_{F}\cdot q) \mbox{ by Lemma \ref{le:Zi1}} \\
&=\sum_{E,F\subset  \{2,\dots, k\}}\left[
\left(a_\al+ \frac{a_\al^2}{b_\al}\de\right)\cdot
\bfz^{J}_{E,F}
\right](Z_{E}\cdot p)(Z_{F}\cdot q)\\
&\phantom{"="}-\sum_{E,F\subset  \{2,\dots, k\}}\left(\frac{a_\al}{b_\al}\de\cdot \bfz^{J}_{E,F}\right)\left[(Z_\al Z_{E}\cdot p)(Z_{F}\cdot q)+(Z_{E}\cdot p)(Z_\al Z_{F}\cdot q)\right]\\
&\phantom{"="}+\sum_{\tiny E,F\subset  \{2,\dots, k\}}\left(\frac{1}{b_\al}\de\cdot \bfz^{J}_{E,F}\right)(Z_\al Z_{E}\cdot p)(Z_\al Z_{F}\cdot q) \mbox{ by Equation \eqref{eq:case3}.}
\end{align*}
Comparing the coefficients with $B^Z_{1}\cdot (\bfz^{J}_{E,F})$ from  \eqref{eq:BZ}, we see that they coincide. The proof then follows by induction.
\end{proof}


The following corollary follows immediately. We see in Section~\ref{se:restriction} that the Leibniz coefficients $\bfz^{I}_{[k],E}$ arise as factors in summands of specific structure constants in Schubert calculus, justifying the name.
\begin{cor}\label{cor:cons}[Generalized Billey's Formula]
Let $I=(i_1,...,i_k)$ be a sequence of indices of simple roots, and denote 
$m_j = s_{i_1}s_{i_2}\cdots s_{i_{j-1}}(a_{i_j})$ and $n_j = s_{i_1}s_{i_2}\cdots s_{i_{j-1}}(b_{i_j})$. For  $E\subset[k]$, we have
$$
\bfz^{I}_{[k],E}=\bfz^{I}_{E, [k]}=(-1)^{k-|E|} \prod\limits_{j\in [k]\setminus E} m_j \prod\limits_{j\in [k]} n_j^{-1}.
$$
\end{cor}

As a consequence of Theorem~\ref{thm:Leibniz} and the coproduct defined in Equation~\ref{eq:coprod}, we obtain the following theorem.
\begin{thm}\label{thm:coprod}
Let $Z_\al=a_\al+b_\al\de_\al\in Q_W$ with $b_\al$ invertible, then for any $I=(i_1,...,i_k)$, we have
$$
\tr(Z_I)=\sum_{E,F\subset[k]}\bfz^I_{E, F}Z_{E}\otimes Z_{F},
$$
where $\bfz^I_{E,F}$ are defined in Definition \ref{def:LeibnizCoeff}. 
\end{thm}


We  specialize Theorem~\ref{thm:Leibniz} to the elements $X_I$ and $Y_I$. For any index $j$, the operators $B^X_j$ and $B^Y_j$ preserve $S$ under the action of $Q_W$ on $Q$, and thus  $B^X_j, B^Y_j \in \bfD_F$ (see \cite[Remark 7.8]{CZZ1}). The first statement in this corollary is the result \cite[Proposition 9.5]{CZZ1}. 

\begin{cor} For the  Demazure elements $X_\al$, and $I=(i_1,...,i_k)$,  we have
\[
X_I\cdot (pq)=\sum_{E,F\subset [k]}\bfx^I_{E,F}(X_E\cdot p) (X_{F}\cdot q), 
\]
where $\bfx^I_{E,F}=(B^X_1 B^X_2\cdots  B^X_k)\cdot 1$ with $B^X_j\in \bfD_F$ defined  in Example~\ref{ex:LeibnizCoeffXY}. Similarly, for the  push-pull elements 
$Y_\al$ and $I=(i_1,...,i_k)$, we have
\[
Y_I\cdot (pq)=\sum_{E,F\subset [k]}\bfy^I_{E,F}(Y_{E}\cdot p) (Y_{F}\cdot q), 
\]
where $\bfy^I_{E, F}=(B^Y_1 B^Y_2\cdots  B^Y_k)\cdot 1$ with $B^Y_j\in \bfD_F$ defined 
in Example~\ref{ex:LeibnizCoeffXY}.
\end{cor}


\junk{
\begin{cor}
Let $I=(i_1,...,i_k)$ be a sequence of indices of simple roots, and denote 
$\be_j=s_{i_1}s_{i_2}\cdots s_{i_{j-1}}(\al_{i_j})$. 
 For  $E\subset[k]$, we have
 \begin{align*}
\bfx^{I_w}_{[k], E}&=\bfx^{I_w}_{E, [k]}=(-1)^{|E|}\prod\limits_{j\in E}x_{\be_{j}}, \mbox{and}\\
\bfy^{I_w}_{[k], E}&=\bfy^{I_w}_{E, [k]}=(-1)^{k-|E|}\prod\limits_{j\in [k]} x_{\be_{j}}(\prod\limits_{j\in [k]\setminus E} x_{-\be_{j}})^{-1}.
\end{align*}
\end{cor}
}


\section{The structure constants of equivariant oriented cohomology of flag varieties}\label{sec:prod}

In this section we prove the main result, i.e., the formulas of  structure constants of $Z_{I_w}^*$, with resulting formulas for the structure constants of $X_{I_w}^*$ and  of $Y_{I_w}^*$.





Let $\{Z_{I_w}^*\}$ be the basis of $Q_W^*$ dual to the basis $\{Z_{I_w}\}$ of $Q_W$.
\begin{thm}\label{thm:prod}
For any $u,v\in W$, the product $ Z_{I_u}^*\ Z_{I_v}^*$ is given by
\[
Z_{I_u}^*\ Z_{I_v}^*=\sum_{w\ge u, v}\bbz^{I_w}_{I_u,I_v}Z^*_{I_w},
\]
where 
$$
\bbz^{I_w}_{I_u, I_v}=\sum_{E, F\subset [\ell(w)]}\bfz^{I_w}_{E,F} c_{E,I_u} c_{F,I_v}\in Q,
$$
 $\bfz^{I_w}_{E, F}\in Q$ are the Leibniz coefficients given in Definition~\ref{def:LeibnizCoeff}. As before, the $Q$ elements $c_{E,I_u}$ and $c_{F,I_v}$ are defined as constants appearing in the expansion
 \begin{equation}
Z_J=\sum_{w\in W}c_{J, I_w}Z_{I_w}. 
\end{equation}
\end{thm}

\begin{ex}Consider the $A_3$-case. Consider $I_u=(2,3,1,2,1), I_v=(1,2,3,2,1)$, then $\bbz^{I_w}_{I_u, I_v}=0$ unless $w=w_0$  is the longest element. Fix $I_{w_0}=(1,2,3,1,2,1)$, in which case we have 
\begin{align*}
\bfz^{I_{w_0}}_{\{2,3,4,5,6\}, \{1,2,3,5,6\}}&=B^Z_1B_2^ZB_3^ZB_4^ZB_5^ZB^Z_6\cdot 1. =\frac{a_{\al_1}a_{\al_2}}{b_{\al_1}b_{\al_2}b_{\al_1+\al_2}b_{\al_2+\al_3}b_{\al_1+\al_2+\al_3}}.\\
c_{\{2,3,4,5,6\}, I_u}&=c_{\{1,2, 3, 5,6\}, I_v}=1.
\end{align*}
Therefore,
\[Z^*_{I_u}\cdot Z^*_{I_v}=\frac{a_{\al_1}a_{\al_2}}{b_{\al_1}b_{\al_2}b_{\al_1+\al_2}b_{\al_2+\al_3}b_{\al_1+\al_2+\al_3}} Z^*_{I_{w_0}}. \]
\end{ex}

