Decorations on Geometric Crystals and Monomial Realizations of Crystal Bases for Classical Groups

0.61 MB·English

by  Toshiki Nakashima

DECORATIONS ON GEOMETRIC CRYSTALS AND MONOMIAL
REALIZATIONS OF CRYSTAL BASES FOR CLASSICAL GROUPS
3
1 TOSHIKINAKASHIMA
0
2
Abstract. We shall describe explicitly the decoration functions for certain decorated geometric
n crystals ofclassical groups andweshall show that they arerepresented interms of monomialreal-
a izationsofcrystalbases.
J

0
3

] Contents
A
1. Introduction 2
Q
2. Preliminaries and Notations 3
.
h 3. Fundamental Representations 4
t 3.1. Type A 4
a n
m 3.2. Type Cn 4
3.3. Type B 5
[ n
3.4. Type D 6
n
1
4. Decorated geometric crystals 7
v
4.1. Definitions 7
1
0 4.2. Characters 8
3 4.3. Positive structure and ultra-discretization 8
7 4.4. Decorated geometric crystal on B 9
w
1. 5. Monomial Realization of Crystals 10
0 5.1. Definitions of Monomial Realization of Crystals 10
3 5.2. Duality on Monomial Realizations 11
1
6. Explicit form of the decoration f for Classical Groups 12
: B
v 6.1. Generalized Minors and the function f 12
B
i 6.2. Bilinear Forms 13
X
6.3. Explicit form of f (tΘ−(c)) for A 13
r B i n
a 7. Explicit form of f (tΘ−(c)) for C 14
B i0 n
7.1. Main theorems 14
7.2. Proof of Theorem 7.1 15
7.3. Proof of Theorem 7.2 19
7.4. Correspondence to the monomial realizations 23
8. Explicit form of f (tΘ−(c)) for B 29
B i0 n
8.1. Main theorems 29
8.2. Proof of Theorem 8.1 30
8.3. Proof of Theorem 8.2 32
8.4. Correspondence to the monomial realizations 32
2010 Mathematics Subject Classification. Primary17B37; 17B67;Secondary81R50;22E46; 14M15.
Key words and phrases. Crystal, decorated geometric crystal, elementary character, monomial realization, funda-
mentalrepresentation, generalizedminor.
supportedinpartbyJSPSGrantsinAidforScientificResearch♯22540031.
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2 TOSHIKINAKASHIMA
8.5. Triangles and ∆ (Θ−(c)) 33
w0snΛn,Λn i0
8.6. Crystal structure on △ 35
n
8.7. Proof of Theorem 8.9 37
9. Explicit form of f (tΘ−(c)) for D 40
B i0 n
9.1. Main Theorems 40
9.2. Proof of Theorem 9.1 41
9.3. Correspondence to the monomial realizations 44
9.4. ∆ (Θ−(c)) and ∆ (Θ−(c)) 45
w0sn−1Λn−1,Λn−1 i0 w0snΛn,Λn i0
9.5. Crystal structure on △′ 47
n
9.6. Proof of Theorem 9.9 49
References 52
1. Introduction
Since the theory of crystal bases was invented by Kashiwara([6, 7]), there have been several kinds
of realizations for crystal bases, e.g., tableaux, paths, polytopes, monomials, etc. In the article, the
monomialrealizationofcrystalbases,whichisintroducedbyNakajima[13]andrefinedbyKashiwara
[9], will be treated and used to describe the decorationfunctions for the decoratedgeometric crystals
([2], see also below). Let Y be the set of Laurant monomials in doubly-indexed variables {Y }
m,i
(m∈Z, i∈I :={1,2,··· ,n}) as follows:
Y :={Y = Ylm,i|l ∈Z\{0} except for finitely many (m,i)}.
m,i m,i
m∈Z,i∈I
Y
We can define the crystal structure on Y by the way as in 5.1 and it is shown that certain connected
component B(Y) ⊂ Y including the highest monomial Y is isomorphic to the crystal B(λ) where
λ=wt(Y) is a dominant integral weight. For example, for type A and any integer m we have
n
Y Y Y 1
B(Λ1)∼={Ym,1, m,2 , m,3 ,··· , m,n , },
Y Y Y Y
m+1,1 m+1,2 m+1,n−1 m,n
where Λ is the first fundamental weight and Y is the highest monomial.
