数理解析研究所講究録
1190 巻 2001 年 95-106

95

SATO-WELTER GAME AND KAC-MOODY LIE
ALGEBRASNORIAKI KAWANAKA$|\mathrm{t}]$

C 触丸.

凝血唄

$5\mathrm{E}$)

ABSTRACT. This is a revised version of [11]. The purpose of thispaper is to give a large class of impartial 2-player games which are“completely solvable” in the sense that they have good formulas orgood algorithms for the Sprague-Grundy numbers. Nim and Sat
Welter game are included here as very special cases. The notionof minuscule elements of Weyl groups, due to D. Peterson, and theclassification of them, due to P. A. Proctor, are essential in ourconstruction.
$(\succ$

1. BASICS ON GAMES

The reference for this section is Conway [2].
1.1. Game graphs
We consider complete information games played by two players. Moreprecisely, we consider only those games which can be represented bya graph as follows. A finite directed graph with the following
properties is called a game graph:
1. The graph has no cycle, i.e. there is no sequenceof.
and that $v_{n}=v_{1}$ unless $n=1$ .
vertices such thatof such that, for
2. There exista a (necessarily unique) vertex
of vertices
any vertex of , there exists a sequenceand that $v_{1}=v_{g},v_{n}=v$ .
such that
Given such a graph , two players can play a game as follows. Place”aat the beginning position . The first player moves the stone
by an edge directed toward . Simito any vertex connected tolarly, the second moves the stone to any vertex connected to by
an edge directed toward , and so on. The player first unable to moveis the loser. The game considered in the present paper is isomorphic
to this game corresponding to a game graph . We shall identify thegame itself and the game graph representing it. It is not difficult to seethat one of the players, the first one or the second one, has a winning
strategy.
$g$$g$$v_{1},v_{2},$ $\ldots,v_{n}$$g$$v_{i}arrow v_{i+1}$$v_{g}$$v$$g$$v_{1},$ $v_{2},$$g$$\ldots,$$v_{n}$$v_{i}arrow v_{i+1}$$g$$‘(\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{e}$$v_{g}$$v$$v$$v_{g}$

$w$

$w$

$g$

$v$

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NORIAKI KAWANAKA

Remark. We can define more general games by considering colors.
We assume each edge of the game graph is colored by one of the twocolors, red or blue, say. Before beginning the game two players choose
their colors; if one of them chooses red, say,then the other must takethe rest, blue. Red (resp. blue) player can play only red (resp. blue)
moves. In [2] games with colors are called partizan and games without
color impartial. Since all the games considered in this paper are impartial, we donot discussgames any more and refer theinterested readers to [2].
$g$$\mathrm{a}\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{Z}\mathrm{a}\mathrm{n};$,1.2. Sums of games
Let and be game graphs. Then the sum $a+b$ of and is a new
game graph defined as follows. The set of vertices of $a+b$ is the setof pairs $(x, y)$ of vertices of and vertices of . Two vertices $(x, y)$and $(x”,y)$ of $a+b$ are connected by an edge directed toward $(x’,y’)$if and only if either $xarrow x’$ and $y=y’$ , or $x=x’$ and $yarrow y’$ . As agame $a+b$ can also be defined as follows. The first player of the game
$a+b$ chooses either
or and plays a first move in the chosen game.
The second player also chooses or and plays a second or first movein the chosen game according as the second chooses the same game as
the first or not.... At the end, one of the player will be unable to play
anymore neither in nor in , which means that the player is the loser.
$a$

$b$

$a$

$x$

$a$

$a$

$y$

$b$

$b$

$b$

$a$

$b$

$b$

$a$

1.3. Energy of games
Let be the game graph of a game. Let
be the set of non-negative
integers. For each vertex $v\in g$ we attach its energy
asfollows. Let
be the set of successors of in . We
define:$g$$\mathrm{N}_{0}$$E(v)\in \mathrm{N}_{0}$$S_{v}=\{w_{1}.w_{2}, \ldots,w_{k}\}$$v$$E(v)=[\mathrm{N}_{0-}\mathrm{f}^{E}(w_{i});w_{i}\in sv\}]$$g$.

