centertableaux,mathmode,boxsize=1em

# Monomial realizations and LS paths of fundamental representations for rank
$2$ Kac-Moody algebras

YUKI KANAKUBO Faculty of Basic Natural Science, Ibaraki University :
<EMAIL_ADDRESS>
###### Abstract

For a Kac-Moody algebra $\mathfrak{g}$ of rank $2$ and a fundamental weight
$\lambda$, we explicitly give an isomorphism between the set of Lakshmibai-
Seshadri paths $\mathbb{B}(\lambda)$ and monomial realization
$\mathcal{M}(\lambda)$. As an application, we also give an explicit form of
monomial realization $\mathcal{M}(\lambda)$ in terms of Weyl groups.

##  1 Introduction

For a symmetrizable Kac-Moody algebra $\mathfrak{g}$, the crystal bases
powerful combinatorial tools to study representations of quantum groups
$U_{q}(\mathfrak{g})$ [Kas90]. A bunch of combinatorial descriptions of
crystal bases have been developed.

In [Nak, Kas03], monomial realizations of crystal bases $B(\lambda)$ of
highest weight representations are invented, which realize elements of
$B(\lambda)$ in terms of Laurent monomials of double indexed variables
$\\{X_{s,i}|s\in\mathbb{Z},i\in I\\}$. Here, $\lambda$ is a dominant integral
weight and $I$ is the index set for simple roots of $\mathfrak{g}$. In
particular, monomial realizations for rank $2$ Kac-Moody algebras are studied
in [JKKS]. In [Lit94, Lit95], a piece-wise linear path $\pi:[0,1]\rightarrow
P\otimes\mathbb{R}$ called Lakshmibai-Seshadri path (LS path) are introduced.
Here, $P$ is a weight lattice. The endpoint $\pi(1)$ is called a weight of
$\pi$ and denoted by ${\rm wt}(\pi)$. One can define operators $f_{k}$ and
$e_{k}$ on LS paths for $k\in I$ called root operators and the set of LS paths
of shape $\lambda$ denoted by $\mathbb{B}(\lambda)$ has a crystal structure by
wt, $f_{k}$, $e_{k}$ and some $\varepsilon$ and $\varphi$-functions. It is
shown in [Kas96, Jos] that $\mathbb{B}(\lambda)$ is isomorphic to
$B(\lambda)$.

In this paper, we will explicitly give an isomorphism between the set of LS
paths $\mathbb{B}(\lambda)$ and monomial realization $\mathcal{M}(\lambda)$
when $\mathfrak{g}$ is of rank $2$ and $\lambda$ is a fundamental weight. Each
LS path is defined by a sequence of rational numbers
$0=a_{0}<a_{1}<\cdots<a_{r}=1$ and Weyl group elements
$\tau_{1}<\cdots<\tau_{r}$. The isomorphism is constructed by these rational
numbers and Weyl group elements (see Sect.5). As an application, we also give
an explicit form of monomial realization in terms of Weyl groups.

The paper is organized as follows. In Sect.2, we review crystals. In Sect.3
and Sect.4, we review LS path models and monomial realizations. We give our
main results in Sect.5 and prove it in Sect.6.

Acknowledgements I would like to thank Daisuke Sagaki for his helpful comments
and fruitful discussions.

##  2 Crystal

Following [Kas93], let us review crystals.

###  2.1 Notation

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra with Cartan subalgebra
$\mathfrak{h}$, generalized Cartan matrix $(a_{i,j})_{i,j\in I}$ and index set
$I=\\{1,2,\cdots,{\rm rank}\ \mathfrak{h}\\}$. Let $P$ and $P^{\vee}$ be the
weight lattice and dual weight lattice with canonical pairing
$\langle,\rangle:P^{\vee}\times P\rightarrow\mathbb{Z}$. Let
$\\{\alpha_{i}\\}_{i\in I}\subset P$ and $\\{h_{i}\\}_{i\in I}\subset
P^{\vee}$ be the sets of simple roots and simple coroots. Let $P^{+}\subset P$
denote the set of dominant integral weights. The fundamental weights
$\Lambda_{i}\in P^{+}$ ($\in I$) are defined as
$\Lambda_{i}(h_{j})=\delta_{i,j}$. We define $W$ as its Weyl group with simple
reflections $s_{i}$ ($i\in I$). For $a,b\in\mathbb{Z}$ with $a\leq b$, let
$[a,b]_{\rm int}:=\\{a,a+1,\cdots,b\\}$ and for $x,y\in\mathbb{R}$ with $x\leq
y$, let $[x,y]$ denote the closed interval from $x$ to $y$.

###  2.2 Crystals

######  Definition 2.1.

[Kas93] A set $\mathcal{B}$ equipped with the maps ${\rm
wt}:\mathcal{B}\rightarrow P$,
$\varepsilon_{k},\varphi_{k}:\mathcal{B}\rightarrow\mathbb{Z}\sqcup\\{-\infty\\}$
and
$\tilde{e}_{k}$,$\tilde{f}_{k}:\mathcal{B}\rightarrow\mathcal{B}\sqcup\\{0\\}$
($k\in I$) is said to be a $P$-crystal if the following holds: For
$b,b^{\prime}\in\mathcal{B}$ and $k\in I$,

  * •

$\varphi_{k}(b)=\varepsilon_{k}(b)+\langle h_{k},{\rm wt}(b)\rangle$,

  * •

${\rm wt}(\tilde{e}_{k}b)={\rm wt}(b)+\alpha_{k}\text{ if
}\tilde{e}_{k}(b)\in\mathcal{B},\quad{\rm wt}(\tilde{f}_{k}b)={\rm
wt}(b)-\alpha_{k}\text{ if }\tilde{f}_{k}(b)\in\mathcal{B},$
$\varepsilon_{k}(\tilde{e}_{k}(b))=\varepsilon_{k}(b)-1,\ \
\varphi_{k}(\tilde{e}_{k}(b))=\varphi_{k}(b)+1\text{ if
}\tilde{e}_{k}(b)\in\mathcal{B},$
$\varepsilon_{k}(\tilde{f}_{k}(b))=\varepsilon_{k}(b)+1,\ \
\varphi_{k}(\tilde{f}_{k}(b))=\varphi_{k}(b)-1\text{ if
}\tilde{f}_{k}(b)\in\mathcal{B},$

  * •

$\tilde{f}_{k}(b)=b^{\prime}$ is equivalent to $b=\tilde{e}_{k}(b^{\prime})$,

  * •

if $\varphi_{k}(b)=-\infty$ then it holds
$\tilde{e}_{k}(b)=\tilde{f}_{k}(b)=0$.

Here, $0$ and $-\infty$ are additional elements which do not belong to
$\mathcal{B}$ and $\mathbb{Z}$, respectively. We call the maps
$\tilde{e}_{k}$,$\tilde{f}_{k}$ Kashiwara operators.

The crystal bases $B(\lambda)$ $(\lambda\in P^{+})$ and $B(\infty)$ have
$P$-crystal structures.

######  Definition 2.2.

We define $P_{\rm cl}:=P/\\{\lambda\in P|\lambda(h_{j})=0\text{ for all }j\in
I\\}$. A set $\mathcal{B}$ equipped with the maps ${\rm
wt}:\mathcal{B}\rightarrow P_{\rm cl}$,
$\varepsilon_{k},\varphi_{k}:\mathcal{B}\rightarrow\mathbb{Z}\sqcup\\{-\infty\\}$
and
$\tilde{e}_{k}$,$\tilde{f}_{k}:\mathcal{B}\rightarrow\mathcal{B}\sqcup\\{0\\}$
($k\in I$) is said to be a $P_{\rm cl}$-crystal if the conditions in
Definition 2.1 hold.

######  Remark 2.3.

  1. (1)

A $P$-crystal $\mathcal{B}$ naturally has a $P_{\rm cl}$-crystal structure by
defining ${\rm wt}:\mathcal{B}\rightarrow P_{\rm cl}$ as the composition of
${\rm wt}:\mathcal{B}\rightarrow P$ and the projection $P\rightarrow P_{\rm
cl}$.

  2. (2)

By an abuse of notation, we use the same symbol $\Lambda_{j}$, $\alpha_{j}\in
P_{\rm cl}$ for the image of $\Lambda_{j}$, $\alpha_{j}\in P$ under the
projection $P\rightarrow P_{\rm cl}$.

######  Definition 2.4.

Let $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ be $P$-crystals (resp. $P_{\rm
cl}$-crystals).

  1. $(1)$

A map
$\psi:\mathcal{B}_{1}\sqcup\\{0\\}\rightarrow\mathcal{B}_{2}\sqcup\\{0\\}$ is
called a strict morphism if $\psi(0)=0$ and the following holds:

     * •

For $k\in I$, if $b\in\mathcal{B}_{1}$ and $\psi(b)\in\mathcal{B}_{2}$ then we
have

${\rm wt}(\psi(b))={\rm
wt}(b),\quad\varepsilon_{k}(\psi(b))=\varepsilon_{k}(b),\quad\varphi_{k}(\psi(b))=\varphi_{k}(b),$

     * •

$\tilde{e}_{k}(\psi(b))=\psi(\tilde{e}_{k}(b))$ and
$\tilde{f}_{k}(\psi(b))=\psi(\tilde{f}_{k}(b))$ for $k\in I$ and
$b\in\mathcal{B}_{1}$, where we set $\tilde{e}_{k}(0)=\tilde{f}_{k}(0)=0$.