1 m,1
A geometric crystal is a sort of geometric lifting of Kashiwara’s crystal bases ([1]), which is gen-
eralized to the affine/Kac-Moody settings ([11, 12, 15]). In this paper we do not treat such general
settings and then we shall consider the simple classicalsettings below. Let g be a simple complex Lie
algebra, G the corresponding complex algebraic group, B± ⊂ G Borel subgroups and U± maximal
unipotent subgroups such that U± ⊂ B±. The notion of decorated geometric crystals has been ini-
tiated by Berenstein and Kazhdan([2]). Let I be the index set of the simple roots. Associated with
the Cartan matrix A = (a ) , the decorated geometric crystal X = (χ,f) is defined as a pair of
i,j i,j∈I
geometriccrystalχ=(X,{e } ,{γ } ,{ε } )anda certainspecialrationalfunctionf onthe algebraic
i i i i i i
variety X satisfying the condition
f(ec(x))=f(x)+(c−1)ϕ (x)+(c−1−1)ε (x),
i i i
foranyi∈I,where ec isthe unitalrationalC× actiononX,andϕ :=ε ·γ is the rationalfunctions
i i i i
on X. The function f is called the decoration (function) of the decorated geometric crystal X. In
[2] the ultra-discretization of the decoration is used to describe the Kashiwara’s crystal base B(λ),
which is quite similar to the polyhedral realizations of crystal bases([14, 19]). This similarity let us
conceive some link between crystal bases and the decorations. Indeed, in [18] we made this link clear
for type A . Here we shall consider another link between them, which is the main purpose of this
n
article. Thepurposehereistopresenttheexplicitformofthedecorationsforthedecoratedgeometric
DECORATIONS ON GEOMETRIC CRYSTALS AND MONOMIAL REALIZATIONS OF CRYSTAL BASES 3
crystalsonB (see 4.4)anddescribethe decorationsintermsofthe monomialrealizationsofcrystal
w0
bases([9, 13]). In [18], we presented the conjecture (see also Conjecture 6.8 below) and gave the
positive answerfor type A . To be more precise,we shall introduce certainpartofthe results in [18].
n
First, we consider the geometric crystal structure on the variety B := TB− where T ⊂ G is the
w0 w0
maximaltorus, w is the longestelement ofthe Weyl groupandB− :=B−∩Uw U. Let χ :U →C
0 w0 0 i
be the elementary characterand η :G→G the positive inverse (see 4.2). Then the decorationf on
B
B is defined by the formula
w0
f (g):= χ (π+(w−1g))+χ (π+(w−1η(g))),
B i 0 i 0
i
X
where π+ : B−U → U is the projection. By the definition of f , it suffices to get the explicit form
B
of χ (π+(w−1g)) and χ (π+(w−1η(g))) for our purpose. Furthermore, the elementary characters are
i 0 i 0
expressed by the “generalizedminors ∆ ” ([3, 4, 5]) and then as in (6.2) we have
γ,δ
∆ (g)+∆ (g)
f (g)= w0Λi,siΛi w0siΛi,Λi ,
B
∆ (g)
i w0Λi,Λi
X
where Λ is the i-th fundamental weight. We shall see the explicit forms of ∆ (g) and
i w0Λi,siΛi
∆ (g) as in (6.8) and (6.9). In most cases except for the spin representations of type B
w0siΛi,Λi n
and D , it is performed by direct calculations. For the cases of the spin representations, we prepare
n
the “triangles”, which has some interesting combinatorial properties and is useful to calculate the
above generalized minors. Then, we can find their relations to the monomial realizations of crystals
(Sect.5).
In [18], we also describe the relations to the polyhedral realizations explicitly for type A though
n
we do not treat that part herein. However, we strongly believe that there exist the relations similar
to the ones for type A . As for the relations to the polyhedral realizations for other classical cases,
n
we shall discuss in forthcoming papers.
