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\title[Deforming motivic Segre classes]{Deforming the motivic Segre classes of Schubert cells in the Grassmannian}

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\author[Raj Gandhi]{
\texorpdfstring{Raj Gandhi\thanks{\href{mailto:rg593@cornell.edu}{rg593@cornell.edu}.}\addressmark{1}}{Raj Gandhi}}
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\address{\addressmark{1}Department of Mathematics, Cornell University, Ithaca, NY 14853, USA}

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\abstract{We $\beta$-deform the motivic Segre classes of Schubert cells in the $d$-step flag variety, conjecturing that the $\beta$-deformations come from analogues of stable envelopes in the equivariant connective $K$-ring of the cotangent bundle of the flag variety. Our main result is a combinatorial formula for the structure constants in the $\beta$-deformed basis in the $d=1$ case using Knutson--Tao puzzles.}

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\keywords{divided difference operators, Knutson--Tao puzzles, connective $K$-theory}

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\section{Introduction}


Consider the $d$-step flag variety $\mrm{Fl}_n(i_1,\dotsc,i_d)$ consisting of flags in $\mbb{C}^n$ of the form
\[(0)\subseteq V_1\subseteq V_2\subseteq \dotsb \subseteq V_{d-1}\subseteq V_d\subseteq \mbb{C}^n,\quad \mrm{dim}(V_j)=i_j.\]
When $d=1$, the flag variety $\mrm{Fl}_n(k)$ is the Grassmannian $\mrm{Gr}(k,n)$ of $k$-planes in $\mbb{C}^n$. The flag variety $\mrm{Fl}_n(i_1,\dotsc,i_d)$ has a distinguished cell decomposition called the Bruhat decomposition, whose cells are indexed by $012\dotsb d$ sequences $\lambda=\lambda_1\dotsb \lambda_n$, where $d$ appears $i_1$ times, $d-1$ appears $i_{2}-i_{1}$ times, and so on. We will denote the set of such sequences by $\Gamma$. The cells defining the Bruhat decomposition are called the Schubert cells $X_\lambda^\circ$, $\lambda\in\Gamma$, and the closures of the Schubert cells are the Schubert varieties $X_\lambda:=\overline{X_\lambda^\circ}$. There is a natural action of the $n$-dimensional complex torus $T$ on $\mrm{Fl}_n(i_1,\dotsc,i_d)$, and the Schubert varieties $X_\lambda$ are invariant under this torus action. The varieties $X_\lambda$ define natural classes $S_\lambda$, called Schubert classes, in the $T$-equivariant cohomology ring $H_T(\mrm{Fl}_n(i_1,\dotsc,i_d))$. In fact, the Schubert classes form a basis for $H_T(\mrm{Fl}_n(i_1,\dotsc,i_d))$ over $H_T(\mrm{pt})\simeq \mbb{Z}[y_1,\dotsc,y_n]$, where $\mrm{pt}$ denotes a point. The Littlewood--Richardson coefficients $c_{\lambda,\mu}^\nu$ are the structure constants in the Schubert basis:
\[S_\lambda\cdot S_\mu=\sum_{\nu}c_{\lambda,\mu}^\nu S_\nu.\]
In the paper \cite{KT03}, Knutson and Tao introduced the first manifestly positive formula for the Littlewood--Richardson coefficients $c_{\lambda,\mu}^\nu$ in the $T$-equivariant cohomology ring ${H}_T(\mrm{Gr}(k,n))$ of the Grassmannian $\mrm{Gr}(k,n)$. The combinatorial objects they used to compute the Littlewood--Richardson coefficients are called \textit{puzzles}. 

In \cite{MO19} and \cite{O17}, Maulik and Okounkov defined natural geometric classes, called \emph{stable classes}, in the $(T\times\mbb{C}^\times)$-equivariant cohomology ring and the $(T\times\mbb{C}^\times)$-equivariant $K$-ring of the cotangent bundle $T^*(\mrm{Fl}_n(i_1,\dotsc,i_d))$ of $\mrm{Fl}_n(i_1,\dotsc,i_d)$. Dividing the stable classes in $H_{T\times\mbb{C}^\times}(T^*(\mrm{Fl}_n(i_1,\dotsc,i_d)))$ and $K_{T\times\mbb{C}^\times}(T^*(\mrm{Fl}_n(i_1,\dotsc,i_d)))$ by an appropriate constant class gives the \emph{Segre--Schwartz--MacPherson (SSM) classes} and \emph{motivic Segre classes} of Schubert cells, respectively. The SSM and motivic Segre classes of Schubert cells are both indexed by $\Gamma$. In \cite{KZJ17} and \cite{KZJ21}, Knutson and Zinn-Justin proved combinatorial formulas for the structure constants in the bases of SSM classes and motivic Segre classes when $d=1,2$ using puzzles by applying the theory of integrable systems.

In this paper, we define a one-parameter ($\beta$-)deformation of the motivic Segre classes of Schubert cells for $T^*(\mrm{Fl}_n(i_1,\dotsc,i_d))$. Evaluating our deformed classes at $\beta=1$ recovers the motivic Segre classes of Schubert cells, and evaluating at $\beta=0$ recovers the SSM classes of Schubert cells. Our main result is \Cref{thm:main}, which is a combinatorial formula for the structure constants in the $\beta$-deformed basis in the case $d=1$ using puzzles. In \Cref{conj:geometric}, we conjecture that our $\beta$-deformed classes arise as analogues of stable classes in the $(T\times\mbb{C}^\times)$-equivariant \emph{connective} $K$-ring of $T^*(\mrm{Fl}_n(i_1,\dotsc,i_d))$.

\section{Preliminaries and notation}



\subsection{Flag varieties}\label{subsection:flag} Let $G=\mrm{GL}_n(\mathbb{C})$. Fix a Borel subgroup $B$ in $G$, and a maximal torus $T$ in $B$. Denote the weight lattice of $T$ by $\Lambda:=\mrm{Hom}(T,\mathbb{C}^\times)$, and denote the root system of $T$ by $\Sigma:=A_{n-1}$. We denote the set of simple roots of $\Sigma$ by $\Delta:=\{\alpha_1,\dotsc,\alpha_{n-1}\}$. The root system $\Sigma$ decomposes as $\Sigma=\Sigma^+\sqcup \Sigma^-$, where $\Sigma^+$ (resp. $\Sigma^-$) is the set of positive (resp. negative) roots of $\Sigma$. For a subset $\Theta:=\{\alpha_{i_1},\dotsc,\alpha_{i_d}\}$, $i_1<i_2<\dotsb<i_d$, of the simple roots $\Delta$, we denote by $P_\Theta$ a parabolic subgroup of $G$ that contains $B$ and has (negative) simple roots $\Delta\setminus \Theta$. The $d$-step flag variety $G/P_\Theta$ is the variety of (partial) flags of vector spaces,
\[(0)\subseteq V_1\subseteq V_2\subseteq \dotsb\subseteq V_{d-1}\subseteq V_d\subseteq \mbb{C}^n, \quad \mrm{dim}(V_j)= i_j.\]
In the special case $\Theta=\Delta$, the flag variety $G/P_\Theta$ is the complete flag variety $G/B$. In the special case $\Theta=\{\alpha_k\}$, the group $P_\Theta$ is a maximal parabolic subgroup of $G$, and $G/P_\Theta$ is the Grassmannian $\mrm{Gr}(k,n)$ of $k$-planes in $\mathbb{C}^n$. A \textbf{negative root} of $G/P_\Theta$ is a root in $\Sigma$ that is \textit{not} a root of $P_\Theta$. Let $\Sigma_\Theta^-$ be the set of negative roots of $G/P_\Theta$, and let $\Sigma_\Theta^+:=-\Sigma_\Theta^-$ be the set of \textbf{positive roots} of $G/P_\Theta$. Define $\Sigma_\Theta:=\Sigma_\Theta^+\sqcup \Sigma_\Theta^-$. We will denote the Weyl group $S_n$ of $G$ by $W$. Let $r_i$ be the simple transposition in $W$ that swaps $i$ and $i+1$ and fixes all other $j$. The subgroup $W_{\Theta}$ of $W$ generated by $\langle r_j \rangle_{\alpha_j\in\Delta\setminus \Theta}$ is the Weyl group of $P_\Theta$. Let $W^\Theta$ be the set of minimal coset representatives of $W/W_\Theta$ in $W$. We will henceforth identify the set $W^\Theta$ with the quotient $W/W_\Theta$. Define $p_j:=i_j-i_{j-1}$. The set $W^\Theta$ is in bijection with the set $\Gamma$ of lists of length $n$ in the alphabet $\{0,\dotsc,d\}$, where $d$ appears $p_1$ times, $d-1$ appears $p_{2}$ times, and so on. Under this identification, the longest word $w_0$ in $W^\Theta$ is identified with the list $d^{p_1} (d-1)^{p_{2}} \dotsb 1^{p_d} 0^{p_{d+1}}$, and $w\cdot w_0\in W^\Theta$ is identified with the list $w(d^{p_1} (d-1)^{p_{2}} \dotsb 1^{p_d} 0^{p_{d+1}})$. 