  2. $(2)$

If a strict morphism
$\psi:\mathcal{B}_{1}\sqcup\\{0\\}\rightarrow\mathcal{B}_{2}\sqcup\\{0\\}$ is
a bijection then $\psi$ is said to be an isomorphism. Then we say
$\mathcal{B}_{1}$ is isomorphic to $\mathcal{B}_{2}$ as $P$-crystals (resp.
$P_{\rm cl}$-crystals).

##  3 LS paths

###  3.1 Definition of LS paths

Following [Lit94], let us review the Lakshmibai-Seshadri paths. For
$\lambda\in P^{+}$, let $W_{\lambda}:=\\{w\in W|w\lambda=\lambda\\}$ be the
stabilizer of $\lambda$ and let $\geq$ be the Bruhat order on $W/W_{\lambda}$.

Let $\underline{\tau}=(\tau_{1},\cdots,\tau_{r})$ be a sequence of elements in
$W/W_{\lambda}$ such that $\tau_{1}>\cdots>\tau_{r}$. Let
$\underline{a}=(a_{0},a_{1},\cdots,a_{r})$ be a sequence of rational numbers
such that $a_{0}:=0<a_{1}<\cdots<a_{r}:=1$. The pair
$\pi=(\underline{\tau};\underline{a})$ is called a rational $W$-path of shape
$\lambda$. The path $\pi$ is identified with a piecewise linear path
$\pi:[0,1]\rightarrow P_{\mathbb{R}}$ defined as

$\pi(t):=\sum_{i=1}^{j-1}(a_{i}-a_{i-1})\tau_{i}(\lambda)+(t-a_{j-1})\tau_{j}(\lambda)\quad\text{for
}a_{j-1}\leq t\leq a_{j},$

where $P_{\mathbb{R}}:=P\otimes_{\mathbb{Z}}\mathbb{R}$. Let
$h_{k}:[0,1]\rightarrow\mathbb{R}$ be a function defined as $t\mapsto\langle
h_{k},\pi(t)\rangle$ for $k\in I$.

For $\tau,\sigma\in W/W_{\lambda}$ such that $\tau>\sigma$ and
$a\in\mathbb{Q}$ such that $0<a<1$, one defines an $a$-chain for the pair
$(\tau,\sigma)$ as

$\kappa_{0}:=\tau>\kappa_{1}:=s_{\beta_{1}}\tau>\kappa_{2}:=s_{\beta_{2}}s_{\beta_{1}}\tau>\cdots>\kappa_{s}:=s_{\beta_{s}}\cdots
s_{\beta_{1}}\tau=\sigma,$

where $\beta_{1},\cdots,\beta_{s}$ are positive real roots satisfying for each
$i=1,\cdots,s$,

$l(\kappa_{i})=l(\kappa_{i-1})-1\quad\text{and
}a\langle\beta_{i}^{\vee},\kappa_{i}(\lambda)\rangle\in\mathbb{Z}.$

######  Definition 3.1.

[Lit94] Let $\pi=(\underline{\tau};\underline{a})$ be a rational $W$-path of
shape $\lambda$ with $\underline{\tau}=(\tau_{1},\cdots,\tau_{r})$ and
$\underline{a}=(a_{0},a_{1},\cdots,a_{r})$. We say $\pi$ is a Lakshmibai-
Seshadri path (in short LS path) of shape $\lambda$ if for each
$i=1,2,\cdots,r-1$, there is an $a_{i}$-chain for the pair
$(\tau_{i},\tau_{i+1})$. Let $\mathbb{B}(\lambda)$ denote the set of
Lakshmibai-Seshadri paths of shape $\lambda$.

###  3.2 Action of Kashiwara operators on LS paths

We fix an index $k\in I$ and a LS path
$\pi=(\underline{\tau};\underline{a})=(\tau_{1},\cdots,\tau_{r};a_{0},a_{1},\cdots,a_{r})$.
Let

$Q:={\rm min}\\{h_{k}(t)|t\in[0,1]\\},\quad R:=h_{k}(1)-Q$ (3.1)

and $q$ (resp. $p$) be the minimal (resp. maximal) number in
$\\{0,1,2,\cdots,r\\}$ such that $h_{k}(a_{q})=Q$ (resp. $h_{k}(a_{p})=Q$). It
is shown $Q\in\mathbb{Z}$ [Lit94]. If $Q<0$ then we define

$y:={\rm max}\\{s\in[0,q]_{\rm int}|h_{k}(t)\geq Q+1\text{ for
}t\in[0,a_{s}]\\}.$

If $R>0$ then we also define

$x:={\rm min}\\{s\in[p,r]_{\rm int}|h_{k}(t)\geq Q+1\text{ for
}t\in[a_{s},1]\\}.$

######  Definition 3.2.

[Lit94]

  1. $($i$)$

If $R>0$ then $f_{k}(\pi)$ equals the following LS path:

$\begin{cases}(\tau_{1},\cdots,\tau_{p-1},s_{k}\tau_{p+1},\cdots,s_{k}\tau_{x},\tau_{x+1},\cdots,\tau_{r};a_{0},\cdots,a_{p-1},a_{p+1},\cdots,a_{r})&\text{
if }\begin{array}[]{l}h_{k}(a_{x})=Q+1,\\\
s_{k}\tau_{p+1}=\tau_{p},\end{array}\\\
(\tau_{1},\cdots,\tau_{p},s_{k}\tau_{p+1},\cdots,s_{k}\tau_{x},\tau_{x+1},\cdots,\tau_{r};a_{0},\cdots,a_{r})&\text{
if }\begin{array}[]{l}h_{k}(a_{x})=Q+1,\\\
s_{k}\tau_{p+1}<\tau_{p},\end{array}\\\
(\tau_{1},\cdots,\tau_{p-1},s_{k}\tau_{p+1},\cdots,s_{k}\tau_{x},\tau_{x},\cdots,\tau_{r};a_{0},\cdots,a_{p-1},a_{p+1},\cdots,a_{x-1},a,a_{x},\cdots,a_{r})&\text{
if }\begin{array}[]{l}h_{k}(a_{x})>Q+1,\\\
s_{k}\tau_{p+1}=\tau_{p},\end{array}\\\
(\tau_{1},\cdots,\tau_{p},s_{k}\tau_{p+1},\cdots,s_{k}\tau_{x},\tau_{x},\cdots,\tau_{r};a_{0},\cdots,a_{x-1},a,a_{x},\cdots,a_{r})&\text{
if }\begin{array}[]{l}h_{k}(a_{x})>Q+1,\\\
s_{k}\tau_{p+1}<\tau_{p}.\end{array}\end{cases}$

Here, $a$ satisfies $a_{x-1}<a<a_{x}$ and $h_{k}(a)=Q+1$. When $p=0$, we
understand $s_{k}\tau_{p+1}<\tau_{p}$.

  2. $($ii$)$

If $Q<0$ then $e_{k}(\pi)$ equals the following LS path:

$\begin{cases}(\tau_{1},\cdots,\tau_{y},s_{k}\tau_{y+1},\cdots,s_{k}\tau_{q},\tau_{q+2},\cdots,\tau_{r};a_{0},\cdots,a_{q},a_{q+2},\cdots,a_{r})&\text{
if }\begin{array}[]{l}h_{k}(a_{y})=Q+1,\\\
s_{k}\tau_{q}=\tau_{q+1},\end{array}\\\
(\tau_{1},\cdots,\tau_{y},s_{k}\tau_{y+1},\cdots,s_{k}\tau_{q},\tau_{q+1},\cdots,\tau_{r};a_{0},\cdots,a_{r})&\text{
if }\begin{array}[]{l}h_{k}(a_{y})=Q+1,\\\
s_{k}\tau_{q}>\tau_{q+1},\end{array}\\\
(\tau_{1},\cdots,\tau_{y+1},s_{k}\tau_{y+1},\cdots,s_{k}\tau_{q},\tau_{q+2},\cdots,\tau_{r};a_{0},\cdots,a_{y},a,a_{y+1},\cdots,a_{q},a_{q+2},\cdots,a_{r})&\text{
if }\begin{array}[]{l}h_{k}(a_{y})>Q+1,\\\
s_{k}\tau_{q}=\tau_{q+1},\end{array}\\\
(\tau_{1},\cdots,\tau_{y+1},s_{k}\tau_{y+1},\cdots,s_{k}\tau_{q},\tau_{q+1},\cdots,\tau_{r};a_{0},\cdots,a_{y},a,a_{y+1},\cdots,a_{r})&\text{
if }\begin{array}[]{l}h_{k}(a_{y})>Q+1,\\\
s_{k}\tau_{q}>\tau_{q+1}.\end{array}\\\ \end{cases}$

Here, $a$ satisfies $a_{y}<a<a_{y+1}$ and $h_{k}(a)=Q+1$. When $q=r$, we
understand $s_{k}\tau_{q}>\tau_{q+1}$.

We define

${\rm wt}(\pi):=\pi(1)=\sum_{i=1}^{r}(a_{i}-a_{i-1})\tau_{i}(\lambda),$ (3.2)
$\varepsilon_{k}(\pi):=\mathbb{Z}\cap{\rm
max}\\{-h_{k}(t)|t\in[0,1]\\},\quad\varphi_{k}(\pi):=\langle
h_{k},\pi(1)\rangle+\varepsilon_{k}(\pi).$ (3.3)

For $\pi\in\mathbb{B}(\lambda)$, it holds $\varepsilon_{k}(\pi)=-Q$ and
$\varphi_{k}(\pi)=R$ with $Q$, $R$ in (3.1).