The organizationof the article is as follows: After the introduction in this sectionand the prelimi-
nariesinSect.2,we reviewthe explicitdescriptionsforthe fundamentalrepresentationofthe classical
LiealgebrasinSect.3. InSect.4,firstweintroducethetheoryofdecoratedgeometriccrystalsfollowing
[2]. Next, we define the decoration by using the elementary characters and certain special positive
decorated geometric crystal on B = TB−. Finally, the ultra-discretization of TB− is described
w w w
explicitly. In Sect.5, the theory of monomial realizations would be introduced and we shall see some
duality on monomial realizations. In Sect.6, we review the generalized minors and their relations
to our elementary characters and certain bilinear forms. The main conjecture will be presented at
the end of the section. In the last three sections, we describe the explicit form of decorations and
express them in terms of the monomial realization of crystal bases, which means that the conjecture
is positively resolvedfor the classical groups.
TheauthorwouldliketoacknowledgeMasakiKashiwarafordiscussionsandhishelpfulsuggestions.
2. Preliminaries and Notations
We list the notations used in this paper. Let A = (a ) be an indecomposable Cartan ma-
ij i,j∈I
trix with a finite index set I (though we can consider more general Kac-Moody setting.). Let
(t,{α } ,{h } ) be the associated root data satisfying α (h ) = a where α ∈ t∗ is a simple
i i∈I i i∈I j i ij i
root and h ∈ t is a simple coroot. Let g = g(A) = ht,e ,f (i ∈ I)i be the simple Lie algebra associ-
i i i
ated with A over C and ∆=∆ ⊔∆ be the root system associated with g, where ∆ is the set of
+ − ±
positive/negative roots. Let P ⊂t∗ be the weight lattice, hh,λi=λ(h) the pairing between t and t∗,
and (α,β) be an inner product on t∗ such that (α ,α ) ∈ 2Z and hh ,λi = 2(αi,λ) for λ ∈ t∗. Let
i i ≥0 i (αi,αi)
P∗ = {h∈ t : hh,Pi⊂ Z} and P :={λ ∈P : hh ,λi ∈Z }. We call an element in P a dominant
+ i ≥0 +
4 TOSHIKINAKASHIMA
integral weight. Let {Λ |i ∈ I} be the set of the fundamental weights satisfying hh ,Λ i = δ is a
i i j i,j
Z-basis of P.
The quantum algebra U (g) is an associative Q(q)-algebra generated by the e , f (i ∈ I), and
q i i
qh (h∈P∗) satisfying the usual relations, where we use the same notations for the generators e and
i
f as the ones for g. The algebra U−(g) is the subalgebra of U (g) generated by the f (i∈I).
i q q i
For the irreducible highest weight module of U (g) with the highest weight λ ∈ P , we denote it
q +
by V(λ) and we denote its crystal base by (L(λ),B(λ)). Similarly, for the crystal base of the algebra
Uq−(g)wedenote(L(∞),B(∞))(see[6,7]). Letπλ :Uq−(g)−→V(λ)∼=Uq−(g)/ iUq−(g)fi1+hhi,λi be
the canonical projection and π : L(∞)/qL(∞) −→ L(λ)/qL(λ) be the induced map from π . Here
λ λ
P
note that π (B(∞))=B(λ)⊔{0}.
λ
By the terminology crystal we mean some combinatorial object obtained by abstracting the
b
propertiesbof crystal bases. Indeed, crystal constitutes a set B and the maps wt : B −→ P,
ε ,ϕ : B −→ Z⊔{−∞} and e˜,f˜ : B ⊔{0} −→ B ⊔{0} (i ∈ I) satisfying several axioms (see
i i i i
[8],[19],[14]). In fact, B(∞) and B(λ) are the typical examples of crystals.
3. Fundamental Representations
3.1. Type A . Let V := V(Λ ) be the vector representation of sl (C) with the standard basis
n 1 1 n+1
{v ,··· ,v }, and {e ,f ,h } the Chevalley generators of sl (C). Their actions on the
1 n+1 i i i i=1,···,n n+1
basis vectors are as follows:
v if j =i,
i
v if j =i+1, v if j =i,
i i+1
(3.1) e v = f v = h v = −v if j =i+1,
i j i j i j  i+1
(0 otherwise, (0 otherwise, 0 otherwise,

3.2. Type C . Let I := {1,2,··· ,n} be the index set of the simple roots of type C . The Cartan
n n
matrix A=(a ) of type C is given by
i,j i,j∈I n
2 if i=j,
−1 if |i−j|=1 and (i,j)6=(n−1,n)
a =
i,j
−2 if (i,j)=(n−1,n),
0 otherwise.