\begin{proof}[Proof of Theorem~\ref{thm:prod}]
The coproduct structure $\Delta$ on $Q_W$ (Equation~\eqref{eq:coprod}) naturally induces a product on $Q_W^*$. 
For all $f, g \in Q_W^*$ and $\sum_{w\in W} q_w \de_w \in Q_W$, 
\begin{align*}
\langle f\ g, \sum\limits_{w\in W} q_w\de_w \rangle  & = \langle f \otimes g, \Delta(\sum_{w\in W} q_w\de_w ) \rangle  \\
&=  \langle f \otimes g,\sum_{w\in W} q_w\de_w \otimes \de_w \rangle\\
& = \sum_{w\in W} q_w   \langle f, \de_w  \rangle   \langle g, \de_w  \rangle.
\end{align*}
Note that this product corresponds to the product on $Q_W^*$ introduced in Section~\ref{sec:prelim} since
\begin{align*}
\langle f_u\ f_v, \sum\limits_{w\in W}  q_w\de_w\rangle &=
\begin{cases}
\langle f_u, \sum q_w\de_w\rangle =  \sum_w q_w\langle f_u, \de_w  \rangle , &\mbox{if $u=v$};\\
0, & \mbox{else},
\end{cases} \\
&= \begin{cases}
q_u, & \mbox{if $u=v$};\\ 
0 ,& \mbox{else}.
\end{cases}
\end{align*}

From Theorem \ref{thm:coprod} we have
\begin{align*}
\tr (Z_{I_w})&=\sum_{E,F\subset [\ell(w)]}\bfz^{I_w}_{E,F}Z_{E}\otimes Z_{F}\\
&=\sum_{E,F\subset [\ell(w)]}\bfz^{I_w}_{E,F}\left[\left(\sum_{u\in W}c_{E, I_u}Z_{I_u}\right)\otimes\left(\sum_{v\in W}c_{F, I_v}Z_{I_v}\right)\right]\\
&=\sum_{u,v\in W}\left[\sum_{E,F\subset [\ell(w)]} \bfz^{I_w}_{E,F} c_{E, I_u} c_{F, I_v}\right] Z_{I_u}\otimes Z_{I_v}\\
&=\sum_{u,v\in W}\bbz^{I_w}_{I_u, I_v}Z_{I_u}\otimes Z_{I_v}.
\end{align*}

Finally we obtain the coefficient by calculating the pairing:
\begin{align*}
\langle Z_{I_u}^*\ Z_{I_v}^*, Z_{I_w}\rangle & =  \langle Z_{I_u}^*\otimes Z_{I_v}^*, \Delta(Z_{I_w})\rangle = \bbz^{I_w}_{I_u, I_v}.
\end{align*}
Since $w=\wt\prod I_w\ge \wt\prod ( I_w|_E)$ for any $E\subset [\ell(w)]$,  by Lemma~\ref{lem:Zcoeff},  $\bbz^{I_w}_{I_u, I_v}=0$ unless $u\le w$ and   $v\le w.$ 
\end{proof}


The coproduct structure on the left $Q$-module $Q_W$ restricts to a coproduct structure on the left $S$-module $\bfD_F$ \cite[Theorem 9.2]{CZZ1}.
 Consequently, the embedding $\bfD_F^*\subset Q_W^*$ is an embedding of subrings. So the structure constants of the $S$-bases $\{X_{I_w}^*\}$ and $\{Y_{I_w}^*\}$ in $\bfD_F^*$ are precisely those of the $Q$-bases $\{X_{I_w}^*\}$ and $\{Y_{I_w}^*\}$ in $Q_W^*$. 




Specializing Theorem \ref{thm:prod} to the $X$-operators, we have
\[
X_{I_u}^*\ X_{I_v}^*=\sum_{w\ge v, w\ge u}\bbx^{I_w}_{I_u, I_v}X^*_{I_w}, 
\]
with 
\begin{align}\label{eq:generalbbx}
\bbx^{I_w}_{I_u, I_v}=\sum_{E, F\subset [\ell(w)]}\bfx^{I_w}_{E,F} c_{I_w|_{E}, I_u} c_{I_w|_{F}, I_v},
\end{align}
where $c_{I,I_v}$ are the coefficients that occur in the expansion $X_I = \sum_v c_{I,I_v} X_{I_v}$.  It follows from \cite[Theorem 9.2 and Proposition 7.7]{CZZ1} that $\bfx^{I_w}_{E,F}\in S$, that $c_{I,I_w}\in S$, so $\bbx^{I_w}_{I_u,I_v} \in S$. Similarly, specializing to the $Y$-operators, the structure constants for $Y_{I_w}^*$ are denoted by $\bby^{I_w}_{I_v,I_u}$ and can be expressed as
$$
\bby^{I_w}_{I_u, I_v}=\sum_{E, F\subset [\ell(w)]}\bfy^{I_w}_{E,F} c_{I_w|_{E}, I_u} c_{I_w|_{F}, I_v},
$$
where now the coefficients $c_{I, I_v}$ are those appearing in the expansion of $Y_I$.  As before, $\bfy^{I_w}_{E, F}\in S$ and $c_{I,I_w}\in S$, so $\bby^{I_w}_{I_u,I_v}\in S$. 
In Section \ref{sec:cohK} we show that these coefficients simplify in the case that $F=F_a$ or $F=F_m$, resulting in Theorem 1 from \cite{GK19}. It is worth noting that the formula \eqref{eq:generalbbx} can be used to construct the  Leray-Hirsch Theorem for flag varieties (see \cite{DZ20}). 



\begin{ex}Assume the root datum is of type $A_1$, then $W=\{e, s_1\}$. 
We calculate the basis change explicitly:
$$
X_e^*=f_e+f_{s_1}, \quad X_{(1)}^*=-x_{\al_1}f_{s_1}, \\
$$
and then we may obtain the products directly:
\begin{align*}
&X_e^*\ X_e^*=X_e^*, && X_e^*\ X_{(1)}^*=X_{(1)}^*,  && X_{(1)}^*\ X_{(1)}^*=-x_{\al_1}X_{(1)}^*, 
\end{align*}
and note that it agrees with Theorem  \ref{thm:prod} with $Z=X$.
\end{ex}


\begin{ex}
Consider the root datum $A_2$, with $W=\{e, s_1, s_2, s_1s_2, s_2s_1, w_o\}$. For the longest element $w_0$, we fix the reduced sequence $I_{w_0}=s_1s_2s_1$. 



