数理解析研究所講究録
第 1795 巻 2012 年 224-233

224

KKR TYPE BIJECTION FOR:ALGORITHMS AND
EXAMPLES$E_{6}^{(1)}$

MASATO OKADO * AND NOBUMASA SANO
ABSTRACT. We review the KKR type bijection for the exceptional affine algebra
obtained in our previous paper [4] by giving concise explanations on$E_{6}^{(1)}$algorithm and more examples.

1. AFFINE ALGEBRA$E_{6}^{(1)}$

AND KR CRYSTAL $B^{1,1}$

We consider the exceptional affie Lie algebra of type
. We label the nodesof the Dynkin diagram as in Figure 1 following [2]. Let be the index set of the
Dynkin nodes, and let
be simple roots, simple, fundamental weights, respectively.
we denote the projection of onto theweight space ofbyand set. Let
stand for the Cartan matrix. For $i,j\in I,$ $i\sim j$ means that the nodes
and are adjacent in the Dynkin diagram. Thenif $i=j,$ $=-1$
if $i\sim j$ , and $=0$ otherwise.
Let us consider the level fundamental representaion ofor simplest
Kirillov-Reshetikhin module corresponding to the node 1. It is known that it hasa crystal basis, denoted by. $F_{0}uow\dot{m}g[1]$ we show the crystal graph ofin Figure 2. Here vertices in the crystal graph signify elements ofand $barrow ib’$
$E_{6}^{(1)}$$I$$\alpha_{i},$$\infty roots$$\alpha_{i}^{\vee},\Lambda_{i}(i\in I)$$Fo\mathbb{I}ow\dot{m}g[2]$$E_{6}$$\overline{\Lambda}_{i}(i\in I_{0})$$\Lambda_{i}$$\overline{P}=\oplus_{i\in I_{0}}Z\overline{\Lambda}_{i},\frac{1}{P^{l}}=\oplus_{i\in I_{O}}Z_{\geq 0}\overline{\Lambda}_{i}$$(C_{ij})_{i,j\in I}$

$i$

$j$

$C_{ j}=(\alpha_{i}|\alpha_{j})=2$$0$$U_{q}’(E_{6}^{(1)})$$B^{1,1}$$B^{1,1}$$B^{1,1}$

123

4

FIGURE 1. Dynkin diagram for5$E_{6}^{(1)}$Date: Apri17, 2012
’Department of Mathematical Science. Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
E-mail: okadoQsigmath.es.osaka-u.ac.jp

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M. OKADO AND N. SANO

$O17$

$O22$

$O24$

$0$

$0$

$0$

$O25$

.

$O26$

$O27$

$0$

$0$

$0$

FIGURE 2. Crystal graph for$B^{1,1}$

stands for the relation$f_{i}b=b’$, or equivalently $b=e_{i}b’$

and
where
are Kashiwara operators. $f_{i}b=0$ resp. $e_{i}b=0)$ signifies that
there is no arrow of color sourcing from (resp. sink into) . Standard notations
are in order. Let $B$ be a crystal. For $b\in B$ we set$e_{i}$$($$f_{i}$$b$$i$$\epsilon_{i}(b)=\max\{m\in \mathbb{Z}_{\geq 0}|e_{i}^{m}b\neq 0\}$$\epsilon(b)=\sum_{i}\epsilon_{i}(b)\Lambda_{i}$$wt(b)=\varphi(b)-\epsilon(b)$

,

.

