\documentclass[final]{FPSAC2022}

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% Title
\title[Quantum Yang--Baxter Moves]{Generalized Quantum Yang--Baxter Moves and Their Application to Schubert Calculus}

% Author
\author[T.~Kouno, C.~Lenart, and S.~Naito]{Takafumi Kouno\thanks{\href{mailto:t.kouno@kurenai.waseda.jp}{t.kouno@kurenai.waseda.jp}. Research Fellow of Japan Society for the Promotion of Science. Supported in part by JSPS Grant-in-Aid for Scientific Research 20J12058 and 22J00874.}\addressmark{1}, 
Cristian Lenart\thanks{\href{mailto:clenart@albany.edu}{clenart@albany.edu}. Supported in part by the NSF grant DMS-1855592.}\addressmark{2}, 
and Satoshi Naito\thanks{\href{mailto:naito@math.titech.ac.jp}{naito@math.titech.ac.jp}. Supported in part by JSPS Grant-in-Aid for Scientific Research (B) 16H03920.}\addressmark{3}}

% Institution
\address{\addressmark{1}Department of Mathematics, Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan \\
\addressmark{2}Department of Mathematics ans Statistics, State University of New York at Albany, NY 12222, U.S.A. \\ 
\addressmark{3}Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152-8551, Japan}

% Submission date
\received{November 19, 2021}
\revised{April 1, 2022}

% English Abstract 
\abstract{The quantum alcove model is a uniform combinatorial model in the representation theory of quantum affine algebras and Schubert calculus on flag manifolds. Given a weight $\lambda$, the model is based on a sequence of roots called a $\lambda$-chain. When $\lambda$ is dominant, the independence of the model from the chosen $\lambda$-chain was shown using certain elementary transformations called quantum Yang--Baxter moves. 
The purpose of the present work is to generalize the quantum Yang--Baxter moves to an arbitrary weight $\lambda$. 
As an application, we give a combinatorial proof of the Chevalley formula 
in the equivariant $K$-group of semi-infinite flag manifolds, first proved by Lenart--Naito--Sagaki. }

% French Abstract 
\resume{Le mod\`ele des alc\^oves quantique est un mod\`ele combinatoire uniforme dans la th\'eorie des repr\'esentations des alg\`ebres affines quantiques et le calcul de Schubert sur les vari\'et\'es de drapeaux. \'Etant donn\'e un poids $\lambda$, le mod\`ele est bas\'e sur une s\'equence de racines appel\'ee $\lambda$-cha\^ine. Lorsque $\lambda$ est dominant, l'ind\'ependance du mod\`ele de la $\lambda$-cha\^ine choisie a \'et\'e montr\'ee en utilisant certaines transformations \'el\'ementaires appel\'ees mouvements de Yang--Baxter quantiques.
Le but de ce travail est de g\'en\'eraliser les mouvements de Yang--Baxter quantiques \`a un poids arbitraire.
Comme application, nous donnons une preuve combinatoire de la formule de Chevalley
dans la $K$-th\'eorie \'equivariante des vari\'et\'es de drapeaux semi-infinies, prouv\'e pour la premi\`ere fois par Lenart--Naito--Sagaki. }

% Keywords
\keywords{quantum Yang--Baxter move, quantum Bruhat graph, quantum alcove model, semi-infinite flag manifold, Chevalley formula}

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\section{Introduction}

The quantum alcove model, introduced by Lenart--Lubovsky \cite{LL1}, is a uniform combinatorial model, which appears in many branches of mathematics related to root systems; 
for example, Schubert calculus of semi-infinite/ordinary flag manifolds, the representation theory of quantum affine algebras, and the theory of Macdonald polynomials. 

Let $\Fg$ be a simple Lie algebra over $\BC$ with Weyl group $W$, weight lattice $P$, and coroot lattice $Q^{\vee}$. 
In the theory of the quantum alcove model, one fixes $w\in W$, $\lambda\in P$, and a certain sequence $\Gamma$ of roots corresponding to $\lambda$, called a \emph{$\lambda$-chain}; then, one considers the collection $\CA(w, \Gamma)$ of so-called \emph{admissible subsets}. 
In general, for a fixed $\lambda \in P$, we have several $\lambda$-chains, 
and hence several collections of admissible subsets (depending on them and some $w\in W$). 
If $\lambda$ is dominant and the $\lambda$-chains are ``reduced'' (defined later), then these collections are known to be mutually isomorphic (as combinatorial models for certain Kashiwara crystals). 
More precisely, Lenart--Lubovsky \cite{LL2} proved that, given reduced $\lambda$-chains $\Gamma_{1}$ and $\Gamma_{2}$ 
such that $\Gamma_{2}$ is obtained from $\Gamma_{1}$ by a certain deformation procedure, called a \emph{Yang--Baxter transformation}, 
there exists a bijection $Y \colon \CA(e, \Gamma_{1}) \rightarrow \CA(e, \Gamma_{2})$, with $e$  the identity of $W$; moreover, 
the bijection preserves some important statistics, including $\wt(\cdot)$ and $\height(\cdot)$. This bijection is called a \emph{quantum Yang--Baxter move}, and the above statement works for any reduced $\lambda$-chains $\Gamma_{1}$ and $\Gamma_{2}$ if we use a sequence of such moves. In~\cite{LL2} it is explained that the quantum Yang--Baxter moves provide a root system generalization of the well-known jeu de taquin on semi-standard Young tableaux. 

The purpose of the present work is to give a generalization of the quantum Yang--Baxter moves to the case of an arbitrary (not necessarily dominant) weight $\lambda \in P$, and an arbitrary $w\in W$. This proves the independence of the quantum alcove model associated with $\lambda$ and $w$ from the chosen $\lambda$-chain. 
Here we should mention that our generalized quantum Yang--Baxter moves are no longer bijections, but \emph{sijections}, \textit{i.e.}, ``signed bijections''; 
recall from \cite{FK} that for 
two signed sets $S$, $T$, a sijection from $S$ to $T$ 
is a triple $(\iota_{S}, \iota_{T}, \varphi)$ consisting of the maps $\iota_{S}$, $\iota_{T}$, and $\varphi$, where 
$\varphi$ is a sign-preserving bijection between subsets $S_{0} \subset S$ and $T_{0} \subset T$, and 
$\iota_{S}$ (resp., $\iota_{T}$) is a sign-reversing involution on $S \setminus S_{0}$ (resp., $T \setminus T_{0}$). 
By regarding collections of admissible subsets as signed sets equipped with certain sign functions, our generalized quantum Yang--Baxter moves can be viewed as sijections. 