For each $w\in W^\Theta$, there is a Schubert cell $X_w^\circ=BwP_\Theta/P_\Theta$ in $G/P_\Theta$. The Schubert cells form a cell decomposition for $G/P_\Theta$. The closures of the Schubert cells $X_w:=\overline{X_w^\circ}$ are Schubert varieties. Let $\mrm{h}^*(-)$ be one of the following (generalized) cohomology theories: singular cohomology $H^*(-)$, $T$-equivariant cohomology $H_T(-)$, $K$-theory $K(-)$, and $T$-equivariant $K$-theory $K_T(-)$. Let $\mrm{pt}$ be a point, so that $H^*(\mrm{pt})\simeq \mbb{Z}$, $H_T(\mrm{pt})\simeq \mbb{Z}[y_1,\dotsc,y_n]$, $K(\mrm{pt})\simeq \mbb{Z}$, and $K_T(\mrm{pt})\simeq \mbb{Z}[e^{\pm y_1},\dotsc,e^{\pm y_n}]$. In $\mrm{h}^*(G/P_\Theta)$ there are natural classes $S_w$, known as Schubert classes, that form a basis for $\mrm{h}^*(G/P_\Theta)$, when we view $\mrm{h}^*(G/P_\Theta)$ as a module over $\mrm{h}^*(\mrm{pt})$. In $K(G/P_\Theta)$ and $K_T(G/P_\Theta)$, the Schubert class $S_w$ is the class of the structure sheaf $[\mcal{O}_{X_w}]$ of $X_w$. In $H^*(G/P_\Theta)$ and $H_T(G/P_\Theta)$, the Schubert class $S_w$ is the class $[X_w]$ defined by the Schubert variety $X_w$. 


\subsection{Cotangent bundles of flag varieties}\label{subsection:cotangent} The cotangent bundle $T^*(G/P_\Theta)$ of $G/P_\Theta$ has a natural action of $\widehat{T}:=T\times \mbb{C}^\times$, where $(t,z)\in \widehat{T}$ acts by $(t,z)\cdot (x,\vec{v}):=(t\cdot x,(t^{-1})^*(z\cdot \vec{v}))$ for all $(x,\vec{v})\in T_x^*(G/P_\Theta)$. For any symplectic resolution $X\to Y$ with an algebraic $\widehat{T}$-action subject to certain conditions, Maulik and Okounkov defined natural geometric bases in $H_{\widehat{T}}^*(X)$ and $K_{\widehat{T}}(X)$ (\cite{MO19}, \cite{O17}), which arise from $\widehat{T}$-invariant cycles in $X$. These bases are called the cohomological and $K$-theoretic stable bases. When $X=T^*(G/P_\Theta)$, the cohomological and $K$-theoretic stable bases are indexed by $W^\Theta$, and we will denote the stable basis in either case by $\{\mrm{St}_\lambda\}_{\lambda\in W^\Theta}$. Consider the localizations $H_{\widehat{T}}^{\mrm{loc}}(\mrm{pt})$ and $K_{\widehat{T}}^{\mrm{loc}}(\mrm{pt})$ of the rings $H_{\widehat{T}}(\mrm{pt})\simeq H_T(\mrm{pt})\otimes_\mbb{Z}\mbb{Z}[\hbar]$ and $K_{\widehat{T}}(\mrm{pt})\simeq K_T(\mrm{pt})\otimes_\mbb{Z} \mbb{Z}[q^{\pm 2}]$ at $\{\hbar+\alpha\}_{\alpha\in\Sigma}$ and $\{1-q^2e^{\alpha}\}_{\alpha\in\Sigma}$, respectively.
Here, $\hbar$ and $q^{2}$ denote the equivariant parameters coming from the $\mbb{C}^\times$-factor of $\widehat{T}$ that dilates the cotangent fibres of $T^*(G/P_\Theta)$.
 Set $H_{\widehat{T}}^{\mrm{loc}}(T^*(G/P_\Theta)):=H_{\widehat{T}}^{\mrm{loc}}(\mrm{pt})\otimes_{H_{\widehat{T}}(\mrm{pt})} H_{\widehat{T}}^*(T^*(G/P_\Theta))$ and $K_{\widehat{T}}^{\mrm{loc}}(T^*(G/P_\Theta)):=K_{\widehat{T}}^{\mrm{loc}}(\mrm{pt})\otimes_{K_{\widehat{T}}(\mrm{pt})} K_{\widehat{T}}(T^*(G/P_\Theta))$. Let $\kappa$ be the class of the zero section of $T^*(G/P_\Theta)\to G/P_\Theta$ in either $H_{\widehat{T}}(T^*(G/P_\Theta))$ or $K_{\widehat{T}}(T^*(G/P_\Theta))$. The elements $S_\lambda:=\tfrac{\mrm{St}_\lambda}{\kappa}$ live in the localization $H_{\widehat{T}}^{\mrm{loc}}(T^*(G/P_\Theta))$ or the localization $K_{\widehat{T}}^{\mrm{loc}}(T^*(G/P_\Theta))$, and $\left\{S_\lambda\right\}_{\lambda\in W^\Theta}$ forms a basis for $H_{\widehat{T}}^{\mrm{loc}}(T^*(G/P_\Theta))$ or $K_{\widehat{T}}^{\mrm{loc}}(T^*(G/P_\Theta))$ over the fraction field $\mrm{Frac}(H_{\widehat{T}}(\mrm{pt}))$ or the fraction field $\mrm{Frac}(K_{\widehat{T}}(\mrm{pt}))$, respectively. The elements $S_\lambda$ in $H_{\widehat{T}}^{\mrm{loc}}(T^*(G/P_\Theta))$ are called \textbf{Segre--Schwartz--MacPherson (SSM) classes}, and the elements $S_\lambda$ in $K_{\widehat{T}}^{\mrm{loc}}(T^*(G/P_\Theta))$ are called \textbf{motivic Segre classes}.