######  Theorem 3.3.

[Kas96, Jos] For a dominant integral weight $\lambda$, the set
$\mathbb{B}(\lambda)$ with the above maps is a $P$-crystal isomorphic to
$B(\lambda)$.

###  3.3 Rank $2$ case

We assume $\mathfrak{g}$ is rank $2$. The generalized Cartan matrix $A$ is in
the form

$A=\begin{pmatrix}2&-a\\\ -b&2\end{pmatrix}$

with $a,b\in\mathbb{Z}_{\geq 0}$.

One defines

$\tau_{2n}^{+}=(s_{2}s_{1})^{n},\quad\tau_{2n+1}^{+}=s_{1}(s_{2}s_{1})^{n},\quad\tau_{2n}^{-}=(s_{1}s_{2})^{n},\quad\tau_{2n+1}^{-}=s_{2}(s_{1}s_{2})^{n}$

for $n\in\mathbb{Z}_{\geq 0}$. We see that

$W/W_{\Lambda_{i}}=\\{\tau_{k}^{\varepsilon(i)}|0\leq k<N\\},$

where $N:=|\langle s_{2}s_{1}\rangle|$ and

$\varepsilon(i)=\begin{cases}+&\text{ if }i=1,\\\ -&\text{ if
}i=2.\end{cases}$

Note that $N=\infty$ when $\mathfrak{g}$ is not finite dimensional. Let $\xi$
be one of solutions for $x^{2}-(ab-2)x+1=0$ and

$[n]:=\xi^{n}+\xi^{n-2}+\cdots+\cdots+\xi^{-n+2}+\xi^{-n}$

for $n\geq 0$ and $[-1]:=0$. We also set

$d_{2n}^{+}:=-b[n-1],\quad d_{2n+1}^{+}:=-[n]-[n-1],\quad
d_{2n}^{-}:=-a[n-1],\quad d_{2n+1}^{-}:=-[n]-[n-1].$

######  Theorem 3.4.

[Sag]

$\mathbb{B}(\Lambda_{i})=\left\\{(\tau_{m_{2}}^{\varepsilon(i)},\tau_{m_{2}-1}^{\varepsilon(i)},\cdots,\tau_{m_{1}+1}^{\varepsilon(i)},\tau_{m_{1}}^{\varepsilon(i)};0,a_{m_{2}},a_{m_{2}-1},\cdots,a_{m_{1}+1},1)|\begin{array}[]{l}N>m_{2}\geq
m_{1}\geq 0,\\\ 0<a_{m_{2}}<\cdots<a_{m_{1}+1}<1,\\\
a_{j}d_{j}^{\varepsilon(i)}\in\mathbb{Z}\text{ for
}j=m_{1}+1,\cdots,m_{2}\end{array}\right\\}.$

######  Example 3.5.

We consider the case the Cartan matrix is of type $B_{2}$:
$A=\begin{pmatrix}2&-1\\\ -2&2\end{pmatrix}$. One obtains

$W/W_{\Lambda_{1}}=\\{e,s_{1},s_{2}s_{1},s_{1}s_{2}s_{1}\\}.$

Using the notation as above, it holds
$\\{\xi,\xi^{-1}\\}=\\{\sqrt{-1},-\sqrt{-1}\\}$ so that

$[n]=\begin{cases}0&\text{ if }n\text{ is odd},\\\ -1&\text{ if }n\equiv 2\
\text{mod }4,\\\ 1&\text{ if }n\equiv 0\ \text{mod }4.\end{cases}$

Hence, $d_{1}^{+}=-1$, $d_{2}^{+}=-2$, $d_{3}^{+}=-1$ so that $a_{1}=a_{3}=1$
and $a_{2}\in\\{1,\frac{1}{2}\\}$ if $a_{j}d_{j}^{+}\in\mathbb{Z}$ for
$j=1,2,3$. Therefore, when $m_{2}>m_{1}$, if $m_{1}=0$ (resp. $m_{2}\geq 3$)
then it contradicts $0<a_{1}<1$ (resp. $0<a_{3}<1$). Thus, if
$(\tau_{m_{2}}^{+},\tau_{m_{2}-1}^{+},\cdots,\tau_{m_{1}+1}^{+},\tau_{m_{1}}^{+};0,a_{m_{2}},a_{m_{2}-1},\cdots,a_{m_{1}+1},1)\in\mathbb{B}(\Lambda_{1})$
then it holds either $0\leq m_{1}=m_{2}\leq 3$ or $m_{2}=2$, $m_{1}=1$ and
$a_{2}=\frac{1}{2}$, that is,

$\mathbb{B}(\Lambda_{1})=\\{(e;0,1),\ (s_{1};0,1),\ (s_{2}s_{1};0,1),\
(s_{1}s_{2}s_{1};0,1),\ (s_{2}s_{1},s_{1};0,\frac{1}{2},1)\\}.$

The crystal graph is as follows:

$(e;0,1)\overset{1}{\rightarrow}(s_{1};0,1)\overset{2}{\rightarrow}(s_{2}s_{1},s_{1};0,\frac{1}{2},1)\overset{2}{\rightarrow}(s_{2}s_{1};0,1)\overset{1}{\rightarrow}(s_{1}s_{2}s_{1};0,1).$

######  Example 3.6.

We consider the case the generalized Cartan matrix is of type $A^{(1)}_{1}$:
$A=\begin{pmatrix}2&-2\\\ -2&2\end{pmatrix}$. A part of the crystal graph for
$\mathbb{B}(\Lambda_{1})$ is as follows:

$\textstyle{(e;0,1)}$$\textstyle{(s_{1};0,1)}$$\textstyle{(s_{2}s_{1},s_{1};0,\frac{1}{2},1)}$$\textstyle{(s_{2}s_{1};0,1)}$$\textstyle{(s_{1}s_{2}s_{1},s_{2}s_{1},s_{1};0,\frac{1}{3},\frac{1}{2},1)}$$\textstyle{(s_{1}s_{2}s_{1},s_{2}s_{1};0,\frac{1}{3},1)}$$\textstyle{(s_{2}s_{1}s_{2}s_{1},s_{1}s_{2}s_{1},s_{2}s_{1},s_{1};0,\frac{1}{4},\frac{1}{3},\frac{1}{2},1)}$$\textstyle{(s_{1}s_{2}s_{1},s_{2}s_{1};0,\frac{2}{3},1)}$$\textstyle{(s_{2}s_{1}s_{2}s_{1},s_{1}s_{2}s_{1},s_{2}s_{1};0,\frac{1}{4},\frac{1}{3},1)}$$\textstyle{(s_{1}s_{2}s_{1}s_{2}s_{1},s_{2}s_{1}s_{2}s_{1},s_{1}s_{2}s_{1},s_{2}s_{1},s_{1};0,\frac{1}{5},\frac{1}{4},\frac{1}{3},\frac{1}{2},1)}$$\textstyle{(s_{1}s_{2}s_{1};0,1)}$$\textstyle{(s_{2}s_{1}s_{2}s_{1},s_{1}s_{2}s_{1},s_{2}s_{1};0,\frac{1}{4},\frac{2}{3},1)}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{2}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{2}$

##  4 Monomial Realizations

Following [Kas03], we recall the notion of monomial realizations for crystal
bases of highest weight representations. One defines a crystal structure on
the set of Laurent monomials

$\displaystyle{\mathcal{Y}}:=\left\\{X=\prod\limits_{s\in\mathbb{Z},\ i\in
I}X_{s,i}^{\zeta_{s,i}}\,\Bigg{|}\,\zeta_{s,i}\in\mathbb{Z},\
\zeta_{s,i}=0\text{ except for finitely many }(s,i)\right\\}.$

For $X=\prod\limits_{s\in\mathbb{Z},\;i\in
I}X_{s,i}^{\zeta_{s,i}}\in{\mathcal{Y}}$, we define
$\text{wt}(X):=\sum\limits_{i,s}\zeta_{s,i}\Lambda_{i}\in P_{\rm cl}$ and

$\displaystyle\varphi_{i}(X):=\max\left\\{\\!\sum\limits_{k\leq
s}\zeta_{k,i}\,|\,s\in\mathbb{Z}\\!\right\\},\\!\quad\varepsilon_{i}(X):=\varphi_{i}(X)-\text{wt}(X)(h_{i}).$

Next, we put

$A_{s,k}:=X_{s,k}X_{s+1,k}\prod\limits_{j\neq k,\
a_{j,k}<0}X_{s+p_{j,k},j}^{a_{j,k}},\ \ \ (s\in\mathbb{Z},\ k\in I)$

where $(a_{j,k})$ is a generalized Cartan matrix of $\mathfrak{g}$ and the
integers $\\{p_{j,k}\\}_{j,k\in I;a_{j,k<0}}$ are defined as

$p_{j,k}=\begin{cases}1&\text{if }k>j,\\\ 0&\text{if }j>k.\end{cases}$

Actions of Kashiwara operators are defined as follows:

$\displaystyle\tilde{f}_{i}X:=\begin{cases}A_{n_{f_{i}},i}^{-1}X&\text{if}\quad\varphi_{i}(X)>0,\\\
0&\text{if}\quad\varphi_{i}(X)=0,\end{cases}\quad\tilde{e}_{i}X:=\begin{cases}A_{n_{e_{i}},i}X&\text{if}\quad\varepsilon_{i}(X)>0,\\\
0&\text{if}\quad\varepsilon_{i}(X)=0,\end{cases}$

where

$\displaystyle
n_{f_{i}}:=\min\left\\{r\in\mathbb{Z}\,\Bigg{|}\,\varphi_{i}(X)=\sum\limits_{k\leq
r}\zeta_{k,i}\right\\},\qquad
n_{e_{i}}:=\max\left\\{r\in\mathbb{Z}\,\Bigg{|}\,\varphi_{i}(X)=\sum\limits_{k\leq
r}\zeta_{k,i}\right\\}.$

Then we get the following:

######  Theorem 4.1.

[Kas03, Nak]

  1. (i)

The set ${\mathcal{Y}}$ with the above maps ${\rm wt}$, $\varepsilon_{i}$,
$\varphi_{i}$ and $\tilde{e}_{i}$, $\tilde{f}_{i}$ $(i\in I)$ is a $P_{\rm
cl}$-crystal.