We use the calculation in Example~\ref{ex:dualbasischange}, and the product structure on $Q_W^*$ to obtain the multiplication table for $\{X_{I_v}\}$.  For example, 
\begin{align*}
X_{I_{w_0}}^*\ X^*_{(1,2)} &= (-x_{\al_1}x_{\al_2}x_{\al_{13}}f_{w_0})\left(x_{\al_1}x_{\al_{13}}(f_{s_1s_2}+f_{w_0})\right)\\
&=-x_{\al_1}^2 x_{\al_2}x_{\al_{13}}^2 f_{w_0}\\
&= x_{\al_1}x_{\al_{13}}X_{I_{w_0}}^*.
\end{align*}
The other products are obtained similarly.
\begin{align*}
& X_{I_{w_0}}^*\ X^*_{I_{w_0}}=-x_1x_2x_{\al_{13}}X_{I_{w_0}}^*, \quad
&&X_{I_{w_0}}^*\  X^*_{(1,2)}=x_1x_{\al_{13}}X_{I_{w_0}}^*, \\
&X_{I_{w_0}}^*\  X^*_{(2,1)}=x_2x_{\al_{13}}X_{I_{w_0}}^*, \quad
&&X_{I_{w_0}}^*\  X^*_{(1)}=-yX_{I_{w_0}}^*, \\
&X_{I_{w_0}}^*\  X_{(2)}^*=-x_{\al_{13}}X_{I_{w_0}}^*, \quad
&&X_{I_{w_0}}^*\  X_e^*=X_{I_{w_0}}^*, \\
&X_{(1,2)}^*\  X_{(1,2)}^*=x_1x_{\al_{13}}X_{(1,2)}^*, \quad
&&X_{(1,2)}^*\  X_{(2,1)}^*=-x_{\al_{13}}X_{I_{w_0}}^*, \\
&X_{(1,2)}^*\  X_{(1)}^*=-x_1X_{(1,2)}^*-\frac{x_1}{x_{-1}}X_{I_{w_0}}^*,\quad
&&X_{(1,2)}^*\  X_{(2)}^*=-x_{\al_{13}}X_{(1,2)}^*,\\
&X_{(1,2)}^*\  X_e^*=X_{(1,2)}^*,\quad
&&X_{(2,1)}^*\  X_{(2,1)}^*=x_2x_{\al_{13}}X_{(2,1)}^*, \\
&X_{(2,1)}^*\  X_{(1)}^*=\frac{y-x_{\al_{13}}}{x_1}X_{I_{w_0}}^*-x_{\al_{13}}X_{(2,1)}^*,\quad
&&X_{(2,1)}^*\  X_{(2)}^*=\frac{x_{\al_{13}}-x_2}{x_1}X_{I_{w_0}}^*-x_2X_{(2,1)}^*, \\
&X_{(2,1)}^*\  X_{e}^*=X_{(2,1)}^*, \quad 
&& X_{(1)}^*\  X_{(2)}^*=X_{(1,2)}^*+X_{(2,1)}^*+\kappa_1X_{I_{w_0}}^*\\
&X_{(2)}^*\  X_{(2)}^*=\frac{x_{\al_{13}}-x_2}{x_1}X^*_{(1,2)}-x_2X_{(2)}^*, 
&& X_{(1)}^*\  X_{(1)}^*=-x_1X_{(1)}^*+\frac{x_{\al_{13}}-x_1}{x_2}X_{(2,1)}^*\\
& &&\phantom{X_{(1)}^*\  X_{(1)}^*}+\frac{x_1y+x_{\al_{13}}^2-x_1x_{\al_{13}}-y^2}{x_1x_2x_{\al_{13}}}X_{I_{w_0}}^*.\\
\end{align*}
One can check that the above coefficients $\bbx_{I_u, I_v}^{I_w}$ agree with the formula \eqref{eq:generalbbx}. 
Note that when computing $\bbx^{I_{w_0}}_{1,1}$, one needs to compute the following coefficients:
\[
\bfx^{I_{w_0}}_{\{3\}, \{3\}}, ~\bfx^{I_{w_0}}_{\{1, 3\}, \{3\}},~\bfx^{I_{w_0}}_{\{3\}, \{1, 3\}},~ \bfx^{I_{w_0}}_{\{1, 3\},\{1,3 \}}.
\]
\end{ex}






As an application, we consider the partial flag variety case. Let $J$ be a subset of $[n]$. Let $P_J$ be the corresponding parabolic subgroup, $W_J<W$ the corresponding subgroup, and $W^J\subset W$ be  the set of minimal length representatives of $W/W_J$.  We say a set of reduced sequences $I_w$ is $J$-compatible if for each $w=uv, u\in W^J, v\in W_J$, we have $I_w=I_u\cup I_v$, i.e., $I_w$ is the concatenation of $I_u$  with $I_v$.

\begin{thm} \label{thm:J} Suppose the set $\{I_w\}$ is $J$-compatible. Then  for any $v,u\in W^J$, we have
\[
X_{I_u}^*\ X_{I_v}^*=\sum_{w\in W^J, w\ge v, w\ge u} \bbx^{I_w}_{I_u, I_v}X^*_{I_w}. 
\]
\end{thm}
\begin{proof} It follows from \cite[Corollary 8.4]{CZZ2} that $X^*_{I_u}, u\in W^J$ is a basis of $(Q_W^*)^{W_J}$. Moreover, from Lemma 4.3 of loc.it., we know $\de_w\bullet (ff')=(\de_w\bullet f)(\de_w \bullet f')$. Therefore, $X_{I_u}^*\ X_{I_v}^*\in (Q_W^*)^{W_J}$, so is a linear combination of $X_{I_w}^*, w\in W^J$. 
\end{proof}

Geometrically, under the assumption of this theorem, it follows from \cite[Corollary 8.4]{CZZ2} that  $\{X_{I_w}^*\}_{w\in W^J}$ is a basis of $(\bfD_F^*)^{W_J}\cong \mathbb{h}_T(G/P_J)$.  So the product  $X_{I_u}^*\ X_{I_v}^*, u,v\in W^J$ is a linear combination of $X_{I_w}^*, w\in W^J$. 

\begin{cor}Let $F=F_a$ or $F_m$, and suppose $u\in W$ satisfies that $u\in W^J$ for some $J$ and $u$ is the longest element in $W^J$. Then for any $v\in W^J$, $\bbx^{w}_{u,v}=0$ for any $w\in W$, unless $w=u$. 
\end{cor}
\begin{proof}In these cases, the braid relations are satisfied, so the structure constants do not depend on the choice of reduced sequences. In other words, fixing $u$ and $J$, we can assume we have chosen  $J$-compatible reduced sequences. Then Theorem \ref{thm:J} applies, which implies that for any $v\in W^J, w\in W$, we have $\bbx^{I_w}_{u,v}=0$ unless $w\in W^J$ and $w\ge u$. Since $u$ is maximal in $W^J$, so $w=u$. 
\end{proof}

\section{Structure constants in singular cohomology and K-theory}\label{sec:cohK}

We can now recover formulas in \cite{GK19} of structure constants of Schubert classes   for singular cohomology ($F=F_a$) and K-theory ($F=F_m$). We first simplify the coefficients $c^X_{I,I_w}$ and $c^Y_{I,I_w}$ in these two cases. 
Recall that, when the formal group law is $F=F_a$ or $F=F_m$, the braid relations are satisfied for $Z_\al=X_\al$ and $Z_i=Y_\al$.



\begin{lem}\label{lem:c}
Let $J$ be a word in the Weyl group. As in Lemma~\ref{lem:Zcoeff}, define coefficients $c_{J,I_w}$  by 
$$
Z_J= \sum_{w\in W}c_{J,I_w}Z_{I_w}.
$$
\begin{enumerate}
\item Let  $F=F_a$. If $Z_\al=X_\al$ or $Z_\al=Y_\al$, then
$$
c_{J,I_w} = \begin{cases}
1 ,&\mbox{if $J$ is  a reduced word for $w$};\\
0, &\mbox{else}.
\end{cases}
$$
\item Let  $F=F_m$. If $Z_\al=X_\al$ or $Z_\al=Y_\al$, then
$$
c_{J,I_w} = \begin{cases}
1 ,&\mbox{if $w=\wt \prod J$};\\
0, &\mbox{else}.
\end{cases}
$$
\end{enumerate}
\end{lem}

\begin{proof} When $F=F_a$ or $F=F_m$, it is well-known that the braid relations are satisfied. We write  $c_{J,w}$ for the coefficient $c_{J,I_w}$.
 When $F=F_a$, $Z_\al^2=0$, so if $J$ is not reduced,  $Z_J=0$. If $J$ is reduced and $\prod J=w$,  then $Z_J=Z_{w}$, so $c_{J,w}=1$ and $c_{J,v}=0$ for $v\neq w$. 