In particular, $E(v)=0$ if is an ending position of , i.e. if no edge
goes outside from . The energy $E(g)$ of a game is defined by:$E(g)=E(v_{g})$ .
In general, for $v\in g$ , the energy $E(v)$ of is the energy of the game
which is “generated” by , i.e. the “sub”game of whose beginning
position is . The importance of the notion of energy in impartial (i.e.
uncolored) games will become apparent by the following:
$v$

$g$

$v$

$g$

$v$

$v$

$g$

$v$

Theorem 1. (R. P. Sprague, P. M. Grundy, see [2])(i) The second player has a winning strategy in the game , if andonly if$g$$E(\mathit{9})=0$.

97

GAMES AND KAC-MOODY LIE ALGEBRAS(ii) Let

$a$

and

$b$

be game graphs. Then
$E(a+b)=E(a)\oplus E(b)$ ,(calledforis defined by the following rules:$m,$ $n\in \mathrm{N}_{0},$$where_{\mathrm{z}}$$n)$$m\oplus n\in \mathrm{N}_{0}$“$n\dot{\iota}m$sum” of $m$ and.

$m\oplus m=0)$$2^{k}\oplus m=2^{k}+m$$m\oplus 0=m$

if $m<2^{k}$ ,,,(commutativity),(associativity).

$m\oplus n=n\oplus m$$(l\oplus m)\oplus n=l\oplus(m\oplus n)$

If one wants to calculate the energy $E(g)$ of a game directly fromthe definition of $E(g)$ , then one needs the whole graph . Hence,
most cases, the actual calculation is hopeless. Thus examples of games
which have good formulas for their energy are very precious. In literatures [2], energy of a game is called the Sprague-Grundy number orGrundy numbers of . We have used the word “energy” deliberately
to stress its importance.
$g$$\mathrm{i}\mathrm{n}\downarrow$

$g$

$g$

$g$

2. SOME EXAMPLES OF GAMES
Here we collect some examples of games.
2.1. Nim ([2])
This is a very well-known game. For a non-negative integer , letbe a totally ordered set with elements. Then$k$

$L_{k}$$k$(finite disjoint union)$L= \bigcup_{i}L_{k_{i}}$of is called a string. The
is a partially ordered set. A component$(0\leq h_{j}<k_{j})$ .
and reduce it toa string
first player)orThe second player also choose a (non-empty) string (and reduce it to a strictly shorter string, and so on. In other words, if
denotes the (trivial) game with just one string , then this gameis the sum$L$$L_{k_{j}}$$\mathrm{c}’ \mathrm{h}_{\mathrm{o}\mathrm{O}\mathrm{S}}\mathrm{e}$$L_{h_{j}}$$L_{k_{j}}$$L_{k_{i}}(i\neq j)$$L_{k}$$L_{k}$$L_{k_{1}}+L_{k_{2}}+\ldots$

of games$L_{k_{i}}$in the sense explained in 1.2. It is easy to see that
$E(L_{k})=k$ .

Hence, by Theorem 1, the energy of the game
$E(L)=k_{1^{\oplus}}k_{2^{\oplus}}\ldots$.

$L$is given by$L_{h_{j}}$98

NORIAKI KAWANAKA

2.2. Sato-Welter game $([2][8][10])$. Welter [10] independently built a beau, M. Sato [8] andIn
tiful theory for the following game.
We have finitely many particles. Each particle is placed in one of theenergy levels labelled by $\{0,1,2,3, \ldots\}$ . We assume a ‘physical law’ asserting that different particles can never be in the same energy lebel.
Each player, in its turn, chooses a particle and reduces its energy to astrictly smaller level. Of course, if an energy level is already occupied
by a particle, then no other particle can move into that energy level.
If a player finds, in its turn, that it can move no particle any more(this is equivalent to say that the energy levels $0,1,2,$ $n-1$ , withnumber of particles, are already occupied), then that player isthe loser.
For our purpose, it is essential toSato-Welter game canalso be played on Young diagrams.
Let $Y$ be a Young diagram. This means $Y$ is a finite subset ofsuch that$1950’ \mathrm{s}$$\mathrm{C}.\mathrm{P}$$\ldots,$$n=\mathrm{t}\mathrm{h}\mathrm{e}$$\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{i}_{\mathrm{C}}\mathrm{e}$that.