  2. (ii)

For a monomial $X\in{\mathcal{Y}}$ such that $\tilde{\varepsilon}_{i}(X)=0$
for all $i\in I$, the set

$\\{\tilde{f}_{j_{m}}\cdots\tilde{f}_{j_{1}}X|m\in\mathbb{Z}_{\geq 0},\
j_{1},\cdots,j_{m}\in I\\}\setminus\\{0\\}$

is isomorphic to $B({\rm wt}(X))$ as $P_{\rm cl}$-crystals.

######  Example 4.2.

We consider the case the Cartan matrix is of type $B_{2}$:
$A=\begin{pmatrix}2&-1\\\ -2&2\end{pmatrix}$.

Then the crystal graph with the highest weight vector $X_{s,1}$, which is
isomorphic to $B(\Lambda_{1})$ is as follows:

$X_{s,1}\overset{1}{\rightarrow}\frac{X_{s,2}^{2}}{X_{s+1,1}}\overset{2}{\rightarrow}\frac{X_{s,2}}{X_{s+1,2}}\overset{2}{\rightarrow}\frac{X_{s+1,1}}{X_{s+1,2}^{2}}\overset{1}{\rightarrow}\frac{1}{X_{s+2,1}}.$

######  Example 4.3.

We consider the case the generalized Cartan matrix is of type $A^{(1)}_{1}$:
$A=\begin{pmatrix}2&-2\\\ -2&2\end{pmatrix}$. Then the crystal graph with the
highest weight vector $X_{s,1}$ is as follows:

$\textstyle{X_{s,1}}$$\textstyle{\frac{X_{s,2}^{2}}{X_{s+1,1}}}$$\textstyle{\frac{X_{s,2}X_{s+1,1}}{X_{s+1,2}}}$$\textstyle{\frac{X_{s+1,1}^{3}}{X_{s+1,2}^{2}}}$$\textstyle{\frac{X_{s,2}X_{s+1,2}}{X_{s+2,1}}}$$\textstyle{\frac{X_{s+1,1}^{2}}{X_{s+2,1}}}$$\textstyle{\frac{X_{s,2}X_{s+2,1}}{X_{s+2,2}}}$$\textstyle{\frac{X_{s+1,1}X_{s+1,2}^{2}}{X_{s+2,1}^{2}}}$$\textstyle{\frac{X_{s+1,1}^{2}X_{s+2,1}}{X_{s+1,2}X_{s+2,2}}}$$\textstyle{\frac{X_{s,2}X_{s+2,2}}{X_{s+3,1}}}$$\textstyle{\frac{X_{s+1,2}^{4}}{X_{s+2,1}^{3}}}$$\textstyle{\frac{X_{s+1,1}X_{s+1,2}}{X_{s+2,2}}}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{2}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{2}$

##  5 Main results

Let $\mathfrak{g}$ be a rank $2$ Kac-Moody algebra. We fix $s\in\mathbb{Z}$
and $i\in I$ and define $i^{\prime}\in I$ as $\\{i,i^{\prime}\\}=\\{1,2\\}=I$.
Let $\mathcal{M}(\Lambda_{i})$ be the monomial realization of $B(\Lambda_{i})$
with the highest weight vector $X_{s,i}$ in Theorem 4.1, that is,

$\mathcal{M}(\Lambda_{i})=\\{\tilde{f}_{j_{m}}\cdots\tilde{f}_{j_{1}}X_{s,i}|m\in\mathbb{Z}_{\geq
0},\ j_{1},\cdots,j_{m}\in I\\}\setminus\\{0\\},$

which is isomorphic to $B(\Lambda_{i})$. For $w\in W$, let $v_{w\Lambda_{i}}$
denote the extremal weight vector in $B(\Lambda_{i})$ with weight
$w\Lambda_{i}\in P$.

For $m\in\mathbb{Z}_{\geq 0}$ such that $2m\leq N-1$, we define integers
$P_{m,i}$, $P_{m,i^{\prime}}$ as follows:

$(s_{i^{\prime}}s_{i})^{m}\Lambda_{i}=P_{m,i}\Lambda_{i}-P_{m,i^{\prime}}\Lambda_{i^{\prime}}\text{
in }P_{\rm cl}.$

Note that when $2m+1\leq N-1$, it holds

$s_{i}(s_{i^{\prime}}s_{i})^{m}\Lambda_{i}=P_{m+1,i^{\prime}}\Lambda_{i^{\prime}}-P_{m,i}\Lambda_{i},$

where if $2m+2\geq N$ then we define $P_{m+1,i^{\prime}}=0$. For
$w=(s_{i^{\prime}}s_{i})^{m}\in W$ as above, we define the following elements
in $\mathcal{Y}$:

$M(v_{w\Lambda_{i}}):=M_{s}(v_{w\Lambda_{i}})=\begin{cases}\left(\frac{X^{P_{m,1}}_{s+m,1}}{X^{P_{m,2}}_{s+m,2}}\right)&{\rm
if}\ i=1,\\\
\left(\frac{X^{P_{m,2}}_{s+m,2}}{X^{P_{m,1}}_{s+m+1,1}}\right)&{\rm if}\
i=2,\end{cases}$ (5.1)
$M(v_{s_{i}w\Lambda_{i}}):=M_{s}(v_{s_{i}w\Lambda_{i}})=\begin{cases}\left(\frac{X^{P_{m+1,2}}_{s+m,2}}{X^{P_{m,1}}_{s+m+1,1}}\right)&{\rm
if}\ i=1,\\\
\left(\frac{X^{P_{m+1,1}}_{s+m+1,1}}{X^{P_{m,2}}_{s+m+1,2}}\right)&{\rm if}\
i=2.\end{cases}$ (5.2)

These are extremal weight vectors in $\mathcal{M}(\Lambda_{i})$ with weight
$w\Lambda_{i}$ and $s_{i}w\Lambda_{i}$, respectively.

We define a map

$\Phi_{s}=\Phi:\mathbb{B}(\Lambda_{i})\rightarrow\mathcal{Y}$

as

$(\tau_{1},\cdots,\tau_{r};a_{0}=0,a_{1},\cdots,a_{r-1},a_{r}=1)\mapsto\prod^{r}_{l=1}\left(M_{s}(v_{\tau_{l}\Lambda_{i}})\right)^{a_{l}-a_{l-1}}.$

######  Theorem 5.1.

The above map is well-defined and induces an isomorphism between
$\mathbb{B}(\Lambda_{i})$ and $\mathcal{M}(\Lambda_{i})$ as $P_{\rm
cl}$-crystals.

Here, well-defined means the image of $\Phi$ is included in $\mathcal{Y}$.
Combining with Theorem 3.4, we get the following expression of Monomial
realization $\mathcal{M}(\Lambda_{i})$.

######  Corollary 5.2.

In the notation of subsection 3.3, we have

$\mathcal{M}(\Lambda_{i})=\left\\{\prod^{m_{2}}_{l=m_{1}}\left(M(v_{\tau_{l}^{\varepsilon(i)}\Lambda_{i}})\right)^{a_{l}-a_{l+1}}|\begin{array}[]{l}N>m_{2}\geq
m_{1}\geq 0,\\\ a_{m_{2}+1}:=0<a_{m_{2}}<\cdots<a_{m_{1}+1}<a_{m_{1}}:=1,\\\
a_{j}d_{j}^{\varepsilon(i)}\in\mathbb{Z}\text{ for
}j=m_{1}+1,\cdots,m_{2}\end{array}\right\\}.$

######  Example 5.3.

We consider the case the Cartan matrix is of type $B_{2}$:
$A=\begin{pmatrix}2&-1\\\ -2&2\end{pmatrix}$. It is the same setting as in
Example 3.5 and 4.2. We obtained

$\mathbb{B}(\Lambda_{1})=\\{(e;0,1),\ (s_{1};0,1),\ (s_{2}s_{1};0,1),\
(s_{1}s_{2}s_{1};0,1),\ (s_{2}s_{1},s_{1};0,\frac{1}{2},1)\\}.$

By the isomorphism $\Phi$, each element goes to an element of
$\mathcal{M}(\Lambda_{1})$:

$(e;0,1)\mapsto
X_{s,1},\quad(s_{1};0,1)\mapsto\frac{X_{s,2}^{2}}{X_{s+1,1}},\quad(s_{2}s_{1};0,1)\mapsto\frac{X_{s+1,1}}{X_{s+1,2}^{2}},\quad(s_{1}s_{2}s_{1};0,1)\mapsto\frac{1}{X_{s+2,1}},$
$(s_{2}s_{1},s_{1};0,\frac{1}{2},1)\mapsto
M(v_{s_{2}s_{1}\Lambda_{1}})^{\frac{1}{2}}M(v_{s_{1}\Lambda_{1}})^{\frac{1}{2}}=\left(\frac{X_{s,2}^{2}}{X_{s+1,1}}\right)^{\frac{1}{2}}\left(\frac{X_{s+1,1}}{X_{s+1,2}^{2}}\right)^{\frac{1}{2}}=\frac{X_{s,2}}{X_{s+1,2}}.$

Hence, the monomials coincide with those in Example 4.2.