,

,

$\varphi_{i}(b)=\max\{m\in \mathbb{Z}_{\geq 0}|f_{i}^{m}b\neq 0\}$$\varphi(b)=\sum_{i}\varphi_{i}(b)\Lambda_{i}$,KKR TYPE BIJECTION FOR$E_{6}^{(1)}$226: ALGORITHMS AND EXAMPLESLet
be crystals. Then the set
endowed with the structure of crystal by$B_{1}\otimes B_{2}=\{b_{1}\otimes b_{2}|b_{1}\in B_{1}, b_{2}\in B_{2}\}$$B_{1},$ $B_{2}$(1.1)$e_{i}(b_{1}\otimes b_{2})=\{\begin{array}{ll}e_{i}b_{1}\otimes b_{2} if \varphi_{i}(b_{1})\geq\epsilon_{i}(b_{2}),b_{1}\otimes e_{i}b_{Q} if \varphi_{i}(b_{1})<\epsilon_{i}(b_{2}),\end{array}$(1.2)$f_{i}(b_{1}\otimes b_{2})=\{\begin{array}{ll}f_{i}b_{1}\otimes b_{2} if \varphi_{i}(b_{1})>\epsilon_{i}(b_{2}),b_{1}\otimes f_{i}b_{2} if \varphi_{i}(b_{1})\leq\epsilon_{i}(b_{2}).\end{array}$$0\otimes b$

and$b\otimes O$

should be understood as$0$. One cheCks that$\epsilon_{i},$$\varphi_{i}$isand wt are givenby$\epsilon_{i}(b_{1}\otimes b_{2})=\max(\epsilon_{i}(b_{1}),\epsilon_{i}(b_{1})+\epsilon_{i}(b_{2})-\varphi_{i}(b_{1}))$,,$\varphi_{i}(b_{1}\otimes b_{2})=\max(\varphi_{i}(b_{2}), \varphi_{i}(b_{1})+\varphi_{i}(b_{2})-\epsilon_{i}(b_{2}))$wt$(b_{1}\otimes b_{2})=$wt $(b_{1})+$ wt$(b_{2})$.

Tensor product of crystals is associative. In order to compute the action of Kashiwara operators
on multiple tensor products, it is convenient to use the mle
called signature rule. See [3, \S 2.1].
$e_{i},$$f_{i}$

2. PATHIn what follows we set $B=B^{1,1}$ . The set of classically restricted paths inof weight
is by definition$B^{\otimes L}$’$\lambda\in\overline{P^{I}}$$\mathcal{P}(\lambda, L)=\{b\in B^{\otimes L}|wt(b)=\lambda$

and $e_{i}b=0$ for all$i\in I_{0}\}$.

Using the tensor product rule (1.1) one can check that the following two conditions
are equivalent forand.
(i) is a classically restricted path of weight
, and
(ii)
wt
is a classically restricted path of weight.
for allwt$\lambda\in\frac{1}{P^{I}}$$b=b_{1}\otimes b_{2}\otimes\cdots\otimes b_{L}\in B^{\Theta L}$$\lambda\in\overline{P^{I}}’$$b$$\lambda-$$b_{1}\otimes\cdots\otimes b_{L-1}$$(b_{L}),$$\epsilon_{\dot{t}}(b_{L})\leq\langle\lambda-$$\alpha_{\check{i}}\}$$(b_{L})$$i\in I_{0}$Example 2.1. The $f_{0}uow\dot{m}g$ three elements of B@6 are classically restricted paths.
The dot.
(1)\copyright \copyright \copyright(2)\copyright(3)@ \copyright
One checks that in (1) is a classically restricted path as follows. Firstly, is theunique classically restricted path in B.
is classically restricted by the above, wt(O)
criterion with $L=2$ , since is classically restricted and.のis classicaUy
is classicaJly restricted with $L=3$ , since, wt. We continue these checks until
restricted and \copyright$\cdot signifies\otimes$$b=\Phi\cdot\emptyset\cdot$$\cdot$$b=\Phi\cdot\emptyset\cdot\theta\cdot$$b=\Phi\cdot\emptyset\cdot\otimes\cdot$$\cdot\Phi$$\cdot$$wt(b)=\overline{\Lambda}_{1}+\overline{\Lambda}_{5}$$\cdot\emptyset\cdot \mathscr{F}$$wt(b)=\overline{\Lambda}_{3}$$\cdot \mathscr{K}$$wt(b)=\overline{\Lambda}_{3}$$\cdot$$b$$\Phi$$\Phi\otimes\emptyset$$\epsilon(\emptyset)=\Lambda r+\Lambda_{1}$$\Phi$$\Lambda_{1}-A0$$\otimes\emptyset\otimes\otimes$$\epsilon$

$($

$=$

$\Phi\otimes\emptyset$$)=\Lambda_{2}$$(\Phi\otimes\emptyset)=\Lambda_{2}-2\Lambda_{0}$$L=6$ .