 \subsection{Equivariant localization and divided difference operators}\label{subsection:localization}

Let $X$ be a smooth complex algebraic variety, and suppose $E$ is a complex linear algebraic group acting on $X$ algebraically with finitely many fixed points $F$. For both $h_E^*(-)=H_E^*(-)$ and $h_E^*(-)=K_E(-)$, the localization map
\[\iota\colon h_E^*(X)\to \bigoplus_{f\in  F}h_E^*(\mrm{pt}),\]
defined by restricting $h_E^*(X)$ to the corresponding fixed point $f$ on each component, is injective.
We will consider the case $X=G/P_\Theta$ and $E=T$, or the case $X=T^*(G/P_\Theta)$ and $E=\widehat{T}$. In either case, the $E$-fixed points of $X$ are indexed by $W^\Theta$. We will now describe how to obtain the images of the classes $S_\lambda$ in the rings $H_T^*(G/P_\Theta)$, $H_{\widehat{T}}^{\text{loc}}(T^*(G/P_\Theta))$, $K_T(G/P_\Theta)$, and $K_{\widehat{T}}^{\text{loc}}(T^*(G/P_\Theta))$ under the localization map. 

Given $\alpha_i\in\Delta$, there is a $\mathbb{Z}$-linear operator $\partial_i$ that acts on $\bigoplus_{\lambda\in W^\Theta}h_E^*(\mrm{pt})$, which we will define explicitly below. Any $w\in W^\Theta$ can be expressed as the reduced product of simple transpositions $w=r_{j_1}\dotsb r_{j_k}$. The operator $\partial_w:=\partial_{j_1}\circ\dotsb\circ \partial_{j_k}$ also acts on $\bigoplus_{\lambda\in W^\Theta}h_E^*(\mrm{pt})$ and is independent of the choice of reduced expression for $w$. We have $\iota(S_{w\cdot w_0})=\partial_w(\iota(S_{w_0}))$, where $w_0$ is the longest word in $W^\Theta$. We will now explicitly describe the restrictions $S_{w_0}|_\lambda$ of $S_{w_0}$ to each fixed point $\lambda\in W^\Theta$, and we will give an explicit formula defining the operator $\partial_i$ in each of the four cases $H_T^*(G/P_\Theta)$, $H_{\widehat{T}}^{\text{loc}}(T^*(G/P_\Theta))$, $K_T(G/P_\Theta)$, and $K_{\widehat{T}}^{\text{loc}}(T^*(G/P_\Theta))$. Given $\alpha=c_1\alpha_1+\dotsb+c_{n-1}\alpha_{n-1}\in\Sigma$, we denote by $y_\alpha$ the element $c_1(y_1-y_2)+c_2(y_2-y_3)+\dots+c_{n-1}(y_{n-1}-y_n)\in\mbb{Z}[y_1,\dotsc,y_n]$. 
\begin{enumerate}
	\item $H_T^*(G/P_\Theta)$. Below are the formulas for $S_{w_0}$ restricted to fixed points $\lambda\in W^\Theta$:	
	\[S_{w_0}|_\lambda=\begin{cases}
		\prod_{\alpha\in \Sigma_\Theta^+}y_{-\alpha},& \lambda=w_0;\\
		0,& \lambda\neq w_0.
	\end{cases}\]		
	There is a natural action of $W$ on the ring $H_T^*(\mrm{pt})\simeq \mbb{Z}[y_1,\dotsc,y_n]$, defined on generators by $w(y_i):=y_{w(i)}$ for all $w\in W$. This $W$-action extends to the ring $\bigoplus_{\lambda\in W^\Theta}H_T^*(\mrm{pt})$ by $w(f_\lambda)_{\lambda}:=(w(f_\lambda))_{w(\lambda)}$ for all $w\in W$. 
	Consider the $\mbb{Z}$-linear operators on $H_T^*(\mrm{pt})$:
	\[\partial_{i}(f)=\frac{f-r_i(f)}{y_{\alpha_i}},\quad f\in H_T^*(\mrm{pt}),\text{ } \alpha_i\in\Delta.\]
	The operators $\partial_i$ naturally define $\mbb{Z}$-linear operators on $\bigoplus_{\lambda\in W^\Theta}H_T^*(\mrm{pt})$. 
	\item  $H_{\widehat{T}}^{\text{loc}}(T^*(G/P_\Theta))$. Below are the formulas for $S_{w_0}$ restricted to fixed points $\lambda\in W^\Theta$:	
	\[S_{w_0}|_\lambda=\begin{cases}
		\prod_{\alpha\in \Sigma_\Theta^+} \tfrac{y_{-\alpha}}{y_{-\alpha}-\hbar},& \lambda=w_0;\\
		0,& \lambda\neq w_0.
	\end{cases}\]		
	There is a natural action of $W$ on the ring $H_{\widehat{T}}^*(\mrm{pt})\simeq \mbb{Z}[y_1,\dotsc,y_n,\hbar]$, defined on generators by $w(y_i):=y_{w(i)}$ and $w(\hbar)=\hbar$ for all $w\in W$. This $W$-action extends  to the ring $\bigoplus_{\lambda\in W^\Theta}H_{\widehat{T}}^{\text{loc}}(\mrm{pt})$ by $w(f_\lambda)_{\lambda}:=(w(f_\lambda))_{w(\lambda)}$ for all $w\in W$. 
	Consider the $\mbb{Z}[\hbar]$-linear operators on $H_{\widehat{T}}^{\text{loc}}(\mrm{pt})$:
	\[\partial_{i}(f)=\frac{\hbar}{y_{\alpha_i}}f+\frac{y_{\alpha_i}-\hbar}{y_{\alpha_i}}r_i(f),\quad f\in H_{\widehat{T}}^{\text{loc}}(\mrm{pt}),\text{ } \alpha_i\in\Delta.\]
	The operators $\partial_i$ naturally define $\mbb{Z}[\hbar]$-linear operators on $\bigoplus_{\lambda\in W^\Theta}H_{\widehat{T}}^{\text{loc}}(\mrm{pt})$. 
	\item $K_T(G/P_\Theta)$. Below are the formulas for $S_{w_0}$ restricted to fixed points $\lambda\in W^\Theta$:	
	\[S_{w_0}|_\lambda=\begin{cases}
		\prod_{\alpha\in\Sigma_\Theta^+} (1-e^{y_{-\alpha}}),& \lambda=w_0;\\
		0,&\lambda\neq w_0.
	\end{cases}\]		
	There is a natural action of $W$ on the ring $K_T(\mrm{pt})\simeq \mbb{Z}[e^{\pm y_1},\dotsc,e^{\pm y_n}]$, defined on generators by $w(e^{\pm y_i}):=e^{\pm y_{w(i)}}$ for all $w\in W$. This $W$-action extends to the ring $\bigoplus_{\lambda\in W^\Theta}K_T(\mrm{pt})$ by $w(f_\lambda)_{\lambda}:=(w(f_\lambda))_{w(\lambda)}$ for all $w\in W$. 
	Consider the $\mbb{Z}$-linear operators on $K_T(\mrm{pt})$:
	\[\partial_{i}(f)=\frac{f}{1-e^{-y_{\alpha_i}}}+\frac{r_i(f)}{1-e^{y_{\alpha_i}}},\quad f\in K_T(\mrm{pt}),\text{ } \alpha_i\in\Delta.\]
	The operators $\partial_i$ naturally define $\mbb{Z}$-linear operators on $\bigoplus_{\lambda\in W^\Theta}K_T(\mrm{pt})$. 
	\item $K_{\widehat{T}}^{\text{loc}}(T^*(G/P_\Theta))$. Below are the formulas for $S_{w_0}$ restricted to fixed points $\lambda\in W^\Theta$:	
	\[S_{w_0}|_\lambda=\begin{cases}
		\prod_{\alpha\in\Sigma_\Theta^+} \tfrac{1-e^{y_{-\alpha}}}{1-q^2e^{y_{-\alpha}}},& \lambda=w_0;\\
		0,&\lambda\neq w_0.
	\end{cases}\]		
	There is a natural action of $W$ on the ring $K_{\widehat{T}}(\mrm{pt})\simeq \mbb{Z}[e^{\pm y_1},\dotsc,e^{\pm y_n},q^{\pm 2}]$, defined on generators by $w(e^{\pm y_i}):=e^{\pm y_{w(i)}}$ and $w(q^{\pm 2})=q^{\pm 2}$ for all $w\in W$. This $W$-action extends to the ring $\bigoplus_{\lambda\in W^\Theta}K_{\widehat{T}}^{\text{loc}}(\mrm{pt})$ by $w(f_\lambda)_{\lambda}:=(w(f_\lambda))_{w(\lambda)}$ for all $w\in W$. 
	Consider the $\mbb{Z}[q^{\pm2}]$-linear operators on $K_{\widehat{T}}^{\text{loc}}(\mrm{pt})$:
	\[\partial_{i}(f)=\frac{1-q^2}{1-e^{-y_{\alpha_i}}}f+\frac{1-q^2e^{y_{\alpha_i}}}{1-e^{y_{\alpha_i}}}r_i(f),\quad f\in K_{\widehat{T}}^{\text{loc}}(\mrm{pt}),\text{ } \alpha_i\in\Delta.\]
	The operators $\partial_i$ naturally define $\mbb{Z}[q^{\pm 2}]$-linear operators on $\bigoplus_{\lambda\in W^\Theta}K_{\widehat{T}}^{\text{loc}}(\mrm{pt})$. 
\end{enumerate}