######  Example 5.4.

We consider the case the generalized Cartan matrix is of type $A^{(1)}_{1}$:
$A=\begin{pmatrix}2&-2\\\ -2&2\end{pmatrix}$, which is the same setting as in
Example 3.6 and 4.3. By the isomorphism $\Phi$, the paths in Example 3.6 go to
the corresponding monomials in Example 4.3. For example, the element

$(s_{1}s_{2}s_{1}s_{2}s_{1},s_{2}s_{1}s_{2}s_{1},s_{1}s_{2}s_{1},s_{2}s_{1},s_{1};0,\frac{1}{5},\frac{1}{4},\frac{1}{3},\frac{1}{2},1)$

goes to

$\displaystyle
M(v_{s_{1}s_{2}s_{1}s_{2}s_{1}\Lambda_{1}})^{\frac{1}{5}-0}M(v_{s_{2}s_{1}s_{2}s_{1}\Lambda_{1}})^{\frac{1}{4}-\frac{1}{5}}M(v_{s_{1}s_{2}s_{1}\Lambda_{1}})^{\frac{1}{3}-\frac{1}{4}}M(v_{s_{2}s_{1}\Lambda_{1}})^{\frac{1}{2}-\frac{1}{3}}M(v_{s_{1}\Lambda_{1}})^{1-\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\left(\frac{X_{s+2,2}^{6}}{X_{s+3,1}^{5}}\right)^{\frac{1}{5}}\left(\frac{X_{s+2,1}^{5}}{X_{s+2,2}^{4}}\right)^{\frac{1}{20}}\left(\frac{X_{s+1,2}^{4}}{X_{s+2,1}^{3}}\right)^{\frac{1}{12}}\left(\frac{X_{s+1,1}^{3}}{X_{s+1,2}^{2}}\right)^{\frac{1}{6}}\left(\frac{X_{s,2}^{2}}{X_{s+1,1}}\right)^{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle\frac{X_{s,2}X_{s+2,2}}{X_{s+3,1}}.$

######  Remark 5.5.

We can not apply our result to Kac-Moody algebras of other ranks and other
weights.

##  6 Proof

[Proof of Theorem 5.1]

Well-definedness and $\varepsilon_{k}(\Phi(\pi))=\varepsilon_{k}(\pi)$

We fix $k\in\\{1,2\\}$. For
$\pi=(\tau_{1},\cdots,\tau_{r};a_{0}=0,a_{1},\cdots,a_{r-1},a_{r}=1)\in\mathbb{B}(\Lambda_{i})$
and $j\in[0,r]_{\rm int}$, we put

$y_{j}:=\sum^{j}_{l=1}\langle
h_{k},\tau_{l}\Lambda_{i}\rangle(a_{l}-a_{l-1})=h_{k}(a_{j}).$

In particular, it holds $y_{r}=\langle h_{k},{\rm
wt}(\pi)\rangle\in\mathbb{Z}$ by (3.2). Let us prove the exponent of variables
in the form $X_{t,k}$ ($t\in\mathbb{Z}$) in $\Phi(\pi)$ is an integer and
$\varepsilon_{k}(\Phi(\pi))=\varepsilon_{k}(\pi)$.

Since $l(\tau_{j})=l(\tau_{j+1})+1$ by Theorem 3.4 and there exists an
$a_{j}$-chain for $(\tau_{j},\tau_{j+1})$ for $j\in[1,r-1]_{\rm int}$, we see
that for $K\in\\{1,2\\}$,

$a_{j}\langle h_{K},\tau_{j}\Lambda_{i}\rangle=-a_{j}\langle
h_{K},\tau_{j+1}\Lambda_{i}\rangle\in\mathbb{Z}\ \ {\rm if}\
s_{K}\tau_{j}=\tau_{j+1}.$ (6.1)

If $j<r$ and $s_{k}\tau_{j}=\tau_{j+1}$ then

$\displaystyle y_{j}$ $\displaystyle=$ $\displaystyle\sum^{j}_{l=1}\langle
h_{k},\tau_{l}\Lambda_{i}\rangle(a_{l}-a_{l-1})$ $\displaystyle=$
$\displaystyle\langle h_{k},\tau_{j}\Lambda_{i}\rangle
a_{j}+\sum^{j-1}_{l=1}\langle
h_{k},\tau_{l}\Lambda_{i}-\tau_{l+1}\Lambda_{i}\rangle a_{l}$ $\displaystyle=$
$\displaystyle\langle h_{k},\tau_{j}\Lambda_{i}\rangle
a_{j}+\sum_{l\in[1,j-1]_{\rm int},\ s_{k}\tau_{l}=\tau_{l+1}}\langle
h_{k},\tau_{l}\Lambda_{i}-\tau_{l+1}\Lambda_{i}\rangle a_{l}$
$\displaystyle+\sum_{l\in[1,j-1]_{\rm int},\
s_{k^{\prime}}\tau_{l}=\tau_{l+1}}\langle
h_{k},\tau_{l}\Lambda_{i}-\tau_{l+1}\Lambda_{i}\rangle a_{l},$

where $k^{\prime}\in\\{1,2\\}\setminus\\{k\\}$. By (6.1), the first term and
second term are in $\mathbb{Z}$. As for the third term, if
$s_{k^{\prime}}\tau_{l}=\tau_{l+1}$ then

$\displaystyle\langle h_{k},\tau_{l}\Lambda_{i}-\tau_{l+1}\Lambda_{i}\rangle
a_{l}$ $\displaystyle=$ $\displaystyle\langle h_{k},\tau_{l}\Lambda_{i}\rangle
a_{l}-\langle h_{k},\tau_{l+1}\Lambda_{i}\rangle a_{l}$ $\displaystyle=$
$\displaystyle\langle h_{k},\tau_{l}\Lambda_{i}\rangle a_{l}-\langle
s_{k^{\prime}}h_{k},\tau_{l}\Lambda_{i}\rangle a_{l}$ $\displaystyle=$
$\displaystyle\langle h_{k},\tau_{l}\Lambda_{i}\rangle a_{l}-\langle
h_{k},\tau_{l}\Lambda_{i}\rangle
a_{l}+\langle\alpha_{k^{\prime}}(h_{k})h_{k^{\prime}},\tau_{l}\Lambda_{i}\rangle
a_{l}$ $\displaystyle=$ $\displaystyle\alpha_{k^{\prime}}(h_{k})\cdot
a_{l}\cdot\langle h_{k^{\prime}},\tau_{l}\Lambda_{i}\rangle.$

By (6.1), it holds $a_{l}\cdot\langle
h_{k^{\prime}},\tau_{l}\Lambda_{i}\rangle\in\mathbb{Z}$ so that $\langle
h_{k},\tau_{l}\Lambda_{i}-\tau_{l+1}\Lambda_{i}\rangle a_{l}\in\mathbb{Z}$. In
this way, we have

$y_{j}\in\mathbb{Z}\text{ if }s_{k}\tau_{j}=\tau_{j+1}\ \text{or }j=r.$ (6.2)

For $j\in[2,r]_{\rm int}$, we see that $-\langle
h_{k},\tau_{j}\Lambda_{i}\rangle\geq 0$ if and only if $\langle
h_{k},\tau_{j-1}\Lambda_{i}\rangle\geq 0$. Thus, $M(v_{\tau_{j}\Lambda_{i}})$
has a denominator $X_{t,k}^{-\langle h_{k},\tau_{j}\Lambda_{i}\rangle}$ with
$-\langle h_{k},\tau_{j}\Lambda_{i}\rangle\geq 0$ in the expressions (5.1) and
(5.2) if and only if $M(v_{\tau_{j-1}\Lambda_{i}})$ has a numerator
$X_{t,k}^{\langle h_{k},\tau_{j-1}\Lambda_{i}\rangle}$ with $\langle
h_{k},\tau_{j-1}\Lambda_{i}\rangle\geq 0$. Hence, the exponent of $X_{t,k}$ in
$\Phi(\pi)$ is equal to

$(a_{j-1}-a_{j-2})\langle
h_{k},\tau_{j-1}\Lambda_{i}\rangle+(a_{j}-a_{j-1})\langle
h_{k},\tau_{j}\Lambda_{i}\rangle=y_{j}-y_{j-2},$

where $y_{0}=0$. In this case, it holds that $s_{k}\tau_{j}=\tau_{j+1}$ (or
$j=r$) and $s_{k}\tau_{j-2}=\tau_{j-1}$ if $j-2\geq 1$ by Theorem 3.4. Thus,
$y_{j}\in\mathbb{Z}$ and $y_{j-2}\in\mathbb{Z}$ by (6.2). Therefore, the
exponent of $X_{t,k}$ in $\Phi(\pi)$ is an integer.