The element of
has a grading called energy. To define it we introduce a$H$local energy function :Z. Since the crystal graph ofis connected,
it is defined uniquely, up to a global additive $\infty nstant$ , by
$B\emptyset L$$B\otimes Barrow$$B\otimes B$$H(e_{i}(b\otimes b’))=H(b\otimes b’)+\{\begin{array}{ll}1 if i=0 and e_{0}(b\otimes b’)=e_{0}b\otimes b’-1 if i=0 and e_{0}(b\otimes b’)=b\otimes e_{0}b’0 otherwise.\end{array}$

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M. OKADO AND N. SANO

We normalize the additive constant by the condition
H の $\otimes\Phi)=0$ .
$($More specifically, the value of $H$ is calculated as follows. Let
be the crystal$B$ .
graph obtained by forgetting arrows of
One knows that the crystal
fromgraph ofdecomposes into three comected components as$B_{0}$$\infty 1or0$$B_{0}\otimes B_{0}$$B_{0}\otimes B_{0}=B(2\overline{\Lambda}_{1})\oplus B(\overline{\Lambda}_{1}+\overline{\Lambda}_{2})\oplus B(\overline{\Lambda}_{1}+\overline{\Lambda}_{5})$

,

where
stands for the highest weight -crystal of highest weight
and
$H$
,の
the highest weight vector of each component is given by の
. isconstant on each component, and takes the value $-1,$ $-2$ , respectively. One canconfirm it from the fact thatbelongandto the second and third component. In fact, the value of $H$ is given by$B(\lambda)$$\lambda\in\frac{1}{P’}$$E_{6}$$\otimes\Phi,\Phi\otimes\emptyset$$\otimes\otimes$$0,$$e_{0}(\Phi\otimes\emptyset)=\Phi\otimes\otimes$$e_{0}(\Phi\otimes\Phi)=\Phi\otimes\copyright$$H(b\otimes c)=\{\begin{array}{ll}-2 if b\otimes c\in S_{1}0 if b\otimes c\in S_{2} ,-1 otherwise\end{array}$where$S_{1}=\{\Phi\otimes\copyright|j\geq 18\}u\{\emptyset\otimes\copyright|j\geq 23\}U\{\theta\otimes\copyright|j\geq 25\}$$U$,$\{@\otimes\copyright, \emptyset\otimes\copyright|j\geq 26\}u\{\otimes\otimes\otimes|i=5,8,10,13,18\}$@ can be reached by following some (possibly zero) arrows &om \copyright }.
With this $H$ the energy function $D$ is defined by$S_{2}=${$\copyright\otimes\copyright|$$D(b_{1} \otimes\cdots\otimes b_{L})=\sum_{j=1}^{L-1}(L-j)H(b_{j}\otimes b_{j+1})$.

Example 2.2. The energies of classically restricted paths in Example 2.1 are givenby(1) $D(b)=5(-1)+4(-2)+3(-1)+2(-1)+1\cdot 0=-18$,(2) $D(b)=5(-1)+4(-1)+3(-1)+2\cdot 0+1(-2)=-14$ ,(3) $D(b)=5(-1)+4(-1)+3(-1)+2(-1)+1(-1)=-15$.