$\mathrm{N}\cross \mathrm{N}$$Y= \bigcup_{i1}^{l}\{=(i,j);1\leq j\leq n_{i}\}$,with$n_{1}\geq n_{2}\geq...$ $\geq n_{l}\geq 0$.

(If $n_{i}=0$ , then the i-th row of is empty.) The diagram
viewed as a partially ordered set by$Y$$(i,j)\geq(i’,j’)$$Y$

can beif $i\leq i’$ and $j\leq j’$

In particular, $(1, 1)$ is the unique maximum element of $Y$ if $Y$ is nonempty. For any $(i,j)\in Y$ , the subset$H(i,j)=$ { $(k,$ $l)\in Y;k=i$ or $l=j,$ $(k,$ $l)\leq(i,j)$ }of $Y$ is called the hook of at $(i,j)$ , and the number $|H(i,j)|$ of elements in $H(i,j)$ the hook-length at $(i,j)$ . We now define the $\zeta removal$of hooks and pushing up’ procedure well-known in the representation
theory of symmetric groups. This gives a procedure to obtain, for a
fixed element $(i,j)$ of a given Young diagram , a smaller Young diagram $Y’$ . If the set theoretical difference $Y-H(i,j)$ is again a Youngdiagram, then we simply put $Y’=Y-H(i,j)$ . In general, we put
$Y$

$Y$

$Y’=$

{

$(k,$

$l)\in Y;k<i$ or $l<j$ } $\cup\{(k-1, l-1);(k, l)\in Y, (k, l)\leq(i+1,j+1)\}$ .

The difference $H(i,j)_{*}=Y-Y’$ is called the rim-hook of at $(i,j)$ .
Note that a rim-hook $I=H(i,j)_{*}$ is an order ideal of , i.e. $a\in I$ and$b<a$ implies $b\in I$ . We now give the Young diagramatic formulationof Sato-Welter game:be a given Young diagram. Each player, in its turn, chooses
Let
$Y$

$Y$

$Y$

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GAMES AND KAC-MOODY LIE ALGEBRAS

an element $(i,j)$ of

and removes the corresponding hook $H(i,j)$and pushes up (or, equivalently, remove the corresponding rim-hook$H(i,j)_{*})$ . At the end, thety Young diagram will be left, and thelast player is the winner.
$Y$$\mathrm{e}\mathrm{m}\mathrm{p}_{l}$Remark. It seems Sato’s first formulation of the game was in termsof Young diagrams. (One of his motivation was Nakayama conjecturein the representation theory of symmetric groups.) This formulation
does not appear in Welter [10] nor in.
Theorem 2. ([8] [10], see also [2])Let be a Young diagram. Then the energy $E(g_{Y})$ of the Sato-Welter
gameplayed on is given by:$\mathrm{C}_{\mathrm{o}\mathrm{n}\mathrm{W}\mathrm{a}\mathrm{y}}[2]$

$Y$

$Y$

$g_{Y}$$E(g_{Y})= \sum_{(i,j)\in Y}\oplus N(|H(i,j)|)$where $N($ .

$)$

(2.1)

,

is the Sato-Welter no $7m$ defined by$N(k)=k\oplus(k-1)$ ,$k\in \mathrm{N}$.

Remark. As Sato [9] remarked, the structure of the above formulafor $E(g_{Y})$ has a striking resemblanse to that of the “hook formula”
for the dimensions of irreducible representations of symmetric groups.
This observation has been crucial for the present work. See Theorem
6 below.
Remark. One can of course play Sato-Welter game on a disjoint
union of a finite number of Young diagrams. (Nim is a very special
case of this.) The energy of such multi-component Sato-Welter gamecan be calculated using Theorems 1 and 2.