We give an application of generalized quantum Yang--Baxter moves to Schubert calculus of semi-infinite flag manifolds. 
Let $G$ be a connected, simply-connected simple algebraic group over $\BC$ with Lie algebra $\Fg$, 
$H \subset G$ a maximal torus of $G$, and $(H \subset) \ B \subset G$ a Borel subgroup with unipotent radical $N$. 
The \emph{semi-infinite flag manifold} $\bQGr$ is the ind-scheme of infinite type 
whose set of $\BC$-valued points is $\bQGr(\BC) = G(\BC(\!(z)\!)) / (H(\BC) \cdot N(\BC(\!(z)\!)))$, 
where $z$ is an indeterminate (for details, see \cite{Kat}). 
For each element $x$ of the affine Weyl group $W_{\af}$ associated to $\Fg$, 
$\bQGr$ has the corresponding orbit under the action of the Iwahori subgroup of $G(\BC[\![z]\!])$; 
the closure of this orbit is denoted by $\bQG(x)$, and called a \emph{semi-infinite Schubert variety}. 
We set $\bQG := \bQG(e)$, and 
let $K_{H \times \BC^{\ast}}(\bQG)$ denote the $(H \times \BC^{\ast})$-equivariant $K$-group of $\bQG$, introduced in \cite{KNS}. 
Let $[\CO(\lambda)] \in \K$ be the class of the line bundle associated to $\lambda \in P$, and 
$[\CO_{x}]$ the class of the structure sheaf of the semi-infinite Schubert variety $\bQG(x)$ corresponding to $x \in W_{\af}$. 
We set $W_{\af}^{\ge 0} := \{ w t_{\xi} \mid w \in W, \ \xi \in Q^{\vee, +} \}$; 
here $Q^{\vee, +} := \{ \xi \in Q^{\vee} \mid \xi \ge 0 \}$, 
and $t_{\xi}$ for $\xi \in Q^{\vee}$ denotes the translation element. 
The expansion of the product $[\CO(\lambda)] \cdot [\CO_{x}]$ in $\K$ of the following form 
plays an important role in the study of the structure of $\K$: 
for $\lambda \in P$ and $x \in W_{\af}^{\ge 0}$, 
\begin{equation} \label{eq:Chevalley_base}
[\CO(\lambda)] \cdot [\CO_{x}] = \sum_{y \in W_{\af}^{\ge 0}} c_{x, \lambda}^{y} [\CO_{y}], 
\end{equation}
where $c_{x, \lambda}^{y} \in \BZ[P](\!(q^{-1})\!)$. 
An explicit description (in terms of the quantum alcove model) of the coefficients $c_{x, \lambda}^{y}$ in equation~\eqref{eq:Chevalley_base} is given by the \emph{Chevalley formula}, which is proved by Lenart--Naito--Sagaki \cite{LNS} 
based on the Yang--Baxter equation for quantum Bruhat operators. 
By making use of generalized quantum Yang--Baxter moves, we can give a combinatorial proof of this Chevalley formula. 

This paper is an extended abstract of our paper \cite{KLN}, 
which is a part of the first author's Ph.~D.~thesis \cite{Kouno}. 


\section{Basic definitions}

First, we recall the definition of the quantum Bruhat graph, which was introduced by Brenti--Fomin--Postnikov \cite{BFP}. 
Take a Cartan subalgebra $\Fh$ of $\Fg$, and set $\Fh^{\ast} := \Hom_{\BC}(\Fh, \BC)$.
We denote by $\pair{\cdot}{\cdot}$ the canonical pairing of $\Fh^{\ast}$ and $\Fh$. 
Let $\Delta$ be the root system of $\Fg$, $\Delta^{+}$ the set of all positive roots, and set $\rho := (1/2) \sum_{\alpha \in \Delta^{+}} \alpha$. 
For each $\alpha \in \Delta$, we denote by $\alpha^{\vee}$ the coroot of $\alpha$, and by $s_{\alpha} \in W$ the corresponding reflection. 
Also, we denote by $\ell(\cdot)$ the length function on $W$. 

\begin{defn}[{\cite[Definition~6.1]{BFP}}]
The \emph{quantum Bruhat graph} $\QBG(W)$ is the $\Delta^{+}$-labeled directed graph whose vertex set is $W$, and whose (directed) edges are defined as follows: 
$x \xrightarrow{\alpha} y$ for $x, y \in W$ and $\alpha \in \Delta^{+}$ if and only if $y = xs_{\alpha}$ and one of the following conditions holds: 
\begin{itemize}
\item[(B)] $\ell(y) = \ell(x) + 1$; 
\item[(Q)] $\ell(y) = \ell(x) - 2\pair{\rho}{\alpha^{\vee}} + 1$. 
\end{itemize}
If condition (B) (resp., (Q)) holds, then the corresponding edge is called a \emph{Bruhat} (resp., \emph{quantum}) \emph{edge}. 
\end{defn}

Next, following \cite{LP1} and \cite{LNS}, we briefly review the quantum alcove model. 
Set $\Fh_{\BR}^{\ast} := P \otimes_{\BZ} \BR$. 
For each $\alpha \in \Delta$ and $k \in \BZ$, we define a hyperplane $H_{\alpha, k}$ by 
\begin{equation*}
H_{\alpha, k} := \{ \nu \in \Fh_{\BR}^{\ast} \mid \pair{\nu}{\alpha^{\vee}} = k \}. 
\end{equation*}
In this setting, connected components of the space 
\begin{equation*}
\Fh_{\BR}^{\ast} \setminus \bigcup_{\alpha \in \Delta, \ k \in \BZ} H_{\alpha, k}
\end{equation*}
are called \emph{alcoves}. In particular, the specific alcove $A_{\circ}$ defined by 
\begin{equation*}
A_{\circ} := \{ \nu \in \Fh_{\BR}^{\ast} \mid \text{$0 < \pair{\nu}{\alpha^{\vee}} < 1$ for all $\alpha \in \Delta^{+}$} \} 
\end{equation*}
is called the \emph{fundamental alcove}. Also, for each $\lambda \in P$, we define $A_{\lambda}$ by 
\begin{equation*}
A_{\lambda} := A_{\circ} + \lambda = \{ \nu + \lambda \mid \nu \in A_{\circ} \}. 
\end{equation*}
For two adjacent alcoves $A$ and $B$ and for $\beta \in \Delta$, we write $A \xrightarrow{\beta} B$ 
if the common wall of $A$ and $B$ is contained in the hyperplane $H_{\beta, k}$ for some $k \in \BZ$, 
and $\beta$ (viewed as a direction vector) points in the direction from $A$ to $B$. 