When $F=F_m$, we have $Z_\al^2=Z_\al$ and thus $Z_J=Z_w$ where $w:={\wt \prod J}$. It follows that $c_{J,w}=1$ and $c_{J,v}=0$ for $v\neq w$. 
\end{proof}

\begin{ex}\label{ex:FaSch}
Recall that $\mathbb{h}_T(G/B)= H_T^*(G/B)$ when  $F=F_a$. As described in Example~\ref{ex:Schubert} and Proposition 2.7, the element $\zeta_w^X$ in $\bfD_F^*$ corresponds under a natural isomorphism
$$
\bfD_{F_a}^*\longrightarrow \mathbb{h}_T(G/B)
$$
to the equivariant cohomology class Poincar\'e dual to $[X(w)]$, where $[X(w)]$ is the homology class of the Schubert variety. Furthermore, the first Chern classes of the corresponding line bundles are $x_{\al} = \al$ for all simple roots $\al$.



For each $w\in W$, fix a reduced sequence $I_w$. From the specialization of Theorem \ref{thm:prod}, we have defining relations
$$
Y_{u}^*\ Y_v^*=\sum_{w\ge u, w\ge v}\bby^{I_w}_{v,u}Y_{w}^*
$$
for  $\bby^{I_w}_{v,u}$ . Then
\begin{align*}
\bby^{I_w}_{u, v}&=\sum_{E,F\subset [\ell(w)]}\bfy^{I_w}_{E, F}c^Y_{E, I_u}c^Y_{F, I_v}\text{ by Theorem~\ref{thm:prod},}\\
&=\sum_{E,F\tiny\text{ reduced}\atop \mbox{for }u,v} \bfy^{I_w}_{E, F}, \text{ by Lemma~\ref{lem:c}}(1)
\end{align*}
where the second sum is over $E,F$ whose corresponding products of reflections are reduced and equal to $u, v$ respectively. 
 Recall that
\[
\bfy^{I_w}_{E, F}=(B^Y_{{1}}B^Y_{{2}}\cdots B^Y_{{\ell(w)}})\cdot 1,
\]
with 
\[
B_j^Y=\left\{\begin{array}{ll}
x_{\be_j}\de_{\be_j}, &\text{ if }j\in E\cap F,\\
\de_{\be_j}, &\text{ if }j\in  E \text{ or }F, \text{ but not both},\\
Y_{\be_j}, &\text{ if }j\not\in E\cup F.\\
 \end{array}\right.
\]
with $\be_j = \al_{i_j}$.

The coefficients $\bby^{I_w}_{u,v}$ coincide with the structure constants $c^w_{uv}$ in \cite[Theorem 1]{GK19}. Note that in this case, $Y_\al=-X_\al$, so $\zeta^Y_w = (-1)^{\ell(w)} \zeta^X_w$, and thus $X_w^*=(-1)^{\ell(w)} Y_w^*$.  Therefore, 
\[
\bbx^{I_w}_{u,v}=(-1)^{\ell(w)+\ell(u)+\ell(v)}\bby^{I_w}_{u,v}. 
\]


\end{ex}
\begin{ex}\label{ex:FmSch} In case of $F=F_m$, we have $x_\alpha = 1-e^{-\alpha}$. 
The action of $X_\alpha$ (resp. $Y_{-\alpha}$) on $S=K_T(pt)$ corresponds to the action of the ordinary (resp.  isobaric) Demazure operator in  \cite{GK19}. 


Fixing a reduced sequence $I_w$ for each $w$, we have \[
X_u^*\ X_v^*=\sum_{w\ge u, w\ge v}\bbx^{I_w}_{u,v}X_w^*=\sum_{w\ge v, w\ge u}\sum_{E,F}\bfx^{I_w}_{E,F}X_w^*,
\]
where by Lemma~\ref{lem:c}(2), the second sum is over all $E,F\subset [\ell(w)]$ such that $\wt \prod E = u$ and $\wt \prod F=v$.
Here, we have
\[
B_j^X=\left\{\begin{array}{ll}
-(1-e^{-\be_j})\de_{\be_j}, &\text{ if }j\in E\cap F,\\
\de_{\be_j}, &\text{ if }j\in  E \text{ or }F, \text{ but not both},\\
X_{\be_j}, &\text{ if }j\not\in E\cup F.\\
 \end{array}\right.
\] with $\be_j = \al_{i_j}$.

The classes $\{\xi_w:\ w\in W\}$ in \cite{GK19} are defined as the dual basis to $[\mathcal{O}_{X(w)}(-\partial X(w)) ]$ under the pairing obtained by taking the equivaraint cap product and pushing forward to a point. Each $\xi_w$ coincides with the Poincar'e dual class to $[\mathcal{O}_{Y(w)}]$.  In Example~\ref{prop:Schubert} we note that $X_w^* = [\mathcal{O}_{Y(w)}]$, and thus $\xi_w=(-1)^{\ell(w)} X_w^*$. Therefore,
\[
\xi_u\ \xi_v=(-1)^{\ell(u)+\ell(v)+\ell(w)}\sum_{w\ge u, v}\bbx^{I_w}_{u,v}\xi_w.
\]
It follows that the coefficients $(-1)^{\ell(u)+\ell(v)+\ell(w)}\bbx^{I_w}_{u,v}$ coincide with $a^w_{uv}$ in  \cite{GK19}, as is clear from the formula.


With the observation that the classes 
$\{{\mathring \xi}_w:\ w\in W\}$ defined in \cite{GK19} satisfy ${\mathring\xi}_w = Y_w^*$,  a similar argument 
implies that $\bby^w_{u,v}$ coincide with the structure constants ${\mathring a}^w_{u,v}$  defined in \cite{GK19}.

\end{ex}


\begin{ex}Let $F=F_a$. Consider the  $A_2$ case. If $I_w=s_1s_2s_1, u=s_1, v=s_1s_2$, then  
\[
\bby^{s_1s_2s_1}_{s_1,s_1s_2}=\bfy^{s_1s_2s_1}_{\{1\}, \{1,2\}}+\bfy^{s_1s_2s_1}_{\{3\}, \{1,2\}}=\begin{pmatrix}
\al_1 \de_1&\de_2& Y_1\\
+\de_1& \de_2&\de_1
\end{pmatrix}\cdot 1=0+1=1. 
\]
Similarly, 
\[\bby^{s_1s_2s_1}_{s_1,s_2s_1}=\bfy^{s_1s_2s_1}_{\{1\}, \{2,3\}}+\bfy^{s_1s_2s_1}_{\{3\}, \{2,3\}}=
\begin{pmatrix}
\de_1&\de_2&\de_1\\
+Y_1&\de_2&\al_1\de_1
\end{pmatrix}\cdot 1=1-1=0.\]

For the $A_3$ case, one can also compute 
\begin{align*}
\bby^{s_1s_2s_3s_1s_2}_{s_2s_3s_2, s_1s_2s_1}&=\bfy^{s_1s_2s_3s_1s_2}_{\{2,3,5\}, \{1,2,4\}}+\bfy^{s_1s_2s_3s_1s_2}_{\{2,3,5\}, \{2,4,5\}}\\
&=\begin{pmatrix}
\de_1&\al_2\de_2&\de_3&\de_1&\de_2\\
+Y_1&\al_2\de_2 &\de_3&\de_1 & \al_2\de_2
\end{pmatrix}\cdot 1\\
&=(\al_1+\al_2)+\al_3. 
\end{align*}
\end{ex}
\begin{ex}Let $F=F_m$. Consider the $A_3$ case, with	 $I_w=s_1s_2s_3s_1s_2, u=s_2s_3s_2, v=s_1s_2s_1$. We have
\begin{align*}
\bbx^{s_1s_2s_3s_1s_2}_{s_2s_3s_2,s_1s_2s_1}&=\bfx^{s_1s_2s_3s_1s_2}_{\{2,3,5\}, \{1,2,4\}}+\bfx^{s_1s_2s_3s_1s_2}_{\{2,3,5\}, \{2,4, 5\}}+\bfx^{s_1s_2s_3s_1s_2}_{\{2,3,5\}, \{1,2,4, 5\}}\\
&=\begin{pmatrix}
\de_1& -x_2\de_2& \de_3& \de_1& \de_2\\
+X_1& -x_2 \de_2&  \de_3 & \de_1&-x_2\de_2\\
+\de_1 &-x_2\de_2&  \de_3&  \de_1 &-x_2\de_2
\end{pmatrix}\cdot 1\\
&=-x_{1+2}+\frac{x_{1+2+3}x_2-x_{2+3}x_{1+2}}{x_1}+x_{2+3}x_{1+2}\\
&=x_{\al_2}-x_{\al_1+2\al_2+\al_3}. 
\end{align*}
\end{ex}