If $M(v_{\tau_{1}\Lambda_{i}})$ has a denominator $X_{t,k}^{-\langle
h_{k},\tau_{1}\Lambda_{i}\rangle}$ with $-\langle
h_{k},\tau_{1}\Lambda_{i}\rangle\geq 0$ in the expressions (5.1) and (5.2)
then the exponent of $X_{t,k}$ in $\Phi(\pi)$ is equal to
$(a_{1}-a_{0})\langle h_{k},\tau_{1}\Lambda_{i}\rangle=y_{1}$. In this case,
it holds $s_{k}\tau_{1}=\tau_{2}$ or $r=1$ so that $y_{1}\in\mathbb{Z}$ by
(6.2).

If $M(v_{\tau_{r}\Lambda_{i}})$ has a numerator $X_{t,k}^{\langle
h_{k},\tau_{r}\Lambda_{i}\rangle}$ with $\langle
h_{k},\tau_{r}\Lambda_{i}\rangle\geq 0$ then the exponent of $X_{t,k}$ in
$\Phi(\pi)$ is equal to

$(1-a_{r-1})\langle h_{k},\tau_{r}\Lambda_{i}\rangle=\langle
h_{k},\tau_{r}\Lambda_{i}\rangle+a_{r-1}\langle
h_{k},\tau_{r-1}\Lambda_{i}\rangle\in\mathbb{Z}.$

Therefore, the exponent of each variable $X_{t,k}$ in $\Phi(\pi)$ is an
integer and $\Phi$ is well-defined.

It holds $\tau_{j+1}\Lambda_{i}(h_{k})>0$ if and only if $y_{j}<y_{j+1}$ for
$j\in[0,r-1]_{\rm int}$. Thus, it follows by (3.3) that

$\displaystyle\varepsilon_{k}(\pi)$ $\displaystyle=$ $\displaystyle{\rm
max}(\mathbb{Z}\cap\\{-\langle h_{k},\pi(t)\rangle|0\leq t\leq 1\\})$
$\displaystyle=$ $\displaystyle-{\rm min}(\mathbb{Z}\cap\\{\langle
h_{k},\pi(t)\rangle|0\leq t\leq 1\\})$ $\displaystyle=$ $\displaystyle-{\rm
min}\\{y_{j}|j\in[0,r-1]_{\rm
int},\tau_{j+1}\Lambda_{i}(h_{k})>0\\}\cup\\{y_{r}\\}.$

For $j\in[1,r]_{\rm int}$, we put

$y^{\prime}_{j}:=\sum^{r}_{l=j}\langle
h_{k},\tau_{l}\Lambda_{i}\rangle(a_{l}-a_{l-1})=y_{r}-y_{j-1}.$

On the other hand, by (5.1), (5.2), the definition of monomial realizations in
Sect.4 and

$\Phi(\pi)=\prod^{r}_{l=1}\left(M(v_{\tau_{l}\Lambda_{i}})\right)^{a_{l}-a_{l-1}},$

we see that

$\varphi_{k}(\Phi(\pi))={\rm max}(\\{y^{\prime}_{l}|l\in[1,r]_{\rm int},\
\tau_{l}\Lambda_{i}(h_{k})>0\\}\cup\\{0\\}).$

Therefore,

$\displaystyle\varepsilon_{k}(\Phi(\pi))$ $\displaystyle=$
$\displaystyle\varphi_{k}(\Phi(\pi))-\langle h_{k},{\rm wt}(\Phi(\pi))\rangle$
$\displaystyle=$ $\displaystyle{\rm max}(\\{y^{\prime}_{l}|l\in[1,r]_{\rm
int},\ \tau_{l}\Lambda_{i}(h_{k})>0\\}\cup\\{0\\})-y_{r}$ $\displaystyle=$
$\displaystyle{\rm max}(\\{-y_{l-1}|l\in[1,r]_{\rm int},\
\tau_{l}\Lambda_{i}(h_{k})>0\\}\cup\\{-y_{r}\\})$ $\displaystyle=$
$\displaystyle{\rm max}(\\{-y_{l}|l\in[0,r-1]_{\rm int},\
\tau_{l+1}\Lambda_{i}(h_{k})>0\\}\cup\\{-y_{r}\\})$ $\displaystyle=$
$\displaystyle\varepsilon_{k}(\pi).$

${\rm wt}(\Phi(\pi))={\rm wt}(\pi)$ in $P_{\rm cl}$

We fix $k\in\\{1,2\\}$ and take
$\pi=(\tau_{1},\cdots,\tau_{r};a_{0}=0,a_{1},\cdots,a_{r-1},a_{r}=1)\in\mathbb{B}(\Lambda_{i})$.
Then

${\rm wt}(\pi)=\sum^{r}_{l=1}(a_{l}-a_{l-1})\tau_{l}\Lambda_{i}.$

Since

$\Phi(\pi)=\prod^{r}_{l=1}\left(M(v_{\tau_{l}\Lambda_{i}})\right)^{a_{l}-a_{l-1}}$

and $M(v_{\tau_{l}\Lambda_{i}})$ has the weight $\tau_{l}\Lambda_{i}\in P_{\rm
cl}$, we get

${\rm wt}(\Phi(\pi))=\sum^{r}_{l=1}(a_{l}-a_{l-1})\tau_{l}\Lambda_{i}={\rm
wt}(\pi).$

$\tilde{f}_{k}(\Phi(\pi))=\Phi(f_{k}\pi)$

Let
$\pi=(\tau_{1},\cdots,\tau_{r};a_{0}=0,a_{1},\cdots,a_{r-1},a_{r}=1)\in\mathbb{B}(\Lambda_{i})$.
We consider the action of $f_{k}$ and use the same notation as in subsection
3.2:

$Q:={\rm min}(\mathbb{Z}\cap\\{\langle h_{k},\pi(t)\rangle|0\leq t\leq
1\\})=-\varepsilon_{k}(\pi),$ $p:={\rm max}\\{l\in[0,r]_{\rm int}|y_{l}=Q\\}.$

We assume that

$\langle h_{k},{\rm wt}(\pi)\rangle>Q=-\varepsilon_{k}(\pi)$

so that $\varphi_{k}(\Phi(\pi))>0$. In particular, it holds $p<r$. Now, we
know $y_{p+2},y_{p+4},y_{p+6},\cdots\in\mathbb{Z}_{\geq Q+1}$ and
$y_{p+1}>y_{p+2}$, $y_{p+3}>y_{p+4},y_{p+2}$, $y_{p+5}>y_{p+6},y_{p+4}$
(Theorem 3.4). Note that if $p>0$ then $\langle
h_{k},\tau_{p}\Lambda_{i}\rangle<0$ and $s_{k}\tau_{p}=\tau_{p+1}$. Therefore,
we see that

$x:={\rm min}\\{l\in[p,r]_{\rm int}|\langle h_{k},\pi(t)\rangle\geq Q+1\ {\rm
if}\ t\geq a_{l}\\}=p+1.$

We consider a division into cases:

Case 1. $p=0$, $h_{k}(a_{1})=y_{1}=Q+1$

Case 2. $p=0$, $h_{k}(a_{1})=y_{1}>Q+1$

Case 3. $p>0$, $h_{k}(a_{p+1})=y_{p+1}=Q+1$

Case 4. $p>0$, $h_{k}(a_{p+1})=y_{p+1}>Q+1$

We define integers $P_{p+1}$ and $P_{p}$ as

$\tau_{p+1}\Lambda_{i}=P_{p+1}\Lambda_{k}-P_{p}\Lambda_{k^{\prime}}.$

Since $\langle h_{k},\tau_{p+1}\Lambda_{i}\rangle>0$, we obtain $P_{p+1}>0$,
$P_{p}\geq 0$.

Case 1. $p=0$, $h_{k}(a_{1})=y_{1}=Q+1$

If $r>1$ then $Q+1=h_{k}(a_{1})>h_{k}(a_{2})\geq Q+1$ by $y_{p+1}>y_{p+2}$,
which is a contradiction. Thus, it holds $r=1$, that is, $\pi=(\tau_{1};0,1)$.
We see $Q=y_{0}=0$ and $y_{1}=\langle h_{k},\tau_{1}\Lambda_{i}\rangle=1$ so
that Definition 3.2 yields

$f_{k}(\pi)=(s_{k}\tau_{1};0,1).$ (6.3)

Case 2. $p=0$, $h_{k}(a_{1})=y_{1}>Q+1$

If $a_{0}=0<a<a_{1}$, we have

$h_{k}(a)=h_{k}(a_{0})+\langle
h_{k},\tau_{1}\Lambda_{i}\rangle(a-a_{0})=Q+P_{1}a.$

Thus, the equation $h_{k}(a)=Q+1$ in Definition 3.2 means $a=\frac{1}{P_{1}}$,
which yields

$f_{k}(\pi)=(s_{k}\tau_{1},\tau_{1},\tau_{2},\cdots,\tau_{r};0,\frac{1}{P_{1}},a_{1},a_{2},\cdots,a_{r-1},1).$
(6.4)