\section{Deformation of the motivic Segre classes}\label{section:deformation}

\subsection{The Rees ring}\label{subsection:Rees} Consider the polynomial ring $(\mbb{Z}[\beta])[x_{\lambda}]_{\lambda\in{\Lambda}}$ over $\mbb{Z}[\beta]$ in variables $x_\lambda$ with $\lambda\in{\Lambda}$. Let $J_\Lambda$ be the ideal $(x_0,\text{ }x_{\lambda+\mu}-x_\lambda-x_\mu+\beta x_\lambda x_\mu,\text{    }\lambda,\mu\in{\Lambda})$ in $(\mbb{Z}[\beta])[x_{\lambda}]_{\lambda\in{\Lambda}}$. The definition of \textit{Rees ring} below is analogous to the definition in \cite[Example 2.1]{LZ17}.

\begin{dfn}
	Set $S_\Lambda:=((\mbb{Z}[\beta])[x_{\lambda}]_{\lambda\in{\Lambda}})/J_\Lambda$.
	The \textbf{Rees ring} $R_\Lambda$ is the ring
	\[R_\Lambda:=\mbb{Z}[q^{\pm2}]\otimes_{\mbb{Z}} S_\Lambda.\]
\end{dfn}

\begin{lem}\label{lem:int-domain}
	The ring $R_\Lambda$ is an integral domain.
\end{lem}
\begin{proof}
	As $R_\Lambda=\mbb{Z}[q^{\pm2}]\otimes_{\mbb{Z}}S_\Lambda$, it is enough to show that $S_\Lambda$ is an integral domain.
	Let $p\colon (\mbb{Z}[\beta])[x_\lambda]_{\lambda\in\Lambda}\to S_\Lambda$ be the projection onto the quotient. Consider the ideal $I_\Lambda$ of $(\mbb{Z}[\beta])[x_\lambda]_{\lambda\in\Lambda}$ generated by the elements $x_\lambda$ with $\lambda\in\Lambda$. Set $I_\Lambda^0:=S_\Lambda$. Then we have $\cap_{i\geq 0}I_\Lambda^i=(0)$. The ideal $J_\Lambda$ is contained in $I_\Lambda$. Therefore, $\cap_{i\geq 0}p(I_\Lambda)^i=(0)$ as well. Form the associated graded ring $G:=\oplus_{i\geq 0}p(I_\Lambda)^{i}/p(I_\Lambda)^{i+1}$. Let $H$ be $H_{T}(\mrm{pt})[\beta]$, the $T$-equivariant cohomology ring of a point extended by $\beta$. The argument in \cite[Lemma 4.2]{CPZ} implies there is a ring isomorphism $H\to G$. As $G$ is isomorphic to the integral domain $H$ and $\cap_{i\geq 0}p(I_\Lambda)^i=(0)$, it follows that $S_\Lambda$ is an integral domain.
\end{proof}
It follows from \Cref{lem:int-domain} that the localization $(R_\Lambda)_\beta$ of $R_\Lambda$ at $\beta$ is an integral domain. There is an isomorphism $(R_\Lambda)_\beta\to K_{\widehat{T}}(\mrm{pt})[\beta,\beta^{-1}]$ given by the following map:
\begin{equation}\label{eqn:ReesKiso}
	(R_\Lambda)_\beta\to K_{\widehat{T}}(\mrm{pt})[\beta,\beta^{-1}],\quad x_\lambda\mapsto \beta^{-1}(1-e^\lambda),\quad q^2\mapsto q^2,\quad \beta\mapsto\beta,\quad \lambda \in \Lambda.
\end{equation}
The inverse sends $ e^\lambda\mapsto 1-\beta x_\lambda$ for all $\lambda\in\Lambda$.
Interestingly, there is an isomorphism of $R_\Lambda/(\beta R_\Lambda)$ with $H_{\widehat{T}}(\mrm{pt})[(1+\hbar)^{-1}]$ induced by the map
\begin{equation}\label{eqn:ReesHiso}
	R_\Lambda/(\beta R_\Lambda)\to H_{\widehat{T}}^*(\mrm{pt})[(1+\hbar)^{-1}],\quad x_\lambda\mapsto \lambda,\quad q^2\mapsto 1+\hbar,\quad \lambda\in\Lambda.
\end{equation}
There is an action of $W$ on $R_\Lambda$ defined by generators by $w\cdot q^2=q^2$, $w\cdot \beta=\beta$, and $w\cdot x_\lambda=x_{w(\lambda)}$ for $w\in W$ and $\lambda\in\Lambda$, which intertwines the isomorphisms \eqref{eqn:ReesKiso} and \eqref{eqn:ReesHiso}.

\subsection{Deformed divided difference operators}\label{subsection:deformed-ddo}

\begin{dfn}\label{dfn:operator}
	For a simple root $\alpha\in\Delta$, we define the following $\mbb{Z}[q^{\pm 2}][\beta]$-linear operator on $R_\Lambda$:
	\[\partial_\alpha=\frac{1-q^2}{x_{-\alpha}}+\frac{1-q^2(1-x_\alpha)}{x_\alpha}r_\alpha.\]
\end{dfn}

\begin{rmk}
	A similar ``Demazure--Lusztig" operator is defined in \cite[p. 61]{ZZ17} (for more general oriented cohomology theories). The authors denote their operators by $T_\alpha^F$.
\end{rmk}


Write $w\in W$ as a reduced product of simple transpositions $w=r_{j_1}\cdots r_{j_k}$. We use the notation $\partial_w:=\partial_{\alpha_{j_1}}\circ\dotsb\circ \partial_{\alpha_{j_k}}$. The lemma below shows that $\partial_w$ is independent of the choice of reduced decomposition for $w$.