\begin{defn}[{\cite[Definitions~5.2, 5.4]{LP1}}]
\mbox{}
\begin{enu}
\item A sequence $(A_{0}, \ldots, A_{r})$ of alcoves is called an \emph{alcove path} if $A_{i-1}$ and $A_{i}$ are adjacent for all $i = 1, \ldots, r$. 
\item An alcove path $(A_{0}, \ldots, A_{r})$ is called \emph{reduced} if $r$ is minimal among all alcove paths from $A_{0}$ to $A_{r}$. 
\item Let $\lambda \in P$. A sequence $\Gamma = (\beta_{1}, \ldots, \beta_{r})$ of roots is called a \emph{$\lambda$-chain} 
if there exists an alcove path $(A_{0}, \ldots, A_{r})$ of the form 
\begin{equation*}
A_{\circ} = A_{0} \xrightarrow{-\beta_{1}} \cdots \xrightarrow{-\beta_{r}} A_{r} = A_{-\lambda}; 
\end{equation*}
if this alcove path is reduced, then $\Gamma$ is said to be \emph{reduced}. 
\end{enu}
\end{defn}

We recall the definition of an admissible subset, which is the main object of study in the theory of the quantum alcove model. 
For $\beta \in \Delta$, we set 
\begin{equation*}
\sgn(\beta) := \begin{cases}
1 & \text{if $\beta \in \Delta^{+}$,} \\  
-1 & \text{if $\beta \in -\Delta^{+}$,}
\end{cases}
\end{equation*}
and set $|\beta| := \sgn(\beta)\beta$. 

\begin{defn}[{\cite[Definition~17]{LNS}, \cite[Definition~3.4]{LL1}}]
Let $\lambda \in P$, and take a $\lambda$-chain $(\beta_{1}, \ldots, \beta_{r})$. Let $w \in W$. 
The subset $A = \{j_{1} < \cdots < j_{t}\} \subset \{1, \ldots, r\}$ is said to be \emph{$w$-admissible} if 
\begin{equation*}
\bp(A) \colon w = w_{0} \xrightarrow{|\beta_{j_{1}}|} \cdots \xrightarrow{|\beta_{j_{t}}|} w_{t} =: \ed(A)
\end{equation*}
is a directed path in $\QBG(W)$. 
We denote by $\CA(w, \Gamma)$ the collection of all $w$-admissible subsets. 
\end{defn}

Following \cite[Equations~(12), (13)]{LNS}, we define some important statistics for admissible subsets. 
Let $\lambda \in P$ and $w \in W$. Let $\Gamma = (\beta_{1}, \ldots, \beta_{r})$ be a $\lambda$-chain. 
For $A = \{j_{1} < \cdots < j_{r}\} \in \CA(w, \Gamma)$, we set 
\begin{equation*}
A^{-} := \left\{ j \in A \ \middle| \ \text{the edge $w_{j-1} \xrightarrow{|\beta_{j}|} w_{j}$ in $\bp(A)$ is a quantum one} \right\}, 
\end{equation*}
and define $\down(A)$ by 
\begin{equation*}
\down(A) := \sum_{j \in A^{-}} |\beta_{j}|^{\vee}. 
\end{equation*}
Let 
\begin{equation*}
A_{\circ} = A_{0} \xrightarrow{-\beta_{1}} \cdots \xrightarrow{-\beta_{r}} A_{r} = A_{-\lambda} 
\end{equation*}
be the alcove path corresponding to $\Gamma$. 
For $i = 1, \ldots, r$, let $l_{i} \in \BZ$ be such that the common wall of $A_{i-1}$ and $A_{i}$ is contained in $H_{\beta_{i}, -l_{i}}$. 
For each $\alpha \in \Delta$ and $k \in \BZ$, we denote by $s_{\alpha, k}$ the reflection with respect to $H_{\alpha, k}$. 
Then $\wt(\cdot)$ and $\height(\cdot)$ are defined as follows: 
\begin{align*}
\wt(A) &:= -ws_{\beta_{j_{1}, -l_{j_{1}}}} \cdots s_{\beta_{j_{t}, -l_{j_{t}}}} (-\lambda), \\ 
\height(A) &:= \sum_{j \in A^{-}} \sgn(\beta_{j}) (\pair{\lambda}{\beta_{j}^{\vee}} - l_{j}). 
\end{align*}

Also, for each $A \in \CA(w, \Gamma)$, we define $n(A)$ by 
\begin{equation*}
n(A) := | \{ j \in A \mid \beta_{j} \in -\Delta^{+} \} |. 
\end{equation*}

\section{Quantum Yang--Baxter moves}
In this section, we explain our generalization of quantum Yang--Baxter moves. 
First, we review quantum Yang--Baxter moves introduced by Lenart--Lubovsky \cite{LL2}. 
For this, we need to recall certain deformation procedures for $\lambda$-chains. 
Let $\lambda \in P$, and take a $\lambda$-chain $\Gamma = (\beta_{1}, \ldots, \beta_{r})$. 
The procedures (YB) and (D) are given as follows (see \cite[Lemma~9.3]{LP1}, and also \cite[Remark~38]{LNS}): 
\begin{itemize}
\item[(YB)] Take a segment $(\beta_{t+1}, \ldots, \beta_{t+q})$ of $\Gamma$ such that 
\begin{itemize}
\item[$\bullet$] $\pair{\beta_{t+1}}{\beta_{t+q}^{\vee}} \le 0$, and 
\item[$\bullet$] if we set $\alpha := \beta_{t+1}$ and $\beta := \beta_{t+q}$, then 
\begin{equation*}
(\beta_{t+1}, \ldots, \beta_{t+q}) = (\alpha, s_{\alpha}(\beta), s_{\alpha}s_{\beta}(\alpha), \ldots, s_{\beta}(\alpha), \beta). 
\end{equation*}
\end{itemize}
Then, we define a new $\lambda$-chain $\Gamma'$ by 
\begin{equation*}
\Gamma' := (\beta_{1}, \ldots, \beta_{t}, \beta_{t+q}, \beta_{t+q-1}, \ldots, \beta_{t+1}, \beta_{t+q+1}, \ldots, \beta_{r}); 
\end{equation*}
namely, we reverse the segment $(\beta_{t+1}, \ldots, \beta_{t+q})$. 