2.3. Playing games on partially ordered setsLet
be a collection of (isomorphism classes of) finite partially ordered sets. Suppose, for any $P\in P$ , and any $p\in P$ , an order idealof , called the rim-hook of at , is assigned in such a$P-H(p)\in
P$ . Then just as in the case of Sato-Welter game
way that(Young diagramatic formulation), we can consider a gameplayedon ; two players alternatively remove rim-hooks.
$P$$H(p)_{*}(\neq\emptyset)$

$P$

$P$

$p$

$g_{P}$$P$

3. THE PARITY CONDITION

3.1. Nim sum for integers. We formally put
Let$m\in \mathrm{N}$$-m=(-1)\oplus(m-1)$ ,100

NORIAKI KAWANAKA

and assume$(-1)\oplus(-1)=0$ .

This enables us to extend the nim, we put
For $m,$$\mathrm{s}\mathrm{u}\mathrm{m}\oplus \mathrm{t}\mathrm{o}$the whole set$\mathrm{Z}$of integers.

$n\in \mathrm{Z}$$(m|n)=m\oplus n\oplus(m\oplus n-1)=N(m\oplus n)=N(m-n)$ .

Theorem 3. Let-valued function$\mathrm{Z}$$a_{1},$ $a_{2},$

$f$

on

$\ldots,$$\mathrm{Z}$$a_{n}$

bybe distinct elements of Z. We define a$f(t)= \sum_{=i1}^{n}\oplus(t|a_{i})$

Then,

$f$

,

$t\in \mathrm{Z}$.

cannot be constant.

3.2. Games satisfying the
Let be a game graph. Let
ncted to the beginning position$\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{i}\dot{\mathrm{t}}\mathrm{y}$$g$condition
be vertices of directly conby edges in . For, we$v_{1},v_{2},$ $\ldots,v_{k}$put$v_{g}$$g$$n\in \mathrm{N}_{0}$$g$$a_{n}=|\{1\leq i\leq k ; E(v_{i})=n\}|$.

Then the definition of the energy $E(g)=E(v_{g})$ implies:$a_{n}\neq 0$

if $n<E(g)$ , and $a_{n}=0$ if $n=E(g)$ .

We say that the game$a_{n}$$g$satisfies the parity condition if

is odd if $n<E(g)$ , and$a_{n}$

is even if $n\geq E(g)$ .

By Theorem 3, this condition is equivalent to say that
following parity equality:$\sum_{i=1}^{k}\oplus(t|E(v_{i}))=t\oplus(t-E(g))$$g$satisfies the

.

(3.1)

If we put $t=E(g)$ in (3.1), then we have$E(g)= \sum_{i=1}k\oplus(E(g)|E(v_{i}))=\sum_{i=1}k\oplus N(E(g)-E(v_{i}))$

.
(3.2)

Compare (3.2) with (2.1).
4.

$P$-GAMES4.1. Miniscule elements of Weyl groups
Let $W$ be the Weyl group of a Kac-Moody Lie algebra [4] with simplylaced Dynkin diagram. Let be a dominant integral weight. Following
D. Peterson (unpublished; but see $[1][6][7]$ ), we say that an element$\lambda$

$w$

101

GAMES AND KAC-MOODY LIE ALGEBRAS

of $W$ is -minuscule if there exists a reduced expression(
the reflection associated to simple root ) of such that$\lambda$$w=s_{i_{1}}\ldots s_{i}s_{i_{1}}2$-$s_{i}$$\neg$$\alpha_{i}$$w$$s_{i_{j}}(_{S_{i}}j-1\ldots \mathit{8}_{i}1\lambda)=(s_{ii_{1}}-\cdots S\lambda)j1-\alpha_{i}j$