\begin{lem}\label{lem:independent}
	The operators $\partial_{\alpha}$ satisfy the following relations:
	\begin{enumerate}
		\item $\partial_{\alpha_i}\circ \partial_{\alpha_j}=\partial_{\alpha_j}\circ \partial_{\alpha_i}$ whenever $|i-j|>1$.
		\item $\partial_{\alpha_i}\circ \partial_{\alpha_{i+1}}\circ\partial_{\alpha_i}=\partial_{\alpha_{i+1}}\circ\partial_{\alpha_{i}}\circ\partial_{\alpha_{i+1}}$ for all $i=1,\dotsc,n-2$.
		\item $(\partial_{\alpha_i}+q^2)\circ(\partial_{\alpha_i}+(q^2\beta-q^2-\beta))=0$ for all $i=1,\dotsc,n-1$.
	\end{enumerate}
\end{lem}

\begin{rmk}
	The operator $\partial_\alpha$ has an ``inverse" 
	\[\widetilde{\partial}_\alpha=\frac{1}{q^2(\beta+q^2-\beta q^2)}\left(\frac{q^2-1}{x_\alpha}+\frac{1-q^2(1-x_\alpha)}{x_\alpha}r_\alpha\right)\]
	in the sense that $\partial_\alpha\circ \widetilde{\partial}_\alpha=\widetilde{\partial}_\alpha\circ\partial_\alpha=1$. The operators $\widetilde{\partial}_\alpha$ satisfy the following relations:
	\begin{enumerate}
		\item $\widetilde{\partial}_{\alpha_i}\circ \widetilde{\partial}_{\alpha_j}=\widetilde{\partial}_{\alpha_j}\circ \widetilde{\partial}_{\alpha_i}$ whenever $|i-j|>1$.
		\item $\widetilde{\partial}_{\alpha_i}\circ \widetilde{\partial}_{\alpha_{i+1}}\circ\widetilde{\partial}_{\alpha_i}=\widetilde{\partial}_{\alpha_{i+1}}\circ\widetilde{\partial}_{\alpha_i}\circ\widetilde{\partial}_{\alpha_{i+1}}$ for all $i=1,\dotsc,n-2$.
		\item $\left(\widetilde{\partial}_{\alpha_i}+\tfrac{1}{q^2}\right)\circ\left(\widetilde{\partial}_{\alpha_i}+\tfrac{1}{q^2\beta-q^2-\beta}\right)=0$ for all $i=1,\dotsc,n-1$.
	\end{enumerate}
\end{rmk}

\subsection{Deformed motivic Segre classes}\label{subsection:motive-Segre}
Let $R_\Lambda^{\mrm{loc}}$ be the localization of $R_\Lambda$ at the elements $\{1-q^2(1-x_\lambda)\}_{\lambda\in \Sigma}$. Consider the rings
$\widetilde{R}_\Lambda:=\bigoplus_{\lambda\in W^\Theta}R_\Lambda$ and	
$\widetilde{R}_\Lambda^{\mrm{loc}}:=\bigoplus_{\lambda\in W^\Theta}R_\Lambda^{\mrm{loc}}$, both with pointwise addition and multiplication. There is a natural action of $W$ on $\widetilde{R}_\Lambda$ given by $w\cdot(f_\lambda)_\lambda=(w(f_\lambda))_{w(\lambda)}$ for all $w\in W$. Moreover, the action of $W$ on $R_\Lambda$ preserves the set $\{1-q^2(1-x_\lambda)\}_{\lambda\in \Sigma}$. Therefore, the action of $W$ on $\widetilde{R}_\Lambda$ induces an action of $W$ on $\widetilde{R}_\Lambda^\mrm{loc}$. The action of $W$ on $\widetilde{R}_\Lambda^\mrm{loc}$ induces an action of $\partial_w$ on $\widetilde{R}_\Lambda^\mrm{loc}$ for all $w\in W$. Consider the element $S_{w_0}\in\widetilde{R}_\Lambda^\mrm{loc}$: 
\[S_{w_0}|_\lambda=\begin{cases}
	\prod_{\alpha\in\Sigma_\Theta^+} \frac{x_{-\alpha}}{1-q^2(1-x_{-\alpha})},& \text{if $\lambda=w_0$};\\
	0,&\text{otherwise.}
\end{cases}\] 
Define $S_{w\cdot w_0}:=\partial_{w}(S_{w_0})$. The set $\{S_\lambda\}_{\lambda\in W^\Theta}$ forms a basis for $\widetilde{R}_\Lambda^\mrm{loc}$ over the fraction field $\mrm{Frac}(R_\Lambda)$.
The image of $S_{w_0}$ in $\oplus_{\lambda\in W^\Theta}\mrm{Frac}(K_{\widehat{T}}(\mrm{pt})[\beta,\beta^{-1}])$ induced by \eqref{eqn:ReesKiso} is
\[S_{w_0}|_\lambda=\begin{cases}
	\prod_{\alpha\in\Sigma_\Theta^+} \frac{1-e^{-
			\alpha}}{\beta(1-q^2)+q^2(1-e^{-\alpha})},& \text{if $\lambda=w_0$};\\
	0,&\text{otherwise.}
\end{cases}\] 

\begin{egg}\label{egg:K}
	After applying the isomorphism \eqref{eqn:ReesKiso} and evaluating $\beta=1$, the operator $\partial_\alpha$ and class $S_{w_0}$ are the following:
	\[\partial_\alpha=\frac{1-q^2}{1-e^{-\alpha}}+\frac{1-q^2e^\alpha}{1-e^{\alpha}}r_\alpha
	\quad\text{and}\quad S_{w_0}|_\lambda=\begin{cases}
		\prod_{\alpha\in\Sigma_\Theta^+} \frac{1-e^{-\alpha}}{1-q^2e^{-\alpha}},& \text{if $\lambda=w_0$};\\
		0,&\text{otherwise.}
	\end{cases}
	\]
	Therefore, $\{S_\lambda\}_{\lambda\in W^\Theta}$ is the set of images of motivic Segre classes of Schubert cells in the ring $K_{\widehat{T}}^{\text{loc}}(T^*(G/P_\Theta))$ under the localization map $\iota$.
\end{egg}

\begin{egg}\label{egg:H}
	After applying the isomorphism \eqref{eqn:ReesHiso}, the operator $\partial_\alpha$ and class $S_{w_0}$ are the following:
	{
 \setlength{\abovedisplayskip}{.5em}
  \setlength{\belowdisplayskip}{.5em}
	\[\partial_\alpha=\frac{\hbar}{\alpha}+\frac{\hbar\alpha+\alpha-\hbar}{\alpha}r_\alpha
	\quad\text{and}\quad S_{w_0}|_\lambda=\begin{cases}
		\prod_{\alpha\in\Sigma_\Theta^+} \frac{\alpha}{\hbar\alpha+\hbar+\alpha},& \text{if $\lambda=w_0$};\\
		0,&\text{otherwise.}
	\end{cases}
	\]}
	Replace $\partial_\alpha$ with $\partial_\alpha^H:=\frac{\partial_\alpha}{\hbar+1}$, replace $S_{w_0}$ with $S_{w_0}^H:=(\hbar+1)^{l(w_0)}S_{w_0}$, and set $\overline{\hbar}:=\tfrac{\hbar}{\hbar+1}$. Then 
	{
 \setlength{\abovedisplayskip}{.5em}
  \setlength{\belowdisplayskip}{.5em}
	\[\partial_\alpha^H=\frac{\overline{\hbar}}{\alpha}+\frac{\alpha-\overline{\hbar}}{\alpha}r_\alpha\quad\text{and}\quad S_{w_0}^H|_\lambda=\begin{cases}
		\prod_{\alpha\in\Sigma_\Theta^+}\frac{\alpha}{\alpha+\overline{\hbar}},& \text{if $\lambda=w_0$};\\
		0,&\text{otherwise.}
	\end{cases}\]}
	Thus, we can view the set of homogenizations $\{(\hbar+1)^{l(\lambda)}S_\lambda\}_{\lambda\in W^\Theta}$ as the set of images of the SSM classes of Schubert cells in $H_{\widehat{T}}^{\text{loc}}(T^*(G/P_\Theta))$ under the localization map $\iota$.
\end{egg}
\section{Structure constants in the \texorpdfstring{$d=1$}{d=1} case}