3. RIGGED CONFIGURATION
We first provide the definition of a rigged configuration that is valid for any
simply-laced affine type and datum , and then restrict and
and
to
and a
the case corresponding to paths we consider in this paper. Fix
matrixof nonnegative integers, almost all zero. Let $\nu=$be another such matrix. Say that is an admissible configuration
if it satisfies$\mathfrak{g}$

$L$

$L$

$\mathfrak{g}$$E_{6}^{(1)}$$\lambda\in\frac{1}{P’}$$L=(L_{i}^{(a)})_{a\in I_{0},i\in \mathbb{Z}>0}$$(m_{i}^{(a)})_{a\in I_{O},i\in \mathbb{Z}>0}$$\nu$$a \in I_{0}a\in I_{0}\sum_{i\in \mathbb{Z}>0}im_{i}^{(a)}\alpha_{a}=\sum_{i\in \mathbb{Z}_{>0}}iL_{i}^{(a)}\overline{\Lambda}_{a}-\lambda$

and$p_{i}^{(a)}\geq 0$

for all$a\in I_{0}$and $i\in Z_{>0}$ ,

where
(3.1)

$p_{i}^{(a)}= \sum_{j\in \mathbb{Z}>0}(L_{j}^{(a)}\min(i,j)-\sum_{b\in I_{0}}C_{ab}\min(i,j)m_{j}^{(b)})$.

228
KKR TYPE BIJECTION FOR$E_{6}^{(1)}$: ALGORITHMS AND EXAMPLES

is called a vacancy number.
Let
be an admissible configuration. We identify with asequence of partitions
such that. For a partition
and $i\in Z_{>0}$ , define$p_{i}^{(a)}$$\nu=(m_{i}^{(a)})_{a\in I_{0},i\in Z>0}$$\nu$$\nu^{(a)}=(1^{m_{1}^{(a)}}2^{m_{2}^{(a)}}\cdots)$$\{\nu^{(a)}\}_{a\in I_{0}}$$\mu$$Q_{i}( \mu)=\sum_{j}\min(\mu_{j},i)$,the area of the corresponding Young diagram in the first columns. Then setting
the vacancy number (3.1) is rewritten as
$i$

$\mu$

$Q_{\dot{*}}^{(a)}=Q_{i}(\nu^{(a)})$(3.2)$p_{i}^{(a)}= \sum_{j\in Z>0}L_{j}^{(a)}\dot{m}n(i,j)+\sum_{b\sim a}Q_{i}^{(b)}-2Q_{i}^{(a)}$,where $b\sim a$ stands for $C_{ba}=-1$ as defined in \S 1.
Let $J=\{J^{(a,i)}\}_{(a,i)\in I_{O}xZ>0}$ be a double sequence of partitions. Then a rigged
configuration is a pairof an admissible configuration and $J=\{J^{(a,i)}\}$ suchthat
is a partition contained in therectangle. The set of rigged
configurations for fixed and is denoted by RC.
We define the charge of a rigged configuration. First, define the charge of anadmissible configuration by$(\nu, J)$$\nu$$J^{(a,i)}$$m_{i}^{(a)}\cross p_{i}^{(a)}$$\lambda$$L$$(\lambda, L)$$\nu$$c( \nu)=\frac{1}{2}\sum_{a,b\in I_{0}}\sum_{j,k\in Z_{>0}}C_{ab}\dot{m}n(j, k)m_{j}^{(a)}m_{k}^{(b)}$$- \sum_{a\in I_{0}}\sum_{j,k\in \mathbb{Z}_{>0}}\dot{m}n(j, k)L_{j}^{(a)}m_{k}^{(a)}$Using (3.1)(3.3)$c(\nu)$is rewritten as$c( \nu)=-\frac{1}{2}(\sum_{a\in I_{0},i\in Z>0}p_{i}^{(a)}m_{i}^{(a)}+\sum_{a\in I_{0},j,k\in Z>0}$

We then define the charge of a rigged configuration(3.4)$c(\nu, J)=c(\nu)+|J|$

where $|J|= \sum_{(a,i)\in I_{0}xZ>0}|J^{(a,i)}|$ .
To retum to our case, we use the Cartan matrix
(3.5)

.