We also say is minuscule if it is -minuscule for some . In his studyof Schubert calculus in the Kac-Moody setting, Peterson proved thefollowing
formula”:$\lambda$$\lambda$$\lambda$$‘(\mathrm{h}\mathrm{o}\mathrm{o}\mathrm{k}$Theorem 4. (D. Peterson; see [1]) Let $w\in W$ be minuscule. Thenthe number of reduced expressions of is equal to$w$,$l(w)!-1 \alpha>\prod_{w(\alpha)<0}h(0\alpha)-1$

where $l(w)$ is the length of , and$w$$h(\alpha)$

is the height$of\backslash$

the root .
$\alpha$If $W$ is of type , then this gives an unusual way of stating the famous
hook formula for the number of standard tableaux for a given Youngdiagram. See the last paragraph of 3.2 below. It is announced $[6][7]$that a -analogue of the above theorem will appear in a forthcoming
paper of Peterson and R. A. Proctor. See also [3].
$\mathrm{A}$$q$

4.2. Classification of minuscule elements of Weyl groups
The results in this subsection is due to R. A. Proctor $[6][7]$ .
Theorem 5. (R. A. Proctor) The classification of minuscule $element_{S}f$in Weyl groups are equivalent to the classification of -complete posets.’
A -complete poset can be explicitly $desc\dot{n}bed,$ as a disjoint union oftlslant sums of $tlirreducib\iota_{e}d$ -complete posets.,
$ttd$

$d$

’;

See [6] and [7] for unexplained terminologies. In short, a d-complete
poset is a finite partially ordered set which satisfy, among other technical conditions, the following:
$P$

1. If

$y,w\in P$ satisfy both $xarrow w$ (i.e.

$x>w$ and no element oflies between and ) and $y-w$ , then there exists a unique$v\in P$ satisfying both
$v-x$ and $varrow y$ .
2. Assume, for $v,w\in P$ with $v>w$ , the interval $[w,v]=\{a\in$$P;w\leq a\leq v\}$ looks like:
$x,$

$P$

$w$

$x$

$v-x$
$\uparrow$$\uparrow$$y^{-w}k^{-\ldots-}w_{2}arrow w_{1}=w$

102

NORIAKI KAWANAKA

for some $k\geq 1$ . Then there exists a unique sequemce $v_{1}>v_{2}>$lookssuch that the interval$...>v_{k}=v$ of elements of
like:
$P$

$[w_{1}, v_{1}]$$v_{1}-v_{2}arrow\ldotsarrow v_{k}arrow X$$\uparrow$$\uparrow$. $(4.1)$$yarrow w_{k}arrow\ldotsarrow w_{2}-w_{1}=w$

The partially ordered set (4.1) is called a double-tailed diamond. Rooted
trees, Young diagrams, shifted Young diagrams (see [5]) and doubletailed diamonds are classical examples of -complete posets. A nonclassical example (which includes diamonds as special cases) is given
in 3.4 below. See [7] for a lot of more exotic examples.
In the classification of -complete posets $P$ , we can assume $P$ is connected with respect to the relation –. Then there exists a uniquesubset $T$ of $P$ called the “top tree” of $P$ (see [7] for the definition),
$d$

$d$

which should be considered as a simply-laced Dynkin diagram endowedwith anstructure of rooted tree. In the case is a rooted tree, its
top tree coincides with itself. In the case of a double-tailed diamond(4.1), the top tree is:$P$$\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{r}\mathrm{a}\sim$$P$$v_{1}arrow v_{2}-\ldots-v_{k}-x$ ,$\uparrow$$y$which is a Dynkin diagram of type D. In the case of a Young diagram:$v_{m}arrow v_{m+1}arrow\ldots$$\uparrow$$\uparrow$$v_{m-1}arrow a$

:

–

..

:
....

$b$

:

$\uparrow$$\uparrow$$\ldotsarrow v_{n}$$\uparrow$

..

..