Assume $d=1$. Define $Q(\beta):=q^2+\beta-q^2\beta$. For $\lambda\in\Lambda$, we will set $y_\lambda:=1-q^2(1-x_\lambda)$. A \textbf{puzzle} with side labels $\lambda$, $\mu$, $\nu$ in $W^\Theta$ is a triangle with side labels $\lambda$, $\mu$, $\nu$ that is tiled by the following \textbf{puzzle pieces} with edge labels $0$, $1$, and $10$. Each puzzle piece is equipped with a function from $\Lambda$ to the fraction field $\mrm{Frac}(\mbb{Z}[q^{\pm1}]\otimes_\mbb{Z} S_\Lambda)$ known as its \textbf{fugacity}.
\[
\begin{gathered}
	\rh0000=1
\end{gathered}\quad
\begin{gathered}
	\rh1111=1
\end{gathered}\quad
\begin{gathered}
	\rh0101=1
\end{gathered}\quad
\begin{gathered}
\rh1{10}1{10}=1
\end{gathered}\quad
\begin{gathered}
	\rh{10}0{10}0=1
\end{gathered}
\]
\[
\begin{gathered}
	\rh1{10}00=\frac{1-q^2}{y_{\lambda}}
\end{gathered}\quad
\begin{gathered}
	\rh{10}011=\frac{1-q^2}{y_{\lambda}}
\end{gathered}\quad 
\begin{gathered}
	\rh1010=\frac{qx_\lambda}{y_{\lambda}}
\end{gathered}\quad
\begin{gathered}
	\rh{10}{10}01=\frac{q(q^2-1)}{y_{\lambda}}
\end{gathered}
\]
\[
\begin{gathered}
	\rh11{10}0=\frac{(1-q^2)(1-\beta x_{\lambda})}{y_\lambda}
\end{gathered}\quad
\begin{gathered}
	\rh001{10}=\frac{(1-q^2)(1-\beta x_{\lambda})}{y_\lambda}
\end{gathered}\quad
\begin{gathered}
	\rh0{10}0{10}= \frac{qQ(\beta)x_\lambda}{y_\lambda}
\end{gathered}
\]
\[\begin{gathered}
	\rh{10}1{10}1=\frac{qQ(\beta)x_\lambda}{y_\lambda}
\end{gathered}\quad
\begin{gathered}
	\rh01{10}{10}=\frac{Q(\beta)(q^2-1)(1-\beta x_\lambda)}{qy_\lambda}
\end{gathered}\quad
\begin{gathered}
	\rh{10}{10}{10}{10}=Q(\beta)
\end{gathered}
\]

\[
 \quad 
\uptri111=1 
\quad 
\uptri01{10}=1\quad
\uptri{10}01=1
\quad 
\uptri1{10}0=1 \quad
\uptri{10}{10}{10}=\frac{-Q(\beta)}{q}
\]

\[
\downtri000=1\quad
\downtri111=1\quad
\downtri01{10}=1\quad
\downtri{10}01=1\quad
\downtri1{10}0=1\quad 
\downtri{10}{10}{10}= - q
\]

The bottom row of a puzzle is always tiled by the triangle puzzle pieces, and the rest of the puzzle is tiled by (not rotated) rhombus puzzle pieces. The \textbf{fugacity of a puzzle} is the product of the fugacities of the rhombi and triangles that tile it. The fugacity of a triangle tile is constant, independent of its position in the puzzle, whereas the fugacity of a rhombus tile depends on its position in the puzzle. A rhombus tile that lies in the $i$-th southwest-to-northeast diagonal and $(j-1)$-th northwest-to-southeast diagonal depends on $\lambda=-\sum_{t=i}^j\alpha_t$.
For example, the fugacity of the puzzle below is $\tfrac{qQ(\beta)x_{-\alpha_1}}{y_{-\alpha_1}}\cdot \tfrac{q(q^2-1)}{y_{-\alpha_1-\alpha_2}}\cdot 1$.
\[\begin{tikzpicture}[math mode,nodes={\mcol},x={(-0.577cm,-1cm)},y={(0.577cm,-1cm)},scale=0.7]
	\draw[thick] (1,0) -- node[pos=0.5] {\ss 0} ++(0,1); \draw[thick] (1,0) -- node[pos=0.5] {\ss 1} ++(1,0); 
	\draw[thick] (1,1) -- node[pos=0.5] {\ss 1} ++(0,1); \draw[thick] (1,1) -- node[pos=0.5] {\ss 10} ++(1,0); 
	\draw[thick] (1,2) -- node[pos=0.5] {\ss 1} ++(0,1); \draw[thick] (1,2) -- node[pos=0.5] {\ss 10} ++(1,0); \draw[thick] (1+1,2) -- node {\ss 0} ++(-1,1); 
	\draw[thick] (2,0) -- node[pos=0.5] {\ss10} ++(0,1); \draw[thick] (2,0) -- node[pos=0.5] {\ss 1} ++(1,0); 
	\draw[thick] (2,1) -- node[pos=0.5] {\ss 1} ++(0,1); \draw[thick] (2,1) -- node[pos=0.5] {\ss 1} ++(1,0); \draw[thick] (2+1,1) -- node {\ss 1} ++(-1,1); 
	\draw[thick] (3,0) -- node[pos=0.5] {\ss 10} ++(0,1); \draw[thick] (3,0) -- node[pos=0.5] {\ss 0} ++(1,0); \draw[thick] (3+1,0) -- node {\ss 1} ++(-1,1); 
\end{tikzpicture}\]
We will use the notation $\tikz[scale=0.8,baseline=0cm]{\uptri{\lambda}{\nu}{\mu}}$ to mean the sum over the fugacities of all possible puzzles with the prescribed boundary labels. The \textbf{homogenization} of $S_\lambda$ by a factor $q$ is $q^{l(\lambda)}S_\lambda$, where $l(\lambda)$ is the length of $\lambda\in W^\Theta$. We can now state our main theorem:

\begin{thm}\label{thm:main}
	The product of two classes $q^{l(\lambda)}S_\lambda$ and $q^{l(\mu)}S_\mu$ is given by the ``puzzle'' formula
	\begin{equation}\label{eq:main}
		(q^{l(\lambda)}S_\lambda)(q^{l(\mu)} S_\mu)=
		%    \sum_\nu \left( \sum \Bigg\{fug(P)\ :
		%      \ \text{puzzles $P$ with boundary }\tikz[scale=1.8,baseline=0.5cm]{\uptri{\lambda}{\nu}{\mu}}\Bigg\} \right)
		\sum_\nu \tikz[scale=1.25,baseline=0.5cm]{\uptri{\lambda}{\nu}{\mu}}
		\ (q^{l(\nu)}S_\nu).
	\end{equation}
\end{thm}
\begin{proof}
	The divided difference operators $\partial_\alpha$ and $\widetilde{\partial}_\alpha$ produce an $R$-matrix recursion that defines the classes $q^{l(\lambda)}S_\lambda$. The fugacities of the rhombus and triangle tiles can be used to define $R$, $U$, and $D$ matrices that satisfy various Yang--Baxter type equations \cite[Proposition 3.4]{KZJ21}. The rest of the proof is essentially the same as that of \cite[Theorem 3.8]{KZJ21}. 
\end{proof}