Case 3. $p>0$, $h_{k}(a_{p+1})=y_{p+1}=Q+1$

In this case, if $p\leq r-2$ then $Q+1=h_{k}(a_{p+1})>h_{k}(a_{p+2})\geq Q+1$,
which is a contradiction. Thus, $p=r-1$ and

$f_{k}(\pi)=(\tau_{1},\cdots,\tau_{r-1};0,a_{1},\cdots,a_{r-2},1).$ (6.5)

Case 4. $p>0$, $h_{k}(a_{p+1})=y_{p+1}>Q+1$

Just as in Case 2, if $a_{p}<a<a_{p+1}$ then by the definition of $h_{k}$ it
holds

$h_{k}(a)=h_{k}(a_{p})+\langle
h_{k},\tau_{p+1}\Lambda_{i}\rangle(a-a_{p})=Q+P_{p+1}(a-a_{p}).$

Hence, the equation $h_{k}(a)=Q+1$ means $a=a_{p}+\frac{1}{P_{p+1}}$ and

$f_{k}(\pi)=(\tau_{1},\tau_{2},\cdots,\tau_{r};0,a_{1},\cdots,a_{p-1},a_{p}+\frac{1}{P_{p+1}},a_{p+1},\cdots,a_{r-1},1).$
(6.6)

Next, let us consider the action of $\tilde{f}_{k}$ on $\Phi(\pi)$. Recall
that

$\displaystyle p$ $\displaystyle=$ $\displaystyle{\rm max}\\{l\in[0,r]_{\rm
int}|y_{l}=Q\\}$ (6.7) $\displaystyle=$ $\displaystyle{\rm
max}\\{l\in[0,r]_{\rm int}|y_{r}-y_{l}=y_{r}-Q(=\varphi_{k}(\pi))\\}.$

Note that $y_{r}-y_{l}$ is equal to $\langle h_{k},{\rm
wt}(\prod^{r}_{j=l+1}M(v_{\tau_{j}\Lambda_{i}})^{a_{j}-a_{j-1}})\rangle$. Let
$n_{f}$ be the left index of numerator of $M(v_{\tau_{p+1}\Lambda_{i}})$, that
is,

$M(v_{\tau_{p+1}\Lambda_{i}})=\frac{X^{P_{p+1}}_{n_{f},k}}{X^{P_{p}}_{n^{\prime},k^{\prime}}}$

with some $n^{\prime}\in\mathbb{Z}$. Since we have shown
$\varphi_{k}(\pi)=\varphi_{k}(\Phi(\pi))$, by the definition of monomial
realizations and (6.7), it follows

$\tilde{f}_{k}\Phi(\pi)=\Phi(\pi)\cdot
A_{n_{f},k}^{-1}=\Phi(\pi)\cdot\frac{X^{-a_{k^{\prime},k}}_{n_{f}+\delta_{2,k},k^{\prime}}}{X_{n_{f},k}X_{n_{f}+1,k}}.$

Case 1. $p=0$, $h_{k}(a_{1})=y_{1}=Q+1$

In this case, $\pi=(\tau_{1};0,1)$, $Q=0$ and $\langle
h_{k},\tau_{1}\Lambda_{i}\rangle=h_{k}(a_{1})=1$. Thus, $P_{1}=1$ so that
$M(v_{\tau_{1}\Lambda_{i}})=\frac{X_{n_{f},k}}{X^{P_{0}}_{n^{\prime},k^{\prime}}}$.
It follows from (6.3) that

$\Phi(f_{k}\pi)=M(v_{s_{k}\tau_{1}\Lambda_{i}}),\quad\tilde{f}_{k}\Phi(\pi)=\tilde{f}_{k}M(v_{\tau_{1}\Lambda_{i}})=M(v_{s_{k}\tau_{1}\Lambda_{i}}).$

Therefore, we get $\Phi(f_{k}\pi)=\tilde{f}_{k}\Phi(\pi)$.

Case 2. $p=0$, $h_{k}(a_{1})=y_{1}>Q+1$

We get

$M(v_{\tau_{1}\Lambda_{i}})=\frac{X^{P_{1}}_{n_{f},k}}{X^{P_{0}}_{n^{\prime},k^{\prime}}}$
(6.8)

and

$M(v_{s_{k}\tau_{1}\Lambda_{i}})=M(v_{\tau_{1}\Lambda_{i}})A^{-P_{1}}_{n_{f},k}.$
(6.9)

Combining with (6.4),

$\Phi(\pi)=M(v_{\tau_{1}\Lambda_{i}})^{a_{1}-a_{0}}M(v_{\tau_{2}\Lambda_{i}})^{a_{2}-a_{1}}\cdots
M(v_{\tau_{r}\Lambda_{i}})^{a_{r}-a_{r-1}},$

(6.8) and (6.9), we have

$\displaystyle\Phi(f_{k}\pi)$ $\displaystyle=$ $\displaystyle
M(v_{s_{k}\tau_{1}\Lambda_{i}})^{\frac{1}{P_{1}}}M(v_{\tau_{1}\Lambda_{i}})^{a_{1}-\frac{1}{P_{1}}}M(v_{\tau_{2}\Lambda_{i}})^{a_{2}-a_{1}}\cdots
M(v_{\tau_{r}\Lambda_{i}})^{a_{r}-a_{r-1}}$ $\displaystyle=$ $\displaystyle
M(v_{s_{k}\tau_{1}\Lambda_{i}})^{\frac{1}{P_{1}}}M(v_{\tau_{1}\Lambda_{i}})^{-\frac{1}{P_{1}}}\Phi(\pi)$
$\displaystyle=$
$\displaystyle(M(v_{\tau_{1}\Lambda_{i}})A^{-P_{1}}_{n_{f},k})^{\frac{1}{P_{1}}}M(v_{\tau_{1}\Lambda_{i}})^{-\frac{1}{P_{1}}}\Phi(\pi)$
$\displaystyle=$
$\displaystyle\Phi(\pi)A^{-1}_{n_{f},k}=\tilde{f}_{k}\Phi(\pi).$

Case 3. $p>0$, $h_{k}(a_{p+1})=y_{p+1}=Q+1$

In this case, the previous discussion says $p=r-1$. The definition of $p$
implies $y_{p}=y_{r-1}=Q$. By the assumption, $y_{r}=Q+1=y_{r-1}+1$ so that

$1=y_{r}-y_{r-1}=(1-a_{r-1})\langle
h_{k},\tau_{r}\Lambda_{i}\rangle=(1-a_{r-1})P_{r}.$ (6.10)

Since
$M(v_{\tau_{r}\Lambda_{i}})=\frac{X^{P_{r}}_{n_{f},k}}{X^{P_{r-1}}_{n^{\prime},k^{\prime}}}$,
it holds

$M(v_{\tau_{r}\Lambda_{i}})A^{-P_{r}}_{n_{f},k}=M(v_{s_{k}\tau_{r}\Lambda_{i}})=M(v_{\tau_{r-1}\Lambda_{i}}).$
(6.11)

It follows by (6.5),

$\Phi(\pi)=M(v_{\tau_{1}\Lambda_{i}})^{a_{1}-a_{0}}\cdots
M(v_{\tau_{r-2}\Lambda_{i}})^{a_{r-2}-a_{r-3}}M(v_{\tau_{r-1}\Lambda_{i}})^{a_{r-1}-a_{r-2}}M(v_{\tau_{r}\Lambda_{i}})^{1-a_{r-1}},$

(6.10) and (6.11) that

$\displaystyle\Phi(f_{k}\pi)$ $\displaystyle=$ $\displaystyle
M(v_{\tau_{1}\Lambda_{i}})^{a_{1}-a_{0}}\cdots
M(v_{\tau_{r-2}\Lambda_{i}})^{a_{r-2}-a_{r-3}}M(v_{\tau_{r-1}\Lambda_{i}})^{1-a_{r-2}}$
$\displaystyle=$
$\displaystyle\Phi(\pi)M(v_{\tau_{r}\Lambda_{i}})^{a_{r-1}-1}M(v_{\tau_{r-1}\Lambda_{i}})^{1-a_{r-1}}$
$\displaystyle=$
$\displaystyle\Phi(\pi)M(v_{\tau_{r}\Lambda_{i}})^{a_{r-1}-1}(M(v_{\tau_{r}\Lambda_{i}})A^{-P_{r}}_{n_{f},k})^{1-a_{r-1}}$
$\displaystyle=$
$\displaystyle\Phi(\pi)A^{-P_{r}(1-a_{r-1})}_{n_{f},k}=\Phi(\pi)A^{-1}_{n_{f},k}=\tilde{f}_{k}\Phi(\pi).$

Case 4. $p>0$, $h_{k}(a_{p+1})=y_{p+1}>Q+1$

In this case, we get

$M(v_{\tau_{p+1}\Lambda_{i}})=\frac{X^{P_{p+1}}_{n_{f},k}}{X^{P_{p}}_{n^{\prime},k^{\prime}}},$
$M(v_{\tau_{p}\Lambda_{i}})=M(v_{s_{k}\tau_{p+1}\Lambda_{i}})=\tilde{f}^{P_{p+1}}_{k}M(v_{\tau_{p+1}\Lambda_{i}})=M(v_{\tau_{p+1}\Lambda_{i}})A^{-P_{p+1}}_{n_{f},k}.$
(6.12)