\section{Structure constants of cohomological stable bases}
\label{sec:cohstab}


In this section, we let $F=F_a$. We recall the definition of cohomological stable basis of Maulik-Okounkov, and generalize Su's formula of structure constants for Segre-Schwartz-MacPherson classes (Theorem~\ref{thm:cohstab}). We use the twisted group algebra language for singular cohomology, whose $K$-theory version was given  in \cite{ SZZ17}. As the framework and proofs are very similar as earlier sections,  we will only review essential properties. Note that some of the notation  is restricted to this section only, unless otherwise mentioned.


Let $R^a=\bbZ[h]$, $S^a=\Sym_{R^a}(\La)$ and $Q^a=\Frac(S^a)$. Define
$$
Q^a_W=Q^a\rtimes_{R^a} R^a[W]
$$ with $Q^a$-basis $\de_w, w\in W$. Denote
\[
\widehat \al=h-\al, \quad \al_{w_0}=\prod_{\al>0}\al, \quad \widehat\al_{w_0}=\prod_{\al>0}(h-\al). \]

For any simple root $\al$, define the element
\[
T_\al=-h\frac{1}{\al}(1-\de_{\al})-\de_{\al}=-\frac{h}{\al}+\frac{\widehat\al}{\al}\de_\al\in Q_W^a.
\]
Then it is known that the braid relations are satisfied, and $T_\al^2=1$. Indeed, the algebra generated by $\{T_\al \}$ is called the degenerate (or graded) Hecke algebra. One can then define $T_w$ to be the product of $T_\al$, and  $T_w, w\in W$ is a basis of $Q_W^a$. Moreover, for any (not necessarily reduced) sequence $I$,  if $\prod I=w$, then $T_I=T_{w}$. 


Let $(Q_W^a)^*$ be the $Q^a$-dual of $Q_W^a$, and let $T_w^*\in (Q_W^a)^*$ be the dual basis. Denote the basis of $(Q_W^a)^*$ dual to $\{\de_w\in Q_W^a\}$ by $\{f_w\}$, as in  \S\ref{sec:prelim}. The identity of the ring $(Q_W^a)^*$ is denoted by $\mathbf{1}=\sum_{w\in W}f_w$.

The ring $Q_W^a$ acts on $(Q_W^a)^*$ via the $\bullet$-action, given as before by
$$
\langle z\bullet q^*, z'\rangle = \langle q^*, z'z\rangle\quad \mbox{for $z, z'\in Q_W^a, q^*\in (Q_W^a)^*$.}
$$

It induces a $W$-action on $(Q_W^a)^*$ via the embedding $W\subset Q_W^a$. Let $((Q_W^a)^*)^W$ denote the Weyl-invariant subgroup of  $(Q_W^a)^*$.

In this section only, denote 
\[
\hY=\sum_{w\in W}\de_w\frac{1}{\al_{w_0}\widehat\al_{w_0}}
=\sum_{w\in W}\de_w\frac{1}{\prod_{\al>0}\al(h-\al)}\in Q_W^a.
\]
The map $\hY\bullet \_: (Q_W^a)^*\to ((Q_W^a)^*)^{W}=Q^a\mathbf{1}$ 
is the algebraic analogue of the composition of the map 
\[Q^a\otimes_{S^a}H^*_{T\times \bbC^*}(T^*G/B)\cong Q^a\otimes_{S^a} H^*_{T\times \bbC^*}(G/B)\to Q^a\otimes_{S^a}H^*_{T\times \bbC^*}(\pt).\]
where the last map is the equivariant pushforward of cohomology class on $G/B$ to a point on the second term. The proofs in  \cite[Lemma 7.1]{CZZ2} and \cite[Lemma 5.1]{SZZ17} easily extend to show that, for any $f, g \in (Q_W^a)^*$,
\[
\hY\bullet ((T_\al\bullet f)\cdot g)=\hY\bullet (f\cdot (T_\al\bullet g)).
\]

\begin{defn} We define two bases of $(Q_W^a)^*$ as a module over $Q^a$.  Let
 \begin{align*}
 \stab^+_w &=T_{w^{-1}}\bullet ( \al_{w_0}f_e)\\
 \stab^-_w &=(-1)^{\ell(w_0)}T_{w^{-1}w_0}\bullet (\al_{w_0}f_{w_0})
 \end{align*}
Then $\{\stab^+_w:\ w\in W\}$  and $\{\stab^-_w:\ w\in W\}$ each form bases for $(Q_W^a)^*$ as a module over $Q^a$.  We call these bases the {\bf  cohomological stable bases.} See \cite{S15} for more details.
\end{defn}
It is immediate from the definition that  $\stab^+_w$ has support on $\{f_v:\ v\leq w\}$ and $\stab^-_w$ has support on $\{f_v: v\geq w\}$.

The following lemma is the analogue of Theorem 5.7 and Lemma 5.6 in \cite{SZZ17}. The first identity was due to Maulik-Okounkov originally.
\begin{lem}\label{lem:Kpairing} We have 
\[
\hY\bullet \left[\stab^+_v\cdot \stab^-_u\right]=(-1)^{\ell(w_0)}\de_{v,u}\mathbf{1}, \quad \hY \bullet \left[\stab^+_v\cdot \widehat \al_{w_0}T_u^* \right]=\de_{v,u}\mathbf{1}.
\]
\end{lem}

Denote \[
\stab^-_u\cdot \stab^-_v=\sum_{w\in W}\bbt^{w}_{u,v}\stab^-_w,
\]
Then
\begin{thm}\label{thm:cohstab}
\begin{enumerate}\item We have $\stab^-_w=(-1)^{\ell(w_0)}\widehat\al_{w_0}T^*_w$. 
\item For each $w\in W$, fix a reduced sequence $I_w$.  Then
\[
\bbt^{w}_{u,v}={\sum_{\tiny\begin{array}{c}E,F\subset[\ell(w)]\\
\prod (I_w|_{E})=u, \prod (I_w|_{F})=v\end{array} }}\widehat\al_{w_0}^2 \bft^{I_w}_{E,F}, 
\]
 where $\bft_{E, F}^{I_w}=(B_1^T B_2^T\cdots B^T_k)\cdot 1$ with  
\[
B_j^T=\left\{\begin{array}{ll}
\frac{\al_{i_j}}{\widehat\al_{i_j}}\de_{i_j}, &\text{ if }j\in E\cap F,\\
\frac{h}{\widehat\al_{i_j}}\de_{i_j}, &\text{ if }j\in  E \text{ or }F, \text{ but not both},\\
-\frac{h}{\al_{i_j}}+\frac{h^2}{\al_{i_j}\widehat\al_{i_j}}\de_{i_j}, &\text{ if }j\not\in E\cup F.\\
 \end{array}\right.
\]
\end{enumerate}
\end{thm}
\begin{proof}(1). This follows from Lemma \ref{lem:Kpairing} above.