$L_{1}^{(a)}=L\delta_{a1}\delta_{i1}$min$(j, k)L_{j}^{(a)}m_{k}^{(a)})$$(\nu, J)$by$(C_{ab})$of.

and set$E_{6}^{(1)}$$(a\in I_{0},i\in \mathbb{Z}>0)$

corraeponding to considering paths in $(B^{1,1})^{@L}$ . Under this constraint, the formulas(3.2) and (3.3) are rewritten as$p_{i}^{(a)}=L \delta_{a1}+\sum_{b\sim a}Q_{i}^{(b)}-2Q_{i}^{(a)}$,$c( \nu)=-\frac{1}{2}(\sum_{a\in I_{0},i\in \mathbb{Z}_{>0}}p_{i}^{(a)}m_{\dot{*}}^{(a)}+L\sum_{k\in Z>0}m_{k}^{(1)})$

By abuse of notation, the set RC
by $RC(\lambda, L)$ .

$(\lambda,L)$for$E_{6}^{(1)}$.

with the restriction (3.5) is denoted

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M. OKADO AND N. SANO

Example 3.1. Three rigged configurations in RC
are illustrated below.

$(\overline{\Lambda}_{1}+\overline{\Lambda}_{5},6)$, RC$(\overline{\Lambda}_{3},6)$, RC$(\overline{\Lambda}_{3},6)$

(1)

1

$0$

$0$

$0$

$0$

$0$

$0ffl_{0}^{o}$(2)$01F_{1}^{o}$(3)$01F_{1}^{0}$

The partitions,
are depicted from left to right as Young diagrams. In the second example, and 1 inon the left signify $p_{2}^{(1)}=0$ and$p_{1}^{(1)}=1$. Looking on the right we see $J_{2}^{(1)}=(0),$ $J_{1}^{(1)}=(1,1,0,0)$ . Rom (3.3) wehave $c(\nu)=-18$ , hence $c(\nu, J)=-14$ . $c(\nu)=-18$ is the same for the other two.
$c(\nu, J)=-18,$ $-15$ for the first and third example.
$\nu^{(6)}$$\nu^{(1)},$ $\nu^{(2)},$$\ldots$$\nu^{(1)}$$0$

4. THE BIJECTION$\Phi$In the previous two sections we have introduced two combinatorial objects$L’\in Z_{\geq 0}$
RC
for. In [4] we showed$(\lambda, L)$$\mathcal{P}(\lambda, L)$$\lambda\in\overline{P’},$

Theorem (Theorem 3.2 of [4]). There exists a bijection$\Phi$: RC$(\lambda, L)arrow \mathcal{P}(\lambda, L)$$satis\ovalbox{\tt\small REJECT} ng$(4.1)$c(\nu, J)=D(\Phi(\nu, J))$.

Leaving the proof to [4], we shall describe the bijection and its inverse.
The bijection is defined by giving a more fundamental procedure denoted by .
For $(\nu, J)\in$ RC
produces a smaller rigged configurationin RC $(\rho,L-1)$
and an element of $B$ . The weight is in fact given by
wt $(b)$ . Write for. We then apply to, obtaining. We eventuaJly arrive at the empty
rigged configurationin RC $(O,0)$ that should correspond tofor all$\ldots,b_{1}\in B$ . Then the image of $(\nu, J)$ by
the empty path in, giving
is determined as.
The inverse bijection
is defined in a completely simnilar mamer. Givena rigged configuration (il, ) in $RC(\rho, L-1)$ and an element of $Bsatis\theta\dot{m}g$, we introduce a procedure to construct a new rigged
for all$(\nu,J)$$(\rho+wt(b),L)$ . Ifconfigurationin RC
is given, we$\Phi^{-1}$$\Phi$$\Phi$$\delta$$(\lambda,L)\delta$$(\tilde{\nu},\tilde{J})$$b$$\rho=\lambda-$$\rho$$b$$\delta$$(\nu^{(a)}=\emptyset$$\mathcal{P}(0,0)$$\Phi$$b_{L}$$b_{L-1}$$(\tilde{\nu},\tilde{J})$$a\in I_{0})$$b_{L},$$b_{L-1},$$b_{1}\otimes\cdots\otimes b_{L-1}\otimes b_{L}\in’\rho(\lambda, L)\subset B^{\otimes L}$$\Phi^{-1}$$b$$\tilde{J}$$\epsilon_{i}(b)\leq\langle\rho,\alpha_{\dot{\}}^{\vee}\rangle$$i\in I_{0}$$\tilde{\delta}$$b_{1}\otimes b_{2}\otimes\cdots b_{L}\in \mathcal{P}(\lambda, L)$,KKR TYPE BIJECTION FOR$E_{6}^{(1)}$230: ALGORITHMS AND EXAMPLES