\item[(D)] Take a segment $(\beta_{t+1}, \beta_{t+2})$ of $\Gamma$ such that $\beta_{t+2} = -\beta_{t+1}$, and we define a new $\lambda$-chain $\Gamma'$ by 
\begin{equation*}
\Gamma' := (\beta_{1}, \ldots, \beta_{t}, \beta_{t+3}, \ldots, \beta_{r}); 
\end{equation*}
namely, we delete the segment $(\beta_{t+1}, \beta_{t+2})$. 
\end{itemize}
The procedure (YB) (resp., (D)) is called a \emph{Yang--Baxter transformation} (resp., \emph{deletion}). 
It is known from \cite[Remark~38]{LNS} (see also \cite[Lemma~9.3]{LP1}) 
that one obtains an arbitrary reduced $\lambda$-chain from any (not necessarily reduced) $\lambda$-chain by repeated application of the procedures (YB) and (D). 

Lenart--Lubovsky proved the following theorem in the case that $\lambda$ is dominant. 

\begin{thm}[{\cite[Sections~3.1, 3.2]{LL2}}]
Let $\lambda \in P^{+}$, and take reduced $\lambda$-chains $\Gamma_{1}$ and $\Gamma_{2}$ 
such that $\Gamma_{2}$ is obtained from $\Gamma_{1}$ by (YB). 
Then, there exists a bijection $Y \colon \CA(e, \Gamma_{1}) \rightarrow \CA(e, \Gamma_{2})$ such that for $A \in \CA(e, \Gamma_{1})$, 
\begin{itemize}
\item $\ed(Y(A)) = \ed(A)$, 
\item $\down(Y(A)) = \down(A)$, 
\item $\wt(Y(A)) = \wt(A)$, and 
\item $\height(Y(A)) = \height(A)$. 
\end{itemize}
\end{thm}

The bijection $Y$ above is called a \emph{quantum Yang--Baxter move}. 
In fact, Lenart--Lubovsky proved that quantum Yang--Baxter moves are isomorphisms of crystals (see \cite[Theorem~3.8]{LL2}). 
By taking composites of quantum Yang--Baxter moves, we see 
that the collections $\CA(e, \Gamma)$ for all reduced $\lambda$-chains $\Gamma$ are mutually isomorphic as crystals (see \cite[Corollary~3.9]{LL2}). 

\begin{rem}
As mentioned in \cite[Corollary~3.14]{LL2}, quantum Yang--Baxter moves for dominant weights realize combinatorial $R$-matrices 
for tensor products of column-shape Kirillov--Reshetikhin crystals; 
also, in type $A$, combinatorial $R$-matrices can be realized by Sch\"{u}tzenberger's jeu de taquin (sliding algorithm) for type $A$ root systems (see \cite[Example~3.1]{LL2}). 
More precisely, assume that $\Fg$ is of type $A_{n-1}$. 
For $r \in \{1, \ldots, n-1\}$, we denote by $B^{r, 1}$ the column-shape Kirillov--Reshetikhin crystal (realized by column-shape semi-standard Young tableaux of height $r$ with entries in $\{ 1, \ldots, n \}$). 
Let $\bp = (p_{1}, \ldots, p_{k})$, with $p_{1}, \ldots, p_{k} \in \{1, \ldots, n-1\}$, be a composition, 
and $\bp' = (p_{1}', \ldots, p_{k}')$ a composition obtained from $\bp$ by permuting its parts. 
It is known (see, for example, \cite[Remarks~2.2\,(2)]{LL2}) that there exists a unique $A_{n-1}^{(1)}$-crystal isomorphism 
\begin{equation*}
B^{\otimes \bp} := B^{p_{1}, 1} \otimes \cdots \otimes B^{p_{k}, 1} \xrightarrow{\sim} B^{\otimes \bp'} := B^{p_{1}', 1} \otimes \cdots \otimes B^{p_{k}', 1}, 
\end{equation*}
called a \emph{combinatorial $R$-matrix}. 
Let $I$ be the index set for the simple roots of $\Fg$; in our case, $I = \{1, \ldots, n-1\}$.  
Let $\alpha_{i}$, $i \in I$, be the simple roots, and $\vpi_{i}$, $i \in I$, the fundamental weights. 
For each $k \in I$, the sequence $\Gamma(k)$ of roots given as 
\begin{equation*}
\begin{array}{lllll}
\Gamma(k) := (\!\!\!\!\!\!& \alpha_{k}, & \alpha_{k} + \alpha_{k+1}, & \cdots, & \alpha_{k} + \cdots + \alpha_{n-1}, \\ 
& \alpha_{k-1} + \alpha_{k}, & \alpha_{k-1} + \alpha_{k} + \alpha_{k+1}, & \cdots, & \alpha_{k-1} + \cdots + \alpha_{n-1}, \\ 
& & & \cdots & \\ 
& \alpha_{1} + \cdots + \alpha_{k}, & \alpha_{1} + \cdots + \alpha_{k+1}, & \cdots, & \alpha_{1} + \cdots + \alpha_{n-1}) 
\end{array}
\end{equation*}
is a $\vpi_{k}$-chain (see \cite[Corollary~15.4]{LP1}). 
Let $\Gamma$ (resp., $\Gamma'$) be the concatenation $\Gamma(p_{1}) \ast \cdots \ast \Gamma(p_{k})$ of $\Gamma(p_{1}), \ldots, \Gamma(p_{k})$ 
(resp., $\Gamma(p_{1}') \ast \cdots \ast \Gamma(p_{k}')$ of $\Gamma(p_{1}'), \ldots, \Gamma(p_{k}')$) in this order. 
Then, there exists a unique $A_{n-1}^{(1)}$-crystal isomorphism between $\CA(e, \Gamma)$ (resp., $\CA(e, \Gamma')$) 
and the subgraph of $B^{\otimes \bp}$ (resp., $B^{\otimes \bp'}$) having only the dual Demazure arrows (\cite[Corollary~3.10 and Remark~3.13]{LL2}). 
Thus the bijection $\CA(e, \Gamma) \xrightarrow{\sim} \CA(e, \Gamma')$, given by quantum Yang--Baxter moves, can be viewed 
as a realization of the combinatorial $R$-matrix $B^{\otimes \bp} \xrightarrow{\sim} B^{\otimes \bp'}$. 
To summarize, we have the following commutative diagram: 