\begin{rmk}\label{rnk:positive}
	Following \cite[Section 6]{KZJ21}, we define a \textbf{positivity monoid} $M$ to be a submonoid of $R_\Lambda^{\text{loc}}$ under addition such that $M\cap (-M)=0$. Consider the submonoid $M$ of $R_\Lambda^{\text{loc}}$, defined as the set of sums of products of the factors over all $\alpha\in\Sigma^+$:
	\[-q^{\pm}\qquad Q(\beta)\qquad  1-\beta x_{-\alpha}\qquad \tfrac{1-q^2}{y_{-\alpha}}\qquad -\tfrac{x_{-\alpha}}{y_{-\alpha}}\]
	Then $M$ is a positivity monoid of $R_\Lambda^{\text{loc}}$. To see it, view $M$ as a submonoid of $\mrm{Frac}(K_{\widehat{T}}(\mrm{pt})[\beta,\beta^{-1}])$ via the isomorphism \eqref{eqn:ReesKiso}, and then evaluate $\beta=1$, $e^{-\alpha_i}=2^i$, and $q=-2^{-n/2}$ to see that every factor is positive. As the structure constants for $\{q^{l(\lambda)}S_\lambda\}_{\lambda\in W^\Theta}$ lie in the positivity monoid $M$, it is in this sense that we consider the structure constants and puzzle formula for them \textbf{positive}.
\end{rmk}
	
	
	\begin{egg}
	We will compute the product $(q \cdot S_{10})\cdot S_{01}$ using puzzles. First, we compute
	\[q \cdot S_{10}=q\cdot \left[0,\tfrac{x_{-\alpha_1}}{y_{-\alpha_1}}\right];\quad\quad\quad S_{01}=(\tfrac{1}{q}\partial_{\alpha_1})(q\cdot S_{10})=\left[1,\tfrac{1-q^2}{y_{-\alpha_1}}\right].\]
	Here, $\alpha_1$ is the positive root of $\mrm{Gr}(1,2)$. Pointwise multiplication of these classes yields the equation $(q\cdot S_{10})\cdot S_{01}=\tfrac{1-q^2}{y_{-\alpha_1}}\cdot (q\cdot S_{10})$. The puzzle rule \eqref{eq:main} gives the same result:
	\[\def\posa{0.5}\def\posb{0.5}\def\thescale{1}
	\begin{tikzpicture}[math mode,nodes={edgelabel},x={(-0.577cm,-1cm)},y={(0.577cm,-1cm)},scale=0.7,baseline=(current  bounding  box.center)]
		\draw[thick] (2,0) -- node[pos=0.5] {\ss 0} ++(0,1); \draw[thick] (2,0) -- node[pos=0.5] {\ss 0} ++(1,0); 
		\draw[thick] (2,1) -- node[pos=0.5] {\ss 1} ++(0,1); \draw[thick] (2,1) -- node[pos=0.5] {\ss 10} ++(1,0); \draw[thick] (2+1,1) -- node {\ss 0} ++(-1,1); 
		\draw[thick] (3,0) -- node[pos=0.5] {\ss 1} ++(0,1); \draw[thick] (3,0) -- node[pos=0.5] {\ss 1} ++(1,0); \draw[thick] (3+1,0) -- node {\ss 1} ++(-1,1); 	
	\end{tikzpicture}
	=\frac{1-q^2}{y_{-\alpha_1}}.\]
	\end{egg}


\section{Towards a geometric interpretation of the classes}


Let $E$ be any linear algebraic group over $\mbb{C}$. Let $\mrm{Sm}_E$ be the category of smooth algebraic varieties over $\mbb{C}$ equipped with an algebraic action of $E$, with morphisms being $E$-equivariant morphisms of varieties. An $E$-equivariant oriented cohomology theory $\mrm{h}_E^*$ is a functor from $\mrm{Sm}_E$ to the category of commutative unital rings satifying several ``cohomological-type" axioms. See \cite{ZZ17} or \cite{CZZ3} for detailed exposition. Let $\Omega_E$ be the $E$-equivariant algebraic cobordism of Krishna \cite{K12} and Malag\'{o}n-L\'{o}pez and Heller \cite{HM13}. The functor $\Omega_E$ is an example of an $E$-equivariant oriented cohomology theory.

\begin{egg}
	The $E$-equivariant Chow theory $\mrm{CH}_E$ of Edidin--Graham \cite{EG98} is an $E$-equivariant oriented cohomology theory. Let $\widehat{\mrm{CH}}_E(\mrm{pt})$ be the completion of $\mrm{CH}_E(\mrm{pt})$ at the \textit{augmentation ideal} (i.e., the ideal generated by algebraic cycles of positive codimensions). Let $X\in \mrm{Sm}_E$. In \cite[Section 7.1]{K12} it is shown there is a canonical map $\Omega_E(\mrm{pt})\to\widehat{\mrm{CH}}_E(\mrm{pt})$ and a ring isomorphism
	\[\Omega_E(X)\otimes_{\Omega_E(\mrm{pt})}\widehat{\mrm{CH}}_E(\mrm{pt})\simeq \mrm{CH}_E(X)\otimes_{{\mrm{CH}}_E(\mrm{pt})} \widehat{\mrm{CH}}_E(\mrm{pt}). \]
	\end{egg}

\begin{egg}
	The $E$-equivariant algebraic $K$-theory $K_E:=K_E^0$ of Thomason \cite{T87} is an $E$-equivariant oriented cohomology theory. Let $\widehat{K}_E(\mrm{pt})$ be the completion of ${K}_E(\mrm{pt})$ at the \textit{augmentation ideal} (i.e., the ideal of virtual representations of rank $0$). Let $X\in \mrm{Sm}_E$. In \cite[Section 7.2]{K12} it is shown that there is a canonical map $\Omega_E(\mrm{pt})\to \widehat{K}_E(\mrm{pt})$ and a ring isomorphism
	\[\Omega_E(X)\otimes_{\Omega_E(\mrm{pt})}\widehat{\mrm{K}}_E(\mrm{pt})\simeq K_E(X)\otimes_{K_E(\mrm{pt})}\widehat{K}_E(\mrm{pt}). \] 
\end{egg}
For any $X\in\mrm{Sm}_E$, we define
\[\widehat{CH}_E(X):=\Omega_E(X)\otimes_{\Omega_E(\mrm{pt})}\widehat{\mrm{CH}}_E(\mrm{pt})\quad\text{and}\quad \widehat{K}_E(X):=\Omega_E(X)\otimes_{\Omega_E(\mrm{pt})}\widehat{{K}}_E(\mrm{pt}).\]