Here α (i6=n) is a short root and α is the long root. Let {h } be the set of the simple co-roots
i n i i∈I
and {Λ } be the set of the fundamental weights satisfying α (h )=a and Λ (h )=δ .
i i∈I j i i,j i j i,j
First, let us describe the vector representation V(Λ ). Set B(n) := {v ,v |i = 1,2,··· ,n.} and
1 i i
define V(Λ ):= Cv. The weight of v is as follows:
1 v∈B(n) i
L
Λ −Λ if i=1,··· ,n,
i i−1
wt(v )=
i
(Λi−1−Λi if i=1,··· ,n,
where Λ =0. The actions of e and f are given by:
0 i i
(3.2) f v =v , f v =v , e v =v , e v =v (1≤i<n),
i i i+1 i i+1 i i i+1 i i i i+1
(3.3) f v =v , e v =v ,
n n n n n n
and the other actions are trivial.
Let Λ be the i-th fundamental weight of type C . As is well-known that the fundamental repre-
i n
sentation V(Λ ) (1 ≤i ≤n) is embedded in V(Λ )⊗i with multiplicity free. The explicit form of the
i 1
DECORATIONS ON GEOMETRIC CRYSTALS AND MONOMIAL REALIZATIONS OF CRYSTAL BASES 5
highest(resp. lowest) weight vector u (resp. v ) of V(Λ ) is realized in V(Λ )⊗i as follows:
Λi Λi i 1
u = sgn(σ)v ⊗···⊗v ,
Λi σ(1) σ(i)
(3.4) σX∈Si
v = sgn(σ)v ⊗···⊗v ,
Λi σ(i) σ(1)
σX∈Si
where S is the i-th symmetric group.
i
We review the crystal B(Λ ) following [10, 16]. Set the order on the set J ={i,i|1≤i≤n} by
k
1<2<···<n−1<n<n<n−1<···<2<1
Then, the crystal B(Λ ) is described:
k
1≤j <···<j ≤1,
(3.5) B(Λ )= [j ,··· ,j ] 1 k
k 1 k if j =j =i, then a+b≤i
a b
(cid:26) (cid:27)
Note that in [10] the vector [j ,··· ,j ] is represented as the column Young tableau. Note also that
1 k
the highest(resp. lowest) weight vector in B(Λ ) is [1,2,··· ,k] (resp. [k,k−1,··· ,2,1]).
k
3.3. Type B . Let I := {1,2,··· ,n} be the index set of the simple roots of type B . The Cartan
n n
matrix A=(a ) of type B is given by
i,j i,j∈I n
2 if i=j,
−1 if |i−j|=1 and (i,j)6=(n,n−1)
a =
i,j
−2 if (i,j)=(n,n−1),
0 otherwise.
Here α (i6=n) is a long root and α is the short root. Let {h } be the set of the simple co-roots
i n i i∈I
and {Λ } be the set of the fundamental weights satisfying α (h )=a and Λ (h )=δ .
i i∈I j i i,j i j i,j
First, let us describe the vector representationV(Λ ) for B . Set B(n) :={v , v |i=1,2,··· ,n}∪
1 n i i
{v } and V(Λ ):= Cv. The weight of v is as follows:
0 1 v∈B(n) i
Lwt(v )=Λ −Λ , wt(v )=Λ −Λ (i=1,··· ,n−1),
i i i−1 i i−1 i
wt(v )=2Λ −Λ , wt(v )=Λ −2Λ , wt(v )=0,
n n n−1 n n−1 n 0
where Λ =0. The actions of e and f are given by:
0 i i
(3.6) f v =v , f v =v , e v =v , e v =v (1≤i<n),
i i i+1 i i+1 i i i+1 i i i i+1
(3.7) f v =v , f v =2v , e v =2v , e v =v ,
n n 0 n 0 n n 0 n n n 0
and the other actions are trivial. For i = 1,2,··· ,n−1, the i-th fundamental representation V(Λ )
i
is realizedin V(Λ )⊗i as the case C and their highest (resp. lowest)weight vector u (resp. v ) is
1 n Λi Λi
given by the formula (3.4).