(2). For each $w\in W$, we fix a reduced decomposition. We have 
\[\stab^-_u\cdot \stab^-_v=(-1)^{\ell(w_0)}\widehat\al_{w_0}T_u^*\cdot (-1)^{\ell(w_0)}\widehat \al_{w_0}T_v^*
=\widehat\al_{w_0}^2T_u^*\cdot T_v^*.\]
Therefore, it suffices to consider the structure constants for $T_u^*$. But  the elements $T_u$ are an instantiation of $Z_{I_u}$ with the coefficient ring $R^a$, with $a_{\al_{i_j}} = -h/\al_{i_j}$ and $b_{\al_{i_j}}=\widehat\al_{i_j}/\al_{i_j}$. Thus Theorem \ref{thm:prod} indicates how to multiply the corresponding dual elements, resulting in $B_j^T$ defined as above.
\end{proof}
\begin{ex}Consider $A_2$-case. If $I_w=s_1s_2s_1, u=v=s_1$, then 
\begin{align*}
\bbt^{s_1s_2s_1}_{s_1, s_1}&=\widehat\al_1\widehat\al_2\widehat\al_{1+2}(\bft^{121}_{\{1\}, \{3\}}+\bft^{121}_{\{3\}, \{1\}}+\bft^{121}_{\{1\}, \{1\}}+\bft^{121}_{\{3\}, \{3\}})\\
&=\widehat\al_1\widehat\al_2\widehat\al_{1+2}\begin{pmatrix}
\frac{h}{\widehat\al_{1}}\de_1&-\frac{h}{\al_2}+\frac{h^2}{\al_2\widehat\al_2}\de_2&\frac{h}{\widehat\al_1}\de_1\\
\frac{h}{\widehat\al_{1}}\de_1&-\frac{h}{\al_2}+\frac{h^2}{\al_2\widehat\al_2}\de_2&\frac{h}{\widehat\al_1}\de_1\\
\frac{\al_1}{\widehat\al_{1}}\de_1&-\frac{h}{\al_2}+\frac{h^2}{\al_2\widehat\al_2}\de_2&-\frac{h}{\al_1}+\frac{h^2}{\al_1\widehat\al_1}\de_1\\
-\frac{h}{\al_1}+\frac{h^2}{\al_1\widehat\al_1}\de_1&-\frac{h}{\al_2}+\frac{h^2}{\al_2\widehat\al_2}\de_2&\frac{\al_1}{\widehat\al_{1}}\de_1\\
\end{pmatrix}\cdot 1=h^2(h+\al_1). 
\end{align*}

If $I_w=s_1s_2s_1, u=s_1, v=s_1s_2$, then
\begin{align*}
\bbt^{s_1s_2s_1}_{s_1,s_1s_2}&=\widehat\al_1\widehat\al_2\widehat\al_{1+2}(\bft^{121}_{\{1\}, \{1,2\}}+\bft^{121}_{\{3\}, \{1,2\}})\\
&=\widehat\al_1\widehat\al_2\widehat\al_{1+2}\begin{pmatrix}
\frac{\al_1}{\widehat\al_{1}}\de_1&\frac{h}{\widehat\al_2}\de_2& -\frac{h}{\al_1}+\frac{h^2}{\al_1\widehat\al_1}\de_1\\
\frac{h}{\widehat\al_1}\de_1& \frac{h}{\widehat\al_2}\de_2&\frac{h}{\widehat\al_1}\de_1
\end{pmatrix}\cdot 1=h^2(h+\al_1).
\end{align*}

Similarly, for $v'=s_2s_1$, we have 
\begin{align*}\bbt^{s_1s_2s_1}_{s_1,s_2s_1}&=\widehat\al_{w_0}^2(\bft^{s_1s_2s_1}_{\{1\}, \{2,3\}}+\bft^{s_1s_2s_1}_{\{3\}, \{2,3\}})\\
&=\widehat\al_1\widehat\al_2\widehat\al_{1+2}\begin{pmatrix}
\frac{h}{\widehat\al_{1}}\de_1&\frac{h}{\widehat\al_2}\de_2& \frac{h}{\widehat\al_1}\de_1\\
-\frac{h}{\al_1}+\frac{h^2}{\al_1\widehat\al_1}\de_1& \frac{h}{\widehat\al_2}\de_2&\frac{\al_1}{\widehat\al_1}\de_1
\end{pmatrix}\cdot 1\\
&=h^3-h^2\widehat\al_{1+2}=h^2(\al_1+\al_2). 
\end{align*}
\end{ex}
\begin{ex}Consider the $A_3$ case. For $I_w=s_1s_2s_3s_1s_2, u=s_2s_3s_2, v=s_1s_2s_1$, we have
\begin{align*}
\bbt^{s_1s_2s_3s_1s_2}_{s_2s_3s_2, s_1s_2s_1}&=\widehat\al_{w_0}(\bft^{s_1s_2s_3s_1s_2}_{\{2,3,5\}, \{1,2,4\}}+\bft^{s_1s_2s_3s_1s_2}_{\{2,3,5\}, \{2,4,5\}})\\
&=\widehat\al_{w_0}\begin{pmatrix}
\frac{h}{\widehat\al_1}\de_1& \frac{\al_2}{\widehat\al_2} \de_2& \frac{h}{\widehat\al_3} \de_3 & \frac{h}{\widehat\al_1}\de_1&\frac{h}{\widehat\al_2}\de_2\\
-\frac{h}{\al_1}+\frac{h^2}{\al_1\widehat\al_1}\de_1& \frac{\al_2}{\widehat\al_2}\de_2&\frac{h}{\widehat\al_3} \de_3&\frac{h}{\widehat\al_1} \de_1& \frac{\al_2}{\widehat\al_2}\de_2
\end{pmatrix}\cdot 1\\
&=h^4\widehat\al_3(\al_1+\al_2)+h^3\widehat\al_3 (h\al_3+\al_1\al_2+\al_2^2+\al_2\al_3)\\
&=h^3\widehat\al_3(h+\al_2)(\al_1+\al_2+\al_3). 
\end{align*}
\begin{align*}
\bbt^{12312}_{232, 1}&=\widehat\al_{w_0}(\bft^{12312}_{\{2,3,5\}, \{1\}}+\bft^{12312}_{\{2,3,5\}, \{4\}})\\
&=h^5(\widehat\al_2+2\widehat\al_3).\\
\bbt^{12312}_{232, 2}&=\widehat\al_{w_2}(\bft^{12312}_{\{2,3,5\},\{2\}}+\bft^{12312}_{\{2,3,5\}, \{5\}})\\
&=h^4(3h^2+h\al_2+(\al_{2+3})\widehat\al_{1+2+3}). 
\end{align*}
\end{ex}
\begin{rem}\label{rem:Su}It is not difficult to see that the formula in \cite[Theorem 1.1]{S19} is the structure constants of $\sigma^*_w\in (Q_W^a)^*$, where 
\[
\sigma_i=\frac{1+\al_i}{\al_i}\de_i-\frac{1}{\al_i}\in Q_W^a. 
\]
\end{rem}


\section{Structure constants for K-theoretic stable bases}
In this section, we give a formula of  the structure constants for K-theory stable basis. Similar as Section \ref{sec:cohstab}, the treatment is done  by using the twisted group algebra method. This method was firstly done in \cite{SZZ17}, so we only recall the definition. Some notations are restricted to this section only, unless otherwise mentioned. 