start from the empty rigged configuration and apply 6 times, thereby obtaining
a rigged configuration in RC.
Before giving the algorithms of and , we recall thatis the crystal graph(see Figure 2) obtained by forgetting arrows of color from . We call a row insingular if its rigging (number on the right in Example 3.1) is equal to thecorresponding vacancy number(number on the left).
$L$$(\lambda,L)$$\tilde{\delta}$$B_{0}$$\delta$$B$$0$$\nu^{(a)}$$p_{i}^{(a)}$4.1. Algorithm. For a rigged configuration

the outputs
and byare are given as follows. First set $b=O$ and. Viewing the crystal graph of(Figure 2), repeat the following process for $j=1,2,$
until stopped. Romproceed by one step through an arrow of. Find the mimimal integer
such that, reset to be the sinkhas a singular row of length and setof the arrow. A singular row selected previously during the process should not be
selected again. If there is no such integer, then set
and stop. If there are
two arrows sourcing from $\infty mpare$ the minimal integers and take the smaller one.
If the integers are the same, either one can be taken. The output of the algorithm
does not depend on the choice. Then is the one just obtained and is obtained$bom(\nu, J)$ by removing the rightmost box ffom every singular row selected by theabove procedure. The new rigging is defined to be the same if the correspondin
row is not selected, and declared to be singular if selected.
$\delta$$(\nu, J)$$(\tilde{\nu},\tilde{J})$$b$$\delta$$\ell_{0}=1$$B_{0}$$b$$\ldots$$\infty 1ora$$\nu^{(a)}$$i\geq\ell_{j-1}$

$i$

$b$

$\ell_{j}=i$$\ell_{j}=\infty$

$b,$

$b$

$\tilde{\nu}$$\tilde{J}$$g$4.2. Algorithm. For a given rigged configuration

and $b\in Bsatis\theta\dot{m}g$, the inverse algorithm of is described as follows.
for all
From $b\in B$ go back the arrow in the crystal graph . Suppose the color of thearrow is . Let the maximal length of the singular row in
be . Repeat thefollowing process for $j=1,2,$
until we arrive at O. Find the maximal integer
such that, reset tohas a singular row of length and set
be the source of the arrow. A singular row selected previously during the process
should not be selected again. If there is no singular row in, then assume there
is a singular row of length in. If there are two arrows ending
and setat , compare the maxtmal integers and take the larger one. If the integers are thesame, either one can be taken. The output of the algorithm does not depend on
the choice. The new configuration is given by addin a box to every singular row
selected by the above procedure. The new rigging is defined to be the same if
the corraeponding row is not selected, and declared to be $sing\iota 4ar$ if selected.
4.3. Examples. We apply successively to the rigged $\infty nfigurations$ in Example
3.1. We see that each rigged configuration corresponds under to the classically
restricted path in Example 2.1. The charge of the rigged $\infty nfiguration$ agrees withthe energy of the corresponding path as guaranteed by our theorem. See Example
2.2.
Let us look at the algorithm in detail in the first example. The top rigged
configuration has no gular row in, henceand $b=$ の．Since no box
is removed by the first , the next rigged configuration is the same as the top one.
However, the vacancy numbers change. Applying one prooeeds inas$\epsilon_{i}(b)\leq(\rho,\alpha_{i}^{\vee}\}$$\tilde{\delta}$$i\in I_{0}$$(\tilde{\nu},\tilde{J})$$\tilde{\delta}$$\delta$$B_{0}$$\nu^{(a)}$$a$$\tilde{l}_{0}$$\ldots$$i\leq\tilde{\ell}_{j-1}$$\nu^{(a)}$$\tilde{\ell}_{j}=i$