Let $I_\Lambda$ be the ideal in $(\mbb{Z}[\beta])[x_\lambda]_{\lambda\in\Lambda}$ generated by the elements $x_\lambda$, with $\lambda\in\Lambda$. Consider the $I_\Lambda$-adic completion $(\mbb{Z}[\beta])\llbracket x_\lambda\rrbracket_{\lambda\in\Lambda}$ of $(\mbb{Z}[\beta])[x_\lambda]_{\lambda\in\Lambda}$. Let $\mcal{J}_\Lambda$ be the closure of the ideal in $(\mbb{Z}[\beta])\llbracket x_\lambda\rrbracket_{\lambda\in\Lambda}$ generated by $x_0$ and $x_{\lambda+\mu}-x_\lambda-x_\mu+\beta x_\lambda x_\mu$ over all $\lambda,\mu\in \Lambda$. Set $\mcal{S}_\Lambda:= (\mbb{Z}[\beta])\llbracket x_\lambda\rrbracket_{\lambda\in\Lambda}/\mcal{J}_\Lambda$.  It follows from \cite[Corollary 2.13]{CPZ} that $\mcal{S}_\Lambda$ is an integral domain. It follows from \cite[Section 2.5]{CPZ} and \cite[Theorem 3.3]{CZZ3} that there are maps from $\mcal{S}_\Lambda$ to both $\widehat{CH}_{{T}}(\mrm{pt})$ and $\widehat{K}_{{T}}(\mrm{pt})$, which are induced by maps of formal group laws, and that there is a map $\Omega_{T}(\mrm{pt})\to \mcal{S}_\Lambda$. The kernel of the composition  $(\mbb{Z}[\beta])[x_\lambda]_{\lambda\in\Lambda}\hookrightarrow (\mbb{Z}[\beta])\llbracket x_\lambda\rrbracket_{\lambda\in\Lambda}\to \mcal{S}_\Lambda$
is precisely the ideal $J_\Lambda$ defined in \Cref{subsection:Rees}. Thus, we can view $S_\Lambda$ as a subring of $\mcal{S}_\Lambda$. Define $\widehat{R}_\Lambda:=\mbb{Z}[\beta]\llbracket x_q\rrbracket \otimes_{\mbb{Z}[\beta]} \mcal{S}_\Lambda$. We can view $\mbb{Z}[ q^{\pm 2}]\otimes_\mbb{Z} S_\Lambda$ as a subring of $\widehat{R}_\Lambda$ via the map $q^2\mapsto 1-x_q$ (observe that $q^{-2}\mapsto 1+x_q+x_q^2+\dotsb$.) As $\Omega_{\widehat{T}}(\mrm{pt})\simeq \Omega_{T}(\mrm{pt})\otimes_{\mbb{L}} \mbb{L}\llbracket x\rrbracket$, the map $\Omega_{{T}}(\mrm{pt})\to \mcal{S}_\Lambda$ induces a map $\Omega_{\widehat{T}}(\mrm{pt})\to \widehat{R}_\Lambda$, $x\mapsto  x_q$.

We expect the definition below to agree with that of \cite{KM22}. We plan to explore this.

\begin{dfn}
	The (completed) \textbf{$\widehat{T}$-equivariant connective $K$-ring} of $T^*(G/P_\Theta)$ is 
	\[CK_{\widehat{T}}(T^*(G/P_\Theta)):=\Omega_{\widehat{T}}(T^*(G/P_\Theta))\otimes_{\Omega_{\widehat{T}}(\mrm{pt})} \widehat{R}_\Lambda.\]
\end{dfn}
There are ring homomorphisms,
\[{CK}_{\widehat{T}}(T^*(G/P_\Theta))\to \widehat{\mrm{CH}}_{\widehat{T}}(T^*(G/P_\Theta))\quad\text{and}\quad {CK}_{\widehat{T}}(T^*(G/P_\Theta))\to \widehat{K}_{\widehat{T}}(T^*(G/P_\Theta)),\]
where $x_q\mapsto -x_{\hbar}$ on the left and $x_q\mapsto x_{q}$ on the right.
Consider the localization $\widehat{R}_\Lambda^{\text{loc}}$ of $\widehat{R}_\Lambda$ at the set $\{ x_q+x_\alpha- x_qx_\alpha\}_{\alpha\in\Sigma}$.
Let $\widehat{\mrm{CH}}_{\widehat{T}}^{\text{loc}}(\mrm{pt})$ be the localization of $\widehat{\mrm{CH}}_{\widehat{T}}(\mrm{pt})$ at the set $\{x_{\hbar}+x_\alpha+x_{\hbar} x_{\alpha}\}_{\alpha\in \Sigma}$, and let $\widehat{K}_{\widehat{T}}^{\text{loc}}(\mrm{pt})$ be the localization of $\widehat{K}_{\widehat{T}}(\mrm{pt})$ at the set $\{x_q+x_\alpha-x_qx_\alpha\}_{\alpha\in \Sigma}$. Define $\widehat{\mrm{CH}}_{\widehat{T}}^{\text{loc}}(T^*(G/P_\Theta)):=\widehat{\mrm{CH}}_{\widehat{T}}(T^*(G/P_\Theta))\otimes_{\widehat{\mrm{CH}}_{\widehat{T}}(\mrm{pt})}\widehat{\mrm{CH}}_{\widehat{T}}^{\text{loc}}(\mrm{pt})$ and $\widehat{K}_{\widehat{T}}^{\text{loc}}(T^*(G/P_\Theta)):=\widehat{K}_{\widehat{T}}(T^*(G/P_\Theta))\otimes_{\widehat{K}_{\widehat{T}}(\mrm{pt})}\widehat{K}_{\widehat{T}}^{\text{loc}}(\mrm{pt})$.


\begin{conj}\label{conj:geometric}
	We conjecture:
	\begin{enumerate}
		\item There are classes $\overline{S}_\lambda\in{{\mrm{CK}}}_{\widehat{T}}(T^*(G/P_\Theta))\otimes_{\widehat{R}_\Lambda} \widehat{R}_\Lambda^{\text{loc}}$ indexed by $\lambda\in W^\Theta$ whose images in $\widehat{\mrm{CH}}_{\widehat{T}}^{\text{loc}}(T^*(G/P_\Theta))$ and $
		\widehat{\mrm{K}}_{\widehat{T}}^{\text{loc}}(T^*(G/P_\Theta))$ can be identified with SSM and motivic Segre classes, respectively. The set $\{\overline{S}_\lambda\}_{\lambda\in W^\Theta}$ is an $\widehat{R}_\Lambda^{\text{loc}}$-basis for ${\mrm{CK}}_{\widehat{T}}(T^*(G/P_\Theta))\otimes_{\widehat{R}_\Lambda} \widehat{R}_\Lambda^{\text{loc}}$.
		\item Under the ``localization map" $\iota\colon {{\mrm{CK}}}_{\widehat{T}}(T^*(G/P_\Theta))\otimes_{\widehat{R}_\Lambda} \widehat{R}_\Lambda^{\text{loc}}\to \oplus_{\lambda\in W^\Theta} {\widehat{R}}_\Lambda^{\text{loc}}$,
		the image $\iota(\overline{S}_\lambda)$ is the class $S_\lambda$ defined in \Cref{subsection:motive-Segre}.
	\end{enumerate}
\end{conj}

\begin{rmk}
	The classes $S_\lambda$ satisfy a ``GKM-type" property. We believe this property, together with a connective $K$-theory analogue of a theorem of Chang and Skjelbred, imply that there exist classes $\overline{S}_\lambda\in {{\mrm{CK}}}_{\widehat{T}}(T^*(G/P_\Theta))\otimes_{\widehat{R}_\Lambda} \widehat{R}_\Lambda^{\text{loc}}$ satisfying \Cref{conj:geometric}. This is work in progress. 
\end{rmk}




\acknowledgements{The author is grateful to his Ph.D. advisor Allen Knutson for proposing the idea that eventually led to this paper and for his attention, guidance, and insights throughout the course of this project. We thank Kirill Zainoulline for introducing the author to connective $K$-theory. This work has benefited from conversations and correspondences with David Anderson, Timothy Miller, Andrei Okounkov, Travis Scrimshaw, Rui Xiong, and Paul Zinn-Justin. This work was partially supported by a Postgraduate Scholarship - Doctoral - from the Natural Sciences and Engineering Research Council of Canada.}

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