(4.2)

$\uparrow$$v_{2}$$\uparrow$$v_{1}$

the top tree is:$v_{1}arrow v_{2}arrow\ldotsarrow v_{m-1}arrow v_{m}arrow v_{m+1}arrow\ldotsarrow v_{n}$,which is a Dynkin diagram of type A. Let $W=W(T)$ be the Weylgroup of the Kac-Moody Lie algebra with Dynkin diagram , and
$T$

103

GAMES AND KAC-MOODY LIE ALGEBRAS

the set of simple reflections of W. Then elements of are in 1-1
correspondence with elements of . This extends uniquely to a correspondence
$T$

$S$

$S$

$\Psi$: $Parrow S$

with the following property: If $v_{1}>w_{1}(\in P)$ are such that the interval
looks like (4.1) for some $k\geq 1$ , then, for any $\Psi(v_{i})=\Psi(w_{i})$ .
(So, e.g. in (4.2), we have $\Psi(a)=\Psi(vm),$ $\Psi(b)=\Psi(vm+1).$ ) Note that,is a totally ordered subset of .
for any $s\in S,$
We now fix a total order on compatible with the original partial order.
In the case of a double-tailed diamond (4.1), the number of ways fixing
such a total order is 2. In the case of a Young diagram, the numberof ways is equal to the number of corresponding standard tableaux,or to the dimension of the corresponding irreducible representation ofsymmetric groups. We now pick elementsof correspondingof in the order fixed above, and take the productto elements$[w_{1}, v_{1}]$

$i,$

$P$

$\Psi^{-1}(s)$$P$$s_{i_{1}},$$p_{1},p_{2},$$s_{i_{2}},$$S$$\ldots$$P$$\ldots$$w=S_{i_{1}}\ldots Si_{2}S_{i}1$

(4.3)

’

where $l=|P|$ . Then is a minuscule element of $W$ independent of
the choice of the total order on , and (4.3) is a reduced expression of. See Theorem 4 and Theorem 5.
$w$

$P$

$w$

4.3. Definitions of hooks and P-gamesLet
be a -complete poset. We are going to introduce a game(which we call the -game) played on . For that purpose, it is enough
$P$

$d$

$g_{P}$$P$$P$

to define, for each $p\in P$ , the corresponding ‘rim-hook’ $H(p)_{*}\subset P$ in
such a way that $H(p)_{*}$ is an order ideal of P. (Then the difference$P-H(p)_{*}$ is again a -complete poset.) See 2.3.
be the top tree, andWe can assume
is connected. Let
and $W=W(T)$ the corresponding root system and Weyl group.
Let$d$$\Sigma=$

$T$

$P$

$\Sigma(T)$$w=s_{i_{\mathrm{t}}}\ldots s_{\dot{x}_{2}}s_{i_{1}}$,be the element of $W$ corresponding to as in (4.3). Let $p\in P$ . If is
the $(=j(p))$ -th element of with respect to the fixed total order of, then we put
$P$

$j$

$p$

$P$

$P$

.

$\alpha^{(p)}=-s_{i\iota i_{l1}}S-\cdots \mathit{8}_{i_{j}}(\alpha i_{j})(\in\Sigma)$

Since this is a positive root, we can write$\alpha^{(p)}=\sum_{i\alpha\in S}ci)(p\alpha_{i}$,$c_{i}^{(p)}\in \mathrm{N}_{0}$

.

104

NORIAKI KAWANAKA

Let$M^{(p)}=\{j(p)\leq m\leq l;s_{i_{m}}s_{i_{m+1}}\ldots s_{i_{1}}(\alpha^{(p)})<S_{i}\ldots si\iota(m+1\alpha^{(})p)\}$.

We put$H(p)=\{p_{m};m\in M^{(p)}\}$and call it the hook of at . Let : $Parrow S$ be as in 3.2. For eachand such that, let
be the minimal element of the totally, andordered set
be the lowest interval ofelements in. (Let $I(p;i)=\emptyset$ if
We put$P$$p$$\Psi$$p$$c_{i}^{(p)}\neq 0$$i$$q_{i_{1}}$$I(p;i)_{*}=[q_{i_{1}}, q_{i_{c}}]=\{q_{i_{1}}, q_{i}2’\ldots, q_{i}c\}(c=c_{i}^{()})p$$\Psi^{-1}(\alpha_{i})$$c_{i}^{(p)}$$\Psi^{-1}(\alpha_{i})$$c_{i}^{(p)}=0.)$$H(p)_{*}= \bigcup_{\alpha\in s^{I}}(ip;i)_{*}$and call it the rim-hook of at . Similarly, we define $I(p;i)^{*}$ to be
the highest interval ofelements in. We put$P$$p$$c_{i}^{(p)}$$\Psi^{-1}(\alpha_{i})$$H(p)^{*}= \bigcup_{\alpha}\in SIi(p;i)^{*}$