\centering
\begin{tikzpicture}[auto, ->]
\node (AG) at (0, 2) {$\CA(e, \Gamma)$}; 
\node (AG') at (7, 2) {$\CA(e, \Gamma')$}; 
\node (Bp) at (0, 0) {$B^{\otimes \bp}$}; 
\node (Bp') at (7, 0) {$B^{\otimes \bp'}$}; 
\draw (AG) -- node[font=\footnotesize] {quantum Yang--Baxter moves} (AG'); 
\draw (Bp) -- node[font=\footnotesize, align=center] {combinatorial $R$-matrix \\ (jeu de taquin)} (Bp'); 
\draw[swap] (AG) -- node {$\simeq$} (Bp); 
\draw (AG') -- node {$\simeq$} (Bp'); 
\end{tikzpicture}
\end{rem}

The purpose of this work is to generalize quantum Yang--Baxter moves to the case of an arbitrary (not necessarily dominant) $\lambda \in P$. 
However, for $\lambda \in P$ and $w \in W$, two collections $\CA(w, \Gamma_{1})$ and $\CA(w, \Gamma_{2})$ of admissible subsets
for $\lambda$-chains $\Gamma_{1}$ and $\Gamma_{2}$ can have different cardinalities; 
hence we cannot construct any bijection $\CA(w, \Gamma_{1}) \rightarrow \CA(w, \Gamma_{2})$ in general. 
For this reason, we need the notion of ``sijection'', \textit{i.e.}, 
signed bijection, to state a generalization of quantum Yang--Baxter moves.

\begin{thm}[{\cite[Theorems~3.2, 3.4; Proposition~5.3]{KLN}}] \label{thm:YB-move}
Let $\lambda \in P$ and $w \in W$. 
Take $\lambda$-chains $\Gamma_{1}$ and $\Gamma_{2}$ such that $\Gamma_{2}$ is obtained from $\Gamma_{1}$ by applying (YB), or by applying (D) 
in which a segment $(\beta, -\beta)$ of $\Gamma_{1}$, with $\beta$ not a simple root, is deleted.  
Then, there exist explicit subsets $\CA_{0}(w, \Gamma_{1}) \subset \CA(w, \Gamma_{1})$ and $\CA_{0}(w, \Gamma_{2}) \subset \CA(w, \Gamma_{2})$ satisfying the following. 
\begin{enu}
\item There exists a bijection $Y \colon \CA_{0}(w, \Gamma_{1}) \rightarrow \CA_{0}(w, \Gamma_{2})$ such that for $A \in \CA_{0}(w, \Gamma_{1})$, 
\begin{itemize}
\item $\ed(Y(A)) = \ed(A)$, 
\item $\down(Y(A)) = \down(A)$, 
\item $\wt(Y(A)) = \wt(A)$, and 
\item $\height(Y(A)) = \height(A)$. 
\end{itemize}
Also, this bijection $Y$ is ``sign-preserving''; that is, $(-1)^{n(Y(A))} = (-1)^{n(A)}$ for all $A \in \CA(w, \Gamma_{k})$. 

\item For $k = 1, 2$, set $\CA_{0}^{C}(w, \Gamma_{k}) := \CA(w, \Gamma_{k}) \setminus \CA_{0}(w, \Gamma_{k})$. 
Then, there exists an involution $I_{k}$ on $\CA_{0}^{C}(w, \Gamma_{k})$ such that 
\begin{itemize}
\item $\ed(I_{k}(A)) = \ed(A)$, 
\item $\down(I_{k}(A)) = \down(A)$, 
\item $\wt(I_{k}(A)) = \wt(A)$, and 
\item $\height(I_{k}(A)) = \height(A)$. 
\end{itemize}
Also, this involution $I_{k}$ is ``sign-reversing''; that is, $(-1)^{n(I_{k}(A))} = -(-1)^{n(A)}$ for all $A \in \CA_{0}^{C}(w, \Gamma_{k})$. 
\end{enu}
\end{thm}

\begin{rem}[{\cite[Remark~3.3]{KLN}}]
We now recall from Fischer--Konvalinka \cite{FK} the notion of sijection. 
Let $S$ (resp., $T$) be a signed set equipped with a sign function $S \rightarrow \{\pm 1\}$ (resp., $T \rightarrow \{\pm 1\}$), 
and $S_{0} \subset S$ (resp., $T_{0} \subset T$) a subset of $S$ (resp., $T$). 
Then, a \emph{sijection} $S \Rightarrow T$ is a triple $(\iota_{S}, \iota_{T}, \varphi)$ consisting of a sign-reversing involution $\iota_{S}$ (resp., $\iota_{T}$) on $S \setminus S_{0}$ (resp., $T \setminus T_{0}$) 
and a sign-preserving bijection $\varphi \colon S_{0} \rightarrow T_{0}$. 
Roughly speaking, a sijection is a ``signed bijection'' between signed sets. 
In this terminology, the maps $I_{1}, I_{2}, Y$ in Theorem~\ref{thm:YB-move} provide a sijection $(I_{1}, I_{2}, Y) \colon \CA(w, \Gamma_{1}) \Rightarrow \CA(w, \Gamma_{2})$. 
This sijection can be thought of as a generalization of the quantum Yang--Baxter move $\CA(e, \Gamma_{1}) \rightarrow \CA(e, \Gamma_{2})$ in the case of a dominant weight. 
\end{rem}