Let $R^m=\bbZ[q^{1/2},q^{-1/2}]$, $S^m=R^m[\La]$. We use the following notations in this section:
\[
x_{\pm\al}=1-e^{\mp\al}, \quad \hx_\al=1-qe^{-\al}, \quad \hx_w=\prod_{\al>0, w^{-1}\al<0}\hx_\al, \quad q_w=q^{\ell(w)}. 
\]
We apply the twisted group algebra construction, with $Q^m=\Frac(S^m)$, $Q^m_W=Q^m\rtimes_{R^m} R^m[W]$. Define 
\[
\tau_\al^-=\frac{q-1}{1-e^\al}+\frac{1-qe^{-\al}}{1-e^\al}\de_\al\in Q_W^m. 
\]
We have $(\tau_\al^-)^2=(q-1)\tau^-_\al+q$. Moreover, it is known that the braid relations are satisfied. Therefore, $\tau_w^-, w\in W$ is a $Q^m$-basis of $Q_W^m$. Moreover, for each sequence $I$, we have 
\[
\tau^-_I=\sum_{w\in W}c^{\tau^-}_{I,w}\tau_w^-,\quad c^{\tau^-}_{I,w}\in S^m.
\]
\begin{lem}\label{lem:Ktau}
\begin{enumerate}
\item We have $c^{\tau^-}_{I,w}=0$ unless $w\le \wt\prod I$.
\item If $I$ is reduced, then $\wt\prod I=\prod I$, so $c^{\tau^-}_{I,w}=0$ unless $w=\prod I$, in which case $c^{\tau^-}_{I,w}=1$. 
\end{enumerate}
\end{lem}
\begin{proof}(1). This follows from the quadratic relation $(\tau^-_i)^2=(q-1)\tau_i^-+q$. 

(2). This follows from the braid relations. 
\end{proof}
Following from Theorem \ref{thm:Leibniz}, we have
\begin{prop}For the element $\tau^-_i$, if $I=(i_1,...,i_k)$, we have 
\[
\tau^-_I\cdot (pq)=\sum_{E,F\subset [k]}P^I_{E,F}(\tau_{I|_{E}}\cdot p) (\tau_{I|_{F}}\cdot q), \quad p,q\in Q.
\]
where $P^I_{E,F}=(B^{\tau^-}_1 B^{\tau^-}_2\cdots  B^{\tau^-}_k)\cdot 1$ with $B^{\tau^-}_j\in Q_W^m$ defined by
\[
B_j^{\tau^-}=\left\{\begin{array}{ll}
\frac{1-e^{\al_{i_j}}}{1-qe^{-\al_{i_j}}}\de_{i_j}, &\text{ if }j\in E\cap F,\\
\frac{1-q}{1-qe^{-\al_{i_j}}}\de_{i_j}, &\text{ if }j\in  E \text{ or }F, \text{ but not both},\\
\frac{q-1}{1-qe^{-\al{i_j}}}\tau^-_{i_j}\de_{i_j}, &\text{ if }j\not\in E\cup F.\\
 \end{array}\right.
\]
\end{prop}

Similar as Section \ref{sec:cohstab}, we take the dual $(Q_W^m)^*$, and $Q_W^m$ acts on $(Q_W^m)^*$ via the $\bullet$-action. Indeed, we have 
\[
(Q_W^m)^*\cong Q^m\otimes_{S^m}K_T(G/B)\cong Q^m\otimes_{S^m}K_T(T^*G/B). 
\]
\begin{defn}\cite[Definition 5.3, Theorem 5.4]{SZZ17} The stable basis is defined as 
\[
\stab^-_w=q_{w_0}q_w^{-1/2}(\tau^-_{w_0w})^{-1}\bullet (\prod_{\al>0}(1-e^{\al})f_{w_0})\in (Q_W^m)^*. 
\]
\end{defn}


Moreover, by \cite[Theorem 5.4, Theorem 6.5]{SZZ17}, we have 
\[
\stab^-_w=q_w^{1/2}\hx_{w_0}(\tau^-_w)^*. 
\]


The following theorem gives the formula of structure constants for K-theory stable basis:
\begin{thm}
\[
\stab_u^-\cdot \stab_v^-=\sum_{w\ge u, w\ge v}p^w_{u,v}\stab_w^-.
\]
Here  
\[
p^w_{u,v}=q^{\frac{1}{2}(\ell(u)+\ell(v)-\ell(w))}\hx_{w_0}\sum P^{I_w}_{E,F}c^\tau_{I_w|_{E},u}c^\tau_{I_w|_{F}, v}, 
\]
where the sum is over all $E, F\subset[\ell(w)]$ such that $\tilde\prod (I_{w}|_{E})\ge u$ and $\tilde\prod (I_w|_{F})\ge v$. 
\end{thm}
\begin{proof}The proof follows similarly as that of Theorem \ref{thm:cohstab}. 
\end{proof}
\begin{rem}Due to the quadratic relation $(\tau_\al^-)^2=(q-1)\tau^-_\al+q$, it is difficult to express the sum in terms of formulas in \S\ref{sec:cohK} and \S\ref{sec:cohstab}. Indeed, this is also the reason why it is difficult to express the restriction formula of $\stab_w^-$ in \cite{SZZ17} in terms of AJS-Billey-Graham-Willems type formula. 
\end{rem}

\section{The restriction formula}\label{se:restriction}

In this section we relate the structure constants of $Z_{I_w}^*$ with its restriction coefficients. This generalizes such relations in cohomology and K-theory from  \cite[(4.32)]{KK86} and \cite[2.25]{KK90}. 

For $Z_\al$ as before, denote  
\[\de_u=\sum_{w\in W}b^Z_{u,I_w}Z_{I_w}, b^Z_{u, I_w}\in Q,\]
 so $Z_{I_v}^*=\sum_{u}b^Z_{u, I_v}f_u$, i.e.,  $Z_{I_v}^*(\de_u)=b^Z_{u, I_v}$. We call $b^Z_{u,I_v}$ the restriction coefficients of $Z_{I_v}^*$.
\begin{thm} \label{thm:res}For any $w\in W$, define the matrix $\fp_w^Z$ with $\fp^Z_{w}(u,v)=\bbz^{I_v}_{I_w, I_u}$, the matrix $\fb^Z$ with $\fb^Z(u,v)=b^Z_{v,I_u}$, and the matrix $\fb^Z_w$ with $\fb^Z_w(u,v)=\de_{u,v}b^Z_{u, I_w}$. Then 
\[
\fp_w^Z=\fb^Z \cdot \fb^Z_w \cdot(\fb^Z)^{-1}. 
\]
\end{thm}
\begin{proof}
We have 
\begin{eqnarray*}
(\fp^Z_w\cdot \fb^Z)(u,v)&=\sum_{z\in W}\fp^Z_w(u,z)\fb^Z(z,v)
&=\sum_{z\in W}\bbz^{I_z}_{I_w,I_u}b^Z_{v,I_z}\\
&=\sum_{z\in W}\bbz^{I_z}_{I_w, I_u}Z^*_{I_z}(\de_v)
&=(Z_{I_u}^*\cdot Z_{I_w}^*)(\de_v)\\
&=Z_{I_u}^*(\de_v)\cdot Z_{I_w}^*(\de_v)
&=b^Z_{v, I_u}b^Z_{v, I_w}\\
&=\sum_{z\in W} b^Z_{z,I_u} \de_{z,v}b^Z_{z,I_w}
&=\sum_{z\in W}\fb^Z(u,z)\fb^Z_w(z,v)\\
&=(\fb^Z\cdot \fb^Z_w)(u,v). 
\end{eqnarray*}
\end{proof}
\begin{cor}\label{cor:res}For any $v,w\in W$, we have
\[
\bbz^{I_v}_{I_w,I_v}=b^Z_{v,I_w}.
\]
In particular, $\bbz^{I_v}_{I_w,I_v}$ does not depend on the choice of $I_v$. 
\end{cor}
\begin{proof}Denote $Z_{I_w}=\sum_{v\le w}a^Z_{I_w,v}\de_v$. Then the matrix $\fa^Z$ with $\fa^Z(u,v)=a^Z_{I_v,u}$ is the inverse of $\fb^Z$. Theorem \ref{thm:res} implies that 
\begin{align*}
\bbz^{I_v}_{I_w, I_v}&=\fp^Z_w(v,v)\\
&=\sum_{z_1,z_2\in W}\fb^Z(v,z_1)\fb^Z_w(z_1,z_2)\fa^Z(z_2,v)\\
&=\sum_{z_1\ge v,z_2\le v}b^Z_{{z_1}, I_v}\de_{z_1,z_2}b^Z_{z_1,I_w}a^Z_{I_v,{z_2}}\\
&=\sum_{v\le z_1\le v}b^Z_{z_1,I_v}b^Z_{z_1,I_w}a^Z_{I_v, z_1}\\
&=b^Z_{v,v}b^Z_{v,I_w}a^Z_{v,v}=b^Z_{v,I_w}=Z^*_{I_w}(\de_v). 
\end{align*}
\end{proof}