$i$

$b$

$\nu^{(a)}$$0$$\nu^{(a)}$$\tilde{\ell}_{j}=0$

$b$

$\nu$

$g$

$J$

$\delta$$\Phi$$\nu^{(1)}$$\dot{s}m$$\ell_{1}=\infty$$\delta$$\delta$$B_{0}$$\Phiarrow\emptysetarrow\oplusarrow@arrow\otimesarrow 8arrow\copyrightarrow\copyrightarrow@$withfor $1\leq j\leq 4,$
for $5\leq j\leq 8,$
for $j=9,$ $obta\dot{\min}gb=@$ .
The boxes in the selected rows are shaded.
The third example differs from the second one just by a rigging in.
$\ell_{j}=1$$\ell_{j}=2$$\ell_{j}=\infty$$\nu^{(1)}$231

M. OKADO AND N. SANO

(1)

1

$0$

1

$0$

$0H_{0}^{0}$の$0\ovalbox{\tt\small REJECT}_{0}^{o}$ts$0\mathscr{H}0$$oo\ovalbox{\tt\small REJECT}_{0}^{0}$$\delta\downarrow$$\mathbb{O}$$oo\ovalbox{\tt\small REJECT}_{0}^{0}$$0\ovalbox{\tt\small REJECT}_{0}^{o}0$$0\ovalbox{\tt\small REJECT}_{0}^{o}0$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\delta\downarrow$$\emptyset$$oEo$$\otimes$$\emptyset$$\delta\downarrow$$\emptyset$$0Ho$$0\ovalbox{\tt\small REJECT}_{0}^{o}0$$\delta\downarrow$$oBo$$oo\ovalbox{\tt\small REJECT}_{0}^{0}$の

KKR TYPE BIJECTION FOR$E_{6}^{(1)}$232: ALGORITHMS AND EXAMPLES(2)$o\prod_{0}$$01P_{1}$@$0\square 0$$oH_{0}^{0}$$0\blacksquare 0$1$\square 1$$lg_{1}$$o\Xi_{0}^{0}$$\delta\downarrow$$0\blacksquare 0$@$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$oE_{0}^{0}$$\delta\downarrow$$o\blacksquare 0$$\emptyset$$\emptyset$$\emptyset$$\delta\downarrow$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\delta\downarrow$$\emptyset$$\emptyset$$\emptyset$

233

M. OKADO AND N. SANO

(3)$o\mathscr{H}0$
$01$

ff

$0$

$\mathscr{F}$$00\ovalbox{\tt\small REJECT}_{0}^{0}$

0110

$oEo$

$\delta\downarrow$$0\mathscr{F}0$$\emptyset$$\emptyset$$\emptyset$$0H_{0}^{o}$$\delta\downarrow$$0\mathscr{F}0$@$\emptyset$$\emptyset$$\emptyset$$\emptyset$$o\Xi_{0}^{0}$$\delta\downarrow$$o\mathscr{F}0$$\emptyset$$\emptyset$

3$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\delta\downarrow$$\emptyset$$\emptyset$$\emptyset$$\delta\downarrow$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$

REFERENCES[1] B. Jones and A. Schilling, Affine structures and a tableau model for
crystals, J. Algebra
324 (2010) 2512-2542.
[2] V. Kac, Infinite dimensional Lie algebras, 3rd ed., Cambridge University Press, 1990.
[3] M. Okado, X $=$ M conjecture, Combinatorial Aspect of Integrable Systems, MSJ Memoirs
17, 43-73 (2007).
, Contemp.
[4] M. Okado and N. Sano, KKR type bijection for the exceptional affine algebra
Math. 565, (2012); http://dx.doi.org/10.1090/conm/565/11181.
$E_{6}$$E_{6}^{(1)}$