Using (6.6) and (6.12), we have

$\displaystyle\Phi(f_{k}(\pi))$ $\displaystyle=$ $\displaystyle
M(v_{\tau_{1}\Lambda_{i}})^{a_{1}-a_{0}}\cdots
M(v_{\tau_{p-1}\Lambda_{i}})^{a_{p-1}-a_{p-2}}$ $\displaystyle
M(v_{\tau_{p}\Lambda_{i}})^{a_{p}+\frac{1}{P_{p+1}}-a_{p-1}}M(v_{\tau_{p+1}\Lambda_{i}})^{a_{p+1}-a_{p}-\frac{1}{P_{p+1}}}$
$\displaystyle M(v_{\tau_{p+2}\Lambda_{i}})^{a_{p+2}-a_{p+1}}\cdots
M(v_{\tau_{r}\Lambda_{i}})^{a_{r}-a_{r-1}}$ $\displaystyle=$
$\displaystyle\Phi(\pi)M(v_{\tau_{p}\Lambda_{i}})^{\frac{1}{P_{p+1}}}M(v_{\tau_{p+1}\Lambda_{i}})^{-\frac{1}{P_{p+1}}}$
$\displaystyle=$
$\displaystyle\Phi(\pi)(M(v_{\tau_{p+1}\Lambda_{i}})A^{-P_{p+1}}_{n_{f},k})^{\frac{1}{P_{p+1}}}M(v_{\tau_{p+1}\Lambda_{i}})^{-\frac{1}{P_{p+1}}}$
$\displaystyle=$
$\displaystyle\Phi(\pi)A^{-1}_{n_{f},k}=\tilde{f}_{k}\Phi(\pi).$

By $\Phi(e;0,1)=M(v_{\Lambda_{i}})=X_{s,i}\in\mathcal{M}(\Lambda_{i})$ and
above argument, we see that

$\Phi(\mathbb{B}(\Lambda_{i}))\subset\mathcal{M}(\Lambda_{i}).$

Let us prove $\Phi$ is injective. Taking

$\pi=(\tau_{1},\cdots,\tau_{r};0,a_{1},\cdots,a_{r-1},1)\in\mathbb{B}(\Lambda_{i}),$
$\pi^{\prime}=(\tau^{\prime}_{1},\cdots,\tau^{\prime}_{m};0,a^{\prime}_{1},\cdots,a^{\prime}_{m-1},1)\in\mathbb{B}(\Lambda_{i})$

and we suppose that $\Phi(\pi)=\Phi(\pi^{\prime})$ and prove
$\pi=\pi^{\prime}$. One may assume $\tau_{1}\neq e$, $\tau^{\prime}_{1}\neq
e$. Let us focus on the denominators of $M(v_{\tau_{1}\Lambda_{i}})$ and
$M(v_{\tau^{\prime}_{1}\Lambda_{i}})$ and put them as

$M(v_{\tau_{1}\Lambda_{i}})=\frac{X^{a}_{b,c}}{X^{P_{t,l}}_{t,l}},\quad
M(v_{\tau^{\prime}_{1}\Lambda_{i}})=\frac{X^{a^{\prime}}_{b^{\prime},c^{\prime}}}{X^{P_{t^{\prime},l^{\prime}}}_{t^{\prime},l^{\prime}}},$

with some $t,t^{\prime}\in\mathbb{Z}$, $l,l^{\prime}\in\\{1,2\\}$,
$P_{t,l},P_{t^{\prime},l^{\prime}}\in\mathbb{Z}_{>0}$ and some monomials
$X^{a}_{b,c}$ and $X^{a^{\prime}}_{b^{\prime},c^{\prime}}$. Considering the
explicit forms of $M(v_{\tau\Lambda_{i}})$ in (5.1), $(\ref{pr-00})$, we see
that $(t,l)$ (resp. $(t^{\prime},l^{\prime})$) is the maximum double index
appearing in $\Phi(\pi)$ (resp. $\Phi(\pi^{\prime})$) in the order

$(s,1)<(s,2)<(s+1,1)<(s+1,2)<(s+2,1)<(s+2,2)<\cdots.$

Since we supposed $\Phi(\pi)=\Phi(\pi^{\prime})$, it holds
$(t,l)=(t^{\prime},l^{\prime})$,
$M(v_{\tau_{1}\Lambda_{i}})^{a_{1}}=M(v_{\tau^{\prime}_{1}\Lambda_{i}})^{a_{1}^{\prime}}$
so that $\tau_{1}=\tau^{\prime}_{1}$ and $a_{1}=a_{1}^{\prime}$. By

$(M(v_{\tau_{1}\Lambda_{i}})^{a_{1}})^{-1}\Phi(\pi)=(M(v_{\tau_{1}\Lambda_{i}})^{a_{1}})^{-1}\Phi(\pi^{\prime})$

and a similar argument, it holds
$M(v_{\tau_{2}\Lambda_{i}})^{a_{2}-a_{1}}=M(v_{\tau^{\prime}_{2}\Lambda_{i}})^{a_{2}^{\prime}-a_{1}^{\prime}}$
so that $\tau_{2}=\tau^{\prime}_{2}$ and $a_{2}=a_{2}^{\prime}$. Repeating
this argument, we see that $r=m$, $a_{j}=a^{\prime}_{j}$,
$\tau_{j}=\tau^{\prime}_{j}$ ($j\in[1,r]_{\rm int}$) and $\pi=\pi^{\prime}$.
We proved $\Phi$ is injective from $\mathbb{B}(\Lambda_{i})$ to
$\mathcal{M}(\Lambda_{i})$.

For any
$\tilde{f}_{i_{t}}\cdots\tilde{f}_{i_{2}}\tilde{f}_{i_{1}}X_{s,i}\in\mathcal{M}(\Lambda_{i})$,
since the map $\Phi$ preserves $\varphi_{k}$, it holds $f_{i_{t}}\cdots
f_{i_{2}}f_{i_{1}}(1;0,1)\in\mathbb{B}(\Lambda_{i})$ and

$\Phi(f_{i_{t}}\cdots
f_{i_{2}}f_{i_{1}}(e;0,1))=\tilde{f}_{i_{t}}\cdots\tilde{f}_{i_{2}}\tilde{f}_{i_{1}}X_{s,i}.$

Thus $\Phi$ is surjective.

For $\pi\in\mathbb{B}(\Lambda_{i})$, it holds $e_{k}\pi=0$ if and only if
$\tilde{e}_{k}\Phi(\pi)=0$. If $e_{k}\pi\neq 0$ then we can write
$e_{k}\pi=f_{i_{t}}\cdots f_{i_{1}}(e;0,1)$ with some $i_{1},\cdots,i_{t}\in
I$. Above argument implies

$\Phi(e_{k}\pi)=\tilde{f}_{i_{t}}\cdots\tilde{f}_{i_{1}}X_{s,i},\quad\tilde{e}_{k}\Phi(\pi)=\tilde{e}_{k}\Phi(f_{k}f_{i_{t}}\cdots
f_{i_{1}}(e;0,1))=\tilde{e}_{k}\tilde{f}_{k}\tilde{f}_{i_{t}}\cdots\tilde{f}_{i_{1}}X_{s,i}=\tilde{f}_{i_{t}}\cdots\tilde{f}_{i_{1}}X_{s,i}$

so that the relation $\Phi(e_{k}\pi)=\tilde{e}_{k}\Phi(\pi)$ also holds. Thus
$\Phi$ is an isomorphism between $\mathbb{B}(\Lambda_{i})$ and
$\mathcal{M}(\Lambda_{i})$.

## References

  * [JKKS] K.Jeong, S.-J.Kang, J.-A.Kim, D.-U.Shin, Crystals and Nakajima monomials for quantum generalized Kac–Moody algebras, J. Algebra, 319, no.9, pp.3732–3751(2008).
  * [Jos] A.Joseph, Quantum Groups and Their Primitive Ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 29, Springer–Verlag, Berlin (1995).
  * [Kas90] M.Kashiwara, Crystalling the $q$-analogue of universal enveloping algebras, Comm. Math. Phys., 133, 249–260 (1990).
  * [Kas93] M.Kashiwara, The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J., 71, no 3, pp.839–858 (1993).
  * [Kas96] M.Kashiwara, Similarity of crystal bases, Lie algebras and their representations (Seoul, 1995), Contemp. Math., 194, American Mathematical Society, Providence, RI, pp.177–186 (1996).
  * [Kas03] M.Kashiwara, Realizations of crystals, Combinatorial and geometric representation theory (Seoul, 2001), Contemp. Math., 325, Amer. Math. Soc., Providence, RI, pp.133–139 (2003).
  * [Lit94] P.Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math.116, no.1-3, pp.329–346 (1994).
  * [Lit95] P.Littelmann, Paths and root operators in representation theory, Ann. of Math. (2)142, no.3, pp.499–525 (1995).
  * [Nak] H.Nakajima, $t$-analogs of $q$-characters of quantum affine algebras of type $A_{n}$, $D_{n}$, Combinatorial and geometric representation theory (Seoul, 2001), Contemp. Math., 325, Amer. Math. Soc., Providence, RI, pp.141–160 (2003).
  * [Sag] D.Sagaki, [Path model and various related results] path model to soreni kanrenshita shokekka (in Japanese), RIMS Kōkyûroku, 1183, pp.35-51 (2001).
  * [San] Y.B.Sanderson, Dimensions of Demazure modules for rank two affine Lie algebras, Compositio Math.101, no.2, pp.115–131 (1996).