\begin{exm} \label{exm:qYB_example}
Assume that $\Fg$ is of type $A_{2}$; in this example, $I = \{1, 2\}$. 
Let $\Gamma_{1} := (-\alpha_{1}-\alpha_{2}, -\alpha_{1}, \alpha_{2}, \alpha_{1}+\alpha_{2})$, 
$\Gamma_{2} := (\alpha_{2}, -\alpha_{1}, -\alpha_{1}-\alpha_{2}, \alpha_{1}+\alpha_{2})$, and 
$\Gamma_{3} := (\alpha_{2}, -\alpha_{1})$. 
We see that $\Gamma_{1}$, $\Gamma_{2}$, and $\Gamma_{3}$ are $(-\vpi_{1}+\vpi_{2})$-chains. 
Also, $\Gamma_{2}$ is obtained from $\Gamma_{1}$ by applying (YB) in which the segment $(-\alpha_{1}-\alpha_{2}, -\alpha_{1}, \alpha_{2})$ in $\Gamma_{1}$ is reversed, 
while $\Gamma_{3}$ is obtained from $\Gamma_{2}$ by applying (D) in which the segment $(-\alpha_{1}-\alpha_{2}, \alpha_{1}+\alpha_{2})$ in $\Gamma_{2}$ is deleted; 
note that $\alpha_{1}+\alpha_{2}$ is not a simple root. 
Below we will describe explicitly generalized quantum Yang--Baxter moves $\CA(s_{1}, \Gamma_{1}) \Rightarrow \CA(s_{1}, \Gamma_{2})$ 
and $\CA(s_{1}, \Gamma_{2}) \Rightarrow \CA(s_{1}, \Gamma_{3})$. 

First of all, we give $A \in \CA(s_{1}, \Gamma_{k})$, $k = 1, 2, 3$, together with $\ed(A)$ and $\down(A)$, in Tables~\ref{tab:list1}, \ref{tab:list2}, and \ref{tab:list3}. 
\begin{table}[htbp]
\begin{minipage}[t]{0.49\hsize}
\centering
\caption{List of $A \in \CA(s_{1}, \Gamma_{1})$}
\label{tab:list1}
\begin{tabular}{|c||ccc|} \hline
$A$ & $n(A)$ & $\ed(A)$ & $\down(A)$ \\ \hline
$\emptyset$ & $0$ & $s_{1}$ & $0$ \\ 
$\{1\}$ & $1$ & $s_{2}s_{1}$ & $0$ \\ 
$\{2\}$ & $1$ & $e$ & $\alpha_{1}^{\vee}$ \\ 
$\{3\}$ & $0$ & $s_{1}s_{2}$ & $0$ \\ 
$\{4\}$ & $0$ & $s_{2}s_{1}$ & $0$ \\ 
$\{1, 2\}$ & $2$ & $s_{2}$ & $\alpha_{1}^{\vee}$ \\ 
$\{1, 3\}$ & $1$ & $s_{1}s_{2}s_{1}$ & $0$ \\ 
$\{2, 3\}$ & $1$ & $s_{2}$ & $\alpha_{1}^{\vee}$ \\ 
$\{1, 2, 3\}$ & $2$ & $e$ & $\alpha_{1}^{\vee} + \alpha_{2}^{\vee}$ \\ 
$\{1, 2, 4\}$ & $2$ & $s_{1}s_{2}$ & $\alpha_{1}^{\vee}$ \\ 
$\{1, 3, 4\}$ & $1$ & $e$ & $\alpha_{1}^{\vee} + \alpha_{2}^{\vee}$ \\ 
$\{2, 3, 4\}$ & $1$ & $s_{1}s_{2}$ & $\alpha_{1}^{\vee}$ \\ \hline
\end{tabular}
\end{minipage}
\hfill
\begin{minipage}[t]{0.49\hsize}
\centering
\caption{List of $A \in \CA(s_{1}, \Gamma_{2})$}
\label{tab:list2}
\begin{tabular}{|c||ccc|} \hline
$A$ & $n(A)$ & $\ed(A)$ & $\down(A)$ \\ \hline
$\emptyset$ & $0$ & $s_{1}$ & $0$ \\ 
$\{1\}$ & $0$ & $s_{1}s_{2}$ & $0$ \\ 
$\{2\}$ & $1$ & $e$ & $\alpha_{1}^{\vee}$ \\ 
$\{3\}$ & $1$ & $s_{2}s_{1}$ & $0$ \\ 
$\{4\}$ & $0$ & $s_{2}s_{1}$ & $0$ \\ 
$\{1, 2\}$ & $1$ & $s_{1}s_{2}s_{1}$ & $0$ \\ 
$\{1, 2, 3\}$ & $2$ & $e$ & $\alpha_{1}^{\vee} + \alpha_{2}^{\vee}$ \\ 
$\{1, 2, 4\}$ & $1$ & $e$ & $\alpha_{1}^{\vee} + \alpha_{2}^{\vee}$ \\ \hline
\end{tabular}

\caption{List of $A \in \CA(s_{1}, \Gamma_{3})$}
\label{tab:list3}
\begin{tabular}{|c||ccc|} \hline
$A$ & $n(A)$ & $\ed(A)$ & $\down(A)$ \\ \hline
$\emptyset$ & $0$ &  $s_{1}$ & $0$ \\ 
$\{1\}$ & $0$ & $s_{1}s_{2}$ & $0$ \\ 
$\{2\}$ & $1$ & $e$ & $\alpha_{1}^{\vee}$ \\ 
$\{1, 2\}$ & $1$ & $s_{1}s_{2}s_{1}$ & $0$ \\ \hline
\end{tabular}
\end{minipage}
\end{table}