\begin{rem}While Theorem \ref{thm:res} is interesting itself, Corollary \ref{cor:res} can also be proved easily by the following argument: since $Z_{I_w}^*(\de_u)=b^Z_{u,I_w}=0$ unless $u\ge w$, so 
\begin{align*}
(Z_{I_u}^*\cdot Z_{I_v}^*)(\de_u)&=(\sum_{w\ge u, w\ge v}\bbz^{I_w}_{I_u, I_v}Z_{I_w}^*)(\de_u)\\
&=\bbz^{I_u}_{I_u, I_v}Z_{I_u}^*(\de_u)=\bbz^{I_u}_{I_u, I_v}b^Z_{u, I_u}.
\end{align*}
On the other hand side, 
\[(Z_{I_u}^*\cdot Z_{I_v}^*)(\de_u)=Z_{I_u}^*(f_u)Z_{I_v}^*(\de_u)=b^Z_{u, I_u}b^Z_{u, I_v}.\] Therefore, $\bbz^{I_u}_{I_u,I_v}=b^Z_{u, I_v}$. 
\end{rem}


\begin{rem}As mentioned in \cite{GK19}, specializing Corollary \ref{cor:res} and Examples \ref{ex:FaSch} and \ref{ex:FmSch} to singular cohomology or K-theory, and $Z_i$ to the the $X_i$ and $Y_i$-operators, one recover the AJS/Billey formula and Graham-Willems formula of restriction coefficients of Schubert classes, which are obtained by using root polynomials.
\end{rem}



\begin{ex}Consider the $A_2$-case with $w=s_1, v=s_1s_2s_1$. We compute $b^X_{v,w}=X_{I_w}^*(\de_v)$. For $\bfx^{I_v}_{[3], E}$, we only need to consider the following three:
\begin{align*}
\bfx^{I_v}_{[3], \{1\}}=-x_1, & \quad \bfx^{I_v}_{[3], \{3\}}=-x_{2},\quad \bfx^{I_v}_{[3], \{1,3\}}=x_{1}x_2.
\end{align*}
On the other hand, $c^X_{I_v|_E,I_w}=1$ when $E=\{1\}, \{3\}$, and $X_1X_1=\ka_1X_1$. So $c^X_{I_v|_{\{1,3\}}, s_1}=\ka_1$. Therefore, 
\begin{align*}
b^X_{w,I_v}=-x_1-x_2+\ka_1x_1x_2.
\end{align*}
In particular, if $F=F_a$, then $b^X_{w,I_v}=-x_1-x_2$, and if $F=F_{m}$, then $b^X_{w,I_v}=-x_1-x_2+x_1x_2=-x_{1+2}$. 
\end{ex}

\begin{ex}Let $w=s_1s_2, v=s_1s_2s_3s_1s_2$. Let us compute $b^X_{v,I_w}=X_{I_w}^*(\de_v)$. We write $X_{ijk\cdots}$ for $X_{i}X_jX_k\cdots$.  To compute $\bfx^{I_v}_{[6], E}$, we only need to consider
\begin{align*}
\bfx^{I_v}_{[6], \{1,2\}}=x_1x_{1+2},& \quad
\bfx^{I_v}_{[6], \{1,5\}}=x_1x_{2+3},\\
\bfx^{I_v}_{[6], \{4,5\}}=x_2x_{2+3},&\quad
\bfx^{I_v}_{[6], \{1,2,5\}}=-x_1x_{1+2}x_{2+3},\\
\bfx^{I_v}_{[6], \{1,4,5\}}=-x_1x_2x_{2+3},&\quad
\bfx^{I_v}_{[6], \{1,2,4,5}\}=x_1x_{1+2}x_2x_{2+3}.
\end{align*}
On the other hand, $c^X_{I_v|_{E}, I_w}=1$ when $E=\{1,2\}, \{1,5\}, \{4,5\}$. Concerning $X_{I_w|_\{1,2,5\}}=X_{122}$, since \[X_1X_2X_2=X_1\ka_2X_2=s_1(\ka_2)X_{12}+\Delta_1(\ka_2)X_{2}=\ka_{1+2}X_{12}+\Delta_1(\ka_2)X_2,\]
so 
\[c^X_{I_v|_{\{1,2,5\}}, I_w}=\kappa_{1+2}.\]
 For $X_{I_w|_{\{1,4,5\}}}=X_{112}$, from $X_1X_1X_2=\ka_1X_{1}X_2$, we get \[c^X_{I_v|_{\{1,4,5\}}, I_w}=\kappa_1.\]
 Lastly, for $X_{I_w|_{\{1,2,4,5\}}}=X_{1122}$, from Lemma \ref{lem:1} we know 
\begin{align*}
X_{1212}&=X_1(X_{121}+\ka_{12}X_1-\ka_{21}X_2)\\
&=\ka_1X_{121}+X_1\ka_{12}X_1-X_1\ka_{21}X_2\\
&=\ka_1X_{121}+s_1(\ka_{12})X_1^2+\Delta_1(\ka_{12})X_1-s_1(\ka_{21})X_{12}-\Delta_1(\ka_{21})X_2\\
&=\ka_1X_{121}+\ka_{-1, 1+2}\ka_1X_1+\Delta_1(\ka_1)X_1-\ka_{1+2,-1}X_{12}-\Delta_1(\ka_{21})X_2, 
\end{align*}
so 
\[c^X_{I_v|_{\{1,2,4,5\}}, I_w}=s_1(\ka_{21})=-\ka_{1+2,-1}.\]
Therefore, 
\begin{align*}
&b^X_{s_1s_2,s_1s_2s_3s_1s_2}\\
&=\bfx^{I_v}_{[6], \{1,2\}}+\bfx^{I_v}_{[6], \{1,5\}}+\bfx^{I_v}_{[6], \{4,5\}}\\
&+
\bfx^{I_v}_{[6], \{1,2,5\}}\ka_{1+2}+\bfx^{I_v}_{[6], \{1,4,5\}}\ka_1+\bfx^{I_v}_{[6], \{1,2,4,5\}}(-\ka_{1+2, -1})\\
&=x_1x_{1+2}+x_1x_{2+3}+x_2x_{2+3}-x_1x_{1+2}x_{2+3}\ka_{1+2}-x_1x_2x_{2+3}\ka_1-x_1x_{1+2}x_{2+3}\ka_{1+2,-1}\\
&=x_1x_{1+2}+x_1x_{2+3}+x_2x_{2+3}-x_{2+3}(x_1+x_2+\frac{x_1}{x_{-1}}x_{1+2}).
\end{align*}
In particular, if $F=F_a$, then 
\[b^X_{s_1s_2,s_1s_2s_3s_1s_2}=\al_1(\al_{1}+\al_{2})+\al_1(\al_{2}+\al_{3})+\al_2(\al_{2}+\al_{3}).\]
 If $F=F_m$, then 
\[b^X_{s_1s_2,s_1s_2s_3s_1s_2}=x_1x_{1+2}+x_1x_{2+3}+x_2x_{2+3}-x_1x_{2+3}(x_{1+2}+x_2).\]
 These agree with the result computed by using root polynomials. 
\end{ex}

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