Then$H(p)^{*}\mathrm{i}_{\mathrm{S}}$a -complete poset.
$d$Remark. If is a Yo\’ung diagram or a shifted
diagram, thenthe above definition of hooks coincide with the known one. See [5],5.1.4, Ex.21 for the graphical definition of hooks of a shifted Youngdiagram. (This exercise gave the motivation for the present work.)$P$$\mathrm{Y}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{g}i$4.4. Main Theorem
Let be a (connected or disconnected) -complete poset, and consider a -game . We have the following generalization of Theorem
2:
$P$

$d$

$P$

$g_{P}$

Theorem 6. The gameits energy $E(g_{P})$ satisfies:$g_{P}$

satisfies the parity condition.

$E(g_{P})= \sum_{p\in P}\oplus N(E(g_{H}(p)*))$$M_{\mathit{0}re\mathit{0}v}er_{f}$.

The proof will appear elsewhere. If is a Young diagram, then we always have $N(E(\mathit{9}H(p)^{*}))=N(|H(p)|)$ (although, in general,$|H(p)|)$ . Hence Theorem 2 is a special case of Theorem 6.
$P$$E(\mathit{9}H(p)*)\neq$

Example. Let“Inset” in [7] .
$)$

$P$

be a -complete poset of the following form (called
$d$

105

GAMES AND KAC-MOODY LIE ALGEBRAS$v_{1}-v_{2}arrow...$$arrow v_{k}arrow xarrow\bulletarrow...\bullet$ $arrow\bulletarrow\bullet$$\uparrow$$\uparrow$$\uparrow$$yarrow\bulletarrow\bulletarrow\ldots$$\uparrow$$\uparrow$$\bulletarrow\bulletarrow\ldots$::$\bulletarrow\ldots$$\uparrow$$\bullet$More precisely,$P$

is the union of a Dynkin diagram of type$v_{1}arrow v_{2}arrow\ldotsarrow v_{k}arrow x$$\mathrm{D}$,$\uparrow$$y$and a Young diagram$Y= \bigcup_{i=1}^{k+1}\{(i,j);1\leq j\leq n_{i}\}$,$n_{1}\geq\ldots\geq n_{k+1}\geq 0$with an identification $x=(1,1)$ if $n_{1}\geq 1$ , and an extra relation$yarrow(2,1)$ if. The top tree of is :$P$$n_{2}\geq 1$$v_{1}-v_{2}arrow\ldots-v_{k}-x-\bulletarrow\ldots\bulletarrow\bulletarrow\bullet$$\uparrow$$y$which is, in general, not a Dynkin diagram of finite type.
By Theorem 6, we get$E(_{\mathit{9}}P)=E(g_{Y}) \oplus\sum_{m1\leq\leq k+1}\oplus N(1+E(g_{Y}m))$where, for $1\leq m\leq k+1,$$Y_{m}$,

is the Young diagram given by:$Y_{m}= \bigcup_{i=1}^{k}\{(i,j);1\leq j\leq n_{i}^{()}m\}$

,

where
$n_{m-}^{(m)}1=n_{m-1}+1$$n_{1}^{(m)}=n_{1}+1,$$n^{(m}2=(m))n2+1,$$n_{m}^{(m)}=n_{m+1},$$\ldots,$$n^{()}m+m+km=n_{k}+1$ .
$n1n=2,$$\ldots,$,

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$game_{\text{ノ}}$

$d$

[7]

$\lambda$

$d$

[8]
[9]

[10]
[11]
[12]

$\mathrm{s}_{\mathrm{u}\mathrm{u}\mathrm{g}}\mathrm{a}\mathrm{k}\mathrm{u}-\mathrm{n}\mathrm{o}_{-}\mathrm{A}\mathrm{y}\mathrm{u}\mathrm{m}\mathrm{i}15- 1$$powe\dot{r}$$\mathrm{z}_{S}$