Let us describe the sijection $\CA(s_{1}, \Gamma_{1}) \Rightarrow \CA(s_{1}, \Gamma_{2})$ as follows. 
We set 
\begin{align*}
\CA_{0}(s_{1}, \Gamma_{1}) &:= \{ \emptyset, \{1\}, \{2\}, \{3\}, \{4\}, \{1, 3\}, \{1, 2, 3\}, \{1, 3, 4\} \}, \\ 
\CA_{0}^{C}(s_{1}, \Gamma_{1}) &:= \{ \{1, 2\}, \{2, 3\}, \{1, 2, 4\}, \{2, 3, 4\}\}, \\ 
\CA_{0}(s_{1}, \Gamma_{2}) &:= \CA(s_{1}, \Gamma_{2}), \\ 
\CA_{0}^{C}(s_{1}, \Gamma_{2}) &:= \emptyset. 
\end{align*}
Then, the bijection $Y \colon \CA_{0}(s_{1}, \Gamma_{1}) \rightarrow \CA_{0}(s_{1}, \Gamma_{2})$ and the involution $I_{1}$ on $\CA_{0}^{C}(s_{1}, \Gamma_{1})$ 
are given in Tables~\ref{tab:Y12} and \ref{tab:I11}; here, the involution $I_{2}$ on $\CA_{0}^{C}(s_{1}, \Gamma_{2}) = \emptyset$ is trivial. 
\begin{table}[htbp]
\begin{minipage}[t]{0.49\hsize}
\centering
\caption{Description of $Y$}
\label{tab:Y12}
\begin{tabular}{|c||c|} \hline
$A \in \CA_{0}(s_{1}, \Gamma_{1})$ & $Y(A) \in \CA_{0}(s_{1}, \Gamma_{2})$ \\ \hline
$\emptyset$ & $\emptyset$ \\ 
$\{1\}$ & $\{3\}$ \\ 
$\{2\}$ & $\{2\}$ \\ 
$\{3\}$ & $\{1\}$ \\ 
$\{4\}$ & $\{4\}$ \\ 
$\{1, 3\}$ & $\{1, 2\}$ \\ 
$\{1, 2, 3\}$ & $\{1, 2, 3\}$ \\ 
$\{1, 3, 4\}$ & $\{1, 2, 4\}$ \\ \hline
\end{tabular}
\end{minipage}
\hfill
\begin{minipage}[t]{0.49\hsize}
\centering
\caption{Description of $I_{1}$}
\label{tab:I11}
\begin{tabular}{|c||c|} \hline
$A \in \CA_{0}^{C}(s_{1}, \Gamma_{1})$ & $I_{1}(A) \in \CA_{0}^{C}(s_{1}, \Gamma_{1})$ \\ \hline
$\{1, 2\}$ & $\{2, 3\}$ \\ 
$\{2, 3\}$ & $\{1, 2\}$ \\ 
$\{1, 2, 4\}$ & $\{2, 3, 4\}$ \\ 
$\{2, 3, 4\}$ & $\{1, 2, 4\}$ \\ \hline
\end{tabular}
\end{minipage}
\end{table}

Let us describe the sijection $\CA(s_{1}, \Gamma_{2}) \Rightarrow \CA(s_{1}, \Gamma_{3})$ as follows. We set 
\begin{align*}
\CA_{0}(s_{1}, \Gamma_{2}) &:= \{ \emptyset, \{1\}, \{2\}, \{1, 2\} \}, \\ 
\CA_{0}^{C}(s_{1}, \Gamma_{2}) &:= \{ \{3\}, \{4\}, \{1, 2, 3\}, \{1, 2, 4\} \}, \\ 
\CA_{0}(s_{1}, \Gamma_{3}) &:= \CA(s_{1}, \Gamma_{3}), \\ 
\CA_{0}^{C}(s_{1}, \Gamma_{3}) &:= \emptyset. 
\end{align*}
Then, the bijection $Y \colon \CA_{0}(s_{1}, \Gamma_{2}) \rightarrow \CA_{0}(s_{1}, \Gamma_{3})$ and the involution $I_{1}$ on $\CA_{0}^{C}(s_{1}, \Gamma_{2})$ 
are given in  Tables~\ref{tab:Y23} and \ref{tab:I12}; here, the involution $I_{2}$ on $\CA_{0}^{C}(s_{1}, \Gamma_{3}) = \emptyset$ is trivial. 
\begin{table}[htbp]
\begin{minipage}[t]{0.49\hsize}
\centering
\caption{Description of $Y$}
\label{tab:Y23}
\begin{tabular}{|c||c|} \hline 
$A \in \CA_{0}(s_{1}, \Gamma_{2})$ & $Y(A) \in \CA_{0}(s_{1}, \Gamma_{3})$ \\ \hline 
$\emptyset$ & $\emptyset$ \\ 
$\{1\}$ & $\{1\}$ \\ 
$\{2\}$ & $\{2\}$ \\ 
$\{1, 2\}$ & $\{1, 2\}$ \\ \hline
\end{tabular}
\end{minipage}
\hfill
\begin{minipage}[t]{0.49\hsize}
\centering
\caption{Description of $I_{1}$}
\label{tab:I12}
\begin{tabular}{|c||c|} \hline
$A \in \CA_{0}^{C}(s_{1}, \Gamma_{2})$ & $I_{1}(A) \in \CA_{0}^{C}(s_{1}, \Gamma_{2})$ \\ \hline 
$\{3\}$ & \{4\} \\ 
$\{4\}$ & \{3\} \\ 
$\{1, 2, 3\}$ & \{1, 2, 4\} \\ 
$\{1, 2, 4\}$ & \{1, 2, 3\} \\ \hline 
\end{tabular}
\end{minipage}
\end{table}
\end{exm}

For details of the construction of $Y$, $I_{1}$, and $I_{2}$, see \cite[Section~4.6 and Proposition~5.3]{KLN}. 

\begin{rem} \label{rem:deletion}
Theorem~\ref{thm:YB-move} for a procedure (D) does not hold if $\beta$ is a simple root; see \cite[Remark~5.4]{KLN}. 
\end{rem}

\section{Schubert calculus of semi-infinite flag manifolds}

In this section, we apply generalized quantum Yang--Baxter moves to the study of the equivariant $K$-group $\K$ of the semi-infinite flag manifold $\bQG$. 
For this purpose, we need additional notation. Following \cite[Sections~4.1, 4.3]{LNS}, for $\lambda = \sum_{i \in I}m_{i}\varpi_{i}$ with $m_{i} \in \BZ$, 
we define $\bPar(\lambda)$ by
\begin{equation*}
\bPar(\lambda) := \left\{ \bchi = (\chi^{(i)})_{i \in I} \ \middle| \ \parbox{19em}{$\chi^{(i)}$ is a partition of length less than or equal to $\max \{m_{i}, 0\}$} \right\}; 
\end{equation*}
here $I$ is the index set for the simple roots of $\Fg$. 
Also, by writing the partition $\chi^{(i)}$ for $i \in I$ as $(\chi_{1}^{(i)} \ge \cdots \ge \chi_{l_{i}}^{(i)})$, we set 
\begin{equation*}
|\bchi| := \sum_{i \in I} \sum_{k = 1}^{l_{i}} \chi_{k}^{(i)}, \quad \iota(\bchi) := \sum_{i \in I} \chi_{1}^{(i)} \alpha_{i}^{\vee}; 
\end{equation*}
if $\chi^{(i)} = \emptyset$, then we understand that $l_{i} = 0$ and $\chi_{1}^{(i)} = 0$. 

By making use of our generalized quantum Yang--Baxter moves, we can give a combinatorial proof of 
the Chevalley formula in the equivariant $K$-group of semi-infinite flag manifolds, first proved by Lenart--Naito--Sagaki based on the Yang--Baxter equation for quantum Bruhat operators. 

\begin{thm}[{\cite[Theorem~33]{LNS}}] \label{thm:Chevalley}
Let $\lambda \in P$ and $x \in W_{\af}^{\ge 0}$. Write $x = wt_{\xi}$ for $w \in W$ and $\xi \in Q^{\vee, +}$. 
Let $\Gamma$ be a reduced $\lambda$-chain. 
Then, in $\K$, the following identity holds: 
\begin{equation*}
\begin{split}
& [\CO(-w_{\circ}\lambda)] \cdot [\CO_{x}] \\ 
&= \sum_{A \in \CA(w, \Gamma)} \sum_{\bchi \in \bPar(\lambda)} (-1)^{n(A)} q^{-\height(A) - \pair{\lambda}{\xi} - |\bchi|} e^{\wt(A)} [\CO_{\ed(A)t_{\xi + \down(A) + \iota(\bchi)}}], 
\end{split}
\end{equation*}
where $w_{\circ} \in W$ denotes the longest element of $W$. 
\end{thm}

To prove this identity, we employ certain generating functions. 

\begin{defn}[{\cite[Definition~5.1]{KLN}}]
Let $\lambda \in P$, and take a $\lambda$-chain $\Gamma$. 
For $x = wt_{\xi} \in W_{\af}$ with $w \in W$ and $\xi \in Q^{\vee}$, we define the \emph{generating function} $\bG_{\Gamma}(x) \in (\BZ[q, q^{-1}])[P][W_{\af}]$ by 
\begin{equation*}
\bG_{\Gamma}(x) := \sum_{A \in \CA(w, \Gamma)} (-1)^{n(A)} q^{-\height(A) - \pair{\lambda}{\xi}} e^{\wt(A)} \ed(A) t_{\xi + \down(A)}. 
\end{equation*}
\end{defn}

The generalized quantum Yang--Baxter moves in Theorem~\ref{thm:YB-move} imply the following preservation of generating functions (Theorem~\ref{thm:generating_function}). 

\begin{thm}[{\cite[Propositions~5.2, 5.3]{KLN}}] \label{thm:generating_function}
Let $\lambda \in P$ and $x \in W_{\af}$. 

\begin{enu}
\item Take $\lambda$-chains $\Gamma_{1}$ and $\Gamma_{2}$ such that $\Gamma_{2}$ is obtained from $\Gamma_{1}$ by applying (YB). 
Then we have $\bG_{\Gamma_{1}}(x) = \bG_{\Gamma_{2}}(x)$. 

\item Take $\lambda$-chains $\Gamma_{1}$ and $\Gamma_{2}$ such that $\Gamma_{2}$ is obtained from $\Gamma_{1}$ by applying (D) 
in which the segment $(\beta, -\beta)$, with $\beta$ not a simple root, is deleted. 
Then we have $\bG_{\Gamma_{1}}(x) = \bG_{\Gamma_{2}}(x)$. 
\end{enu}
\end{thm}

\begin{rem}
In the same way as Theorem~\ref{thm:YB-move}, Theorem~\ref{thm:generating_function}\,(2) does not hold if $\beta$ is a simple root; 
see \cite[Remark~5.4]{KLN}. 
\end{rem}

\begin{proof}[Proof of Theorem~\ref{thm:Chevalley}]
Theorem~\ref{thm:generating_function} plays a crucial role in our proof of Theorem~\ref{thm:Chevalley}. 
First, we write $\lambda$ as the sum $\lambda^{-} + \lambda^{+}$, with $\lambda^{-} \in -P^{+}$ and $\lambda^{+} \in P^{+}$. 
Then, we take a ``lex'' $\lambda^{\pm}$-chain $\Gamma^{\pm}$ (see \cite[Proposition~4.2]{LP2} and \cite[Section~4.2]{LNS}). 
Note that we already know from \cite[Theorems~29, 32]{LNS}
(see also \cite[Theorem~5.13]{KNS} and \cite[Corollaries~C.3, 9.1]{NOS}) the Chevalley formulas for the dominant weight $\lambda^{+}$ and the anti-dominant weight $\lambda^{-}$.
Let $\Gamma_{0}$ denote the concatenation of $\Gamma^{-}$ and $\Gamma^{+}$ in this order. 
Then we deduce that $\bG_{\Gamma^{-}} \circ \bG_{\Gamma^{+}}(x) = \bG_{\Gamma_{0}}(x)$ (see \cite[Theorem~5.10]{KLN}). 
Here, by \cite[Claim~46.1]{LNS} and \cite[Proposition~5.8]{KNS}, we obtain the given reduced $\lambda$-chain $\Gamma$ from the $\Gamma_{0}$ 
by repeated application of (YB), and by that of (D) in which the segment $(\beta, -\beta)$, with $\beta$ not a simple root, is deleted. 
Therefore, Theorem~\ref{thm:generating_function} yields the equality $\bG_{\Gamma_{0}}(x) = \bG_{\Gamma}(x)$. 
This equality immediately implies Theorem~\ref{thm:Chevalley}. 
For the details of this proof, see \cite[Section~5]{KLN}. 
\end{proof}

% Acknowledgements
%\acknowledgements{
%T.~Kouno was supported in part by JSPS Grant-in-Aid for Scientific Research 20J12058 and 22J00874. 
%C.~Lenart was supported in part by the NSF grant DMS-1855592. 
%S.~Naito was supported in part by JSPS Grant-in-Aid for Scientific Research (B) 16H03920. 
%}
% biblatex
\printbibliography

\end{document}