The last fundamental representation V(Λ ) is called the “spin representation”whose dimension is
n
2n. It is realized as follows: Set V(n) := Cǫ where
sp ǫ∈Bs(np)
L
B(n) :={(ǫ ,··· ,ǫ )|ǫ ∈{+,−}(i=1,2,··· ,n)}.
sp 1 n i
6 TOSHIKINAKASHIMA
Define the explicit actions of h , e and f on V(n) by
i i i sp
ǫi·1−ǫi+1·1(ǫ ,··· ,ǫ ), if i6=n,
(3.8) h (ǫ ,··· ,ǫ ) = 2 1 n
i 1 n
(ǫn(ǫ1,··· ,ǫn) if i=n,
i i+1
(··· ,−, +,···) if ǫ =+, ǫ =−, i6=n,
i i+1
(3.9) fi(ǫ1,··· ,ǫn) = (········· ,−n) if ǫn =+, i=n,
 0 otherwise
(··· ,+i,i−+1,···) if ǫi =−, ǫi+1 =+, i6=n,
(3.10) ei(ǫ1,··· ,ǫn) = (········· ,+n) if ǫn =−, i=n,
 0 otherwise.
Then the module V(n) is isomorphicto V(Λ ) as a B -module.
sp n n
Remark. We can associate the crystal structure on the set B(n) by setting f˜ = f and e˜ = e in
sp i i i i
(3.9) and (3.10) respectively, which is also denoted by B(n) and is isomorphic to B(Λ ).
sp n
3.4. Type D . Let I := {1,2,··· ,n} be the index set of the simple roots of type D . The Cartan
n n
matrix A=(a ) of type D is as follows:
i,j i,j∈I n
2 if i=j,
a = −1 if |i−j|=1 and (i,j)6=(n,n−1), (n−1,n), or (i,j)=(n−2,n), (n,n−2)
i,j 
0 otherwise.
Let {h } be the set of the simple co-roots and {Λ } be the set of the fundamental weights
i i∈I i i∈I
satisfying α (h )=a and Λ (h )=δ .
j i i,j i j i,j
First, let us describe the vector representation V(Λ ) for D . Set B(n) := {v ,v |i = 1,2,··· ,n}.
1 n i i
The weight of v is as follows:
i
wt(v )=Λ −Λ , wt(v )=Λ −Λ (i=1,··· ,n−1),
i i i−1 i i−1 i
wt(v )=Λ +Λ −Λ , wt(v )=Λ −Λ +Λ ,
n n−1 n n−2 n n−2 n−1 n
where Λ =0. The actions of e and f are given by:
0 i i
(3.11) f v =v , f v =v , e v =v , e v =v (1≤i<n),
i i i+1 i i+1 i i i+1 i i i i+1
(3.12) f v =v , f v =v , e v =v , e v =v ,
n n n−1 n−1 n n−1 n−1 n−1 n n n−1 n
and the other actions are trivial. For i = 1,2,··· ,n−2, the i-th fundamental representation V(Λ )
i
is realizedinV(Λ )⊗i as the casesB andC andtheir highest(resp.lowest)weightvectoru (resp.
1 n n Λi
v ) is given by the formula (3.4).
Λi
The last two fundamental representations V(Λ ) and V(Λ ) are also called the “spin repre-
n−1 n
sentations” whose dimensions are 2n−1. They are realized as follows: Set V(+,n)(resp. V(−,n))
sp sp
:= ǫ∈Bs(+p,n)(resp. Bs(−p,n))Cǫ where
L
B(+,n)(resp. B(−,n)):={(ǫ ,··· ,ǫ )|ǫ ∈{+,−},ǫ ···ǫ =+(resp. −)}.
sp sp 1 n i 1 n
DECORATIONS ON GEOMETRIC CRYSTALS AND MONOMIAL REALIZATIONS OF CRYSTAL BASES 7
Define the explicit actions of h , e and f on V(±,n) by
i i i sp
ǫi·1−ǫi+1·1(ǫ ,··· ,ǫ ), if i6=n,
(3.13) h (ǫ ,··· ,ǫ ) = 2 1 n
i 1 n (ǫn−1·12+ǫn·1(ǫ1,··· ,ǫn) if i=n,
i i+1
(··· ,−, +,···) if ǫ =+, ǫ =−, i6=n,
i i+1
(3.14) fi(ǫ1,··· ,ǫn) = (········· ,n−−1,−n) if ǫ =+,ǫ =+, i=n,
 0 othenr−w1ise n
(··· ,+i,i−+1,···) if ǫ =−, ǫ =+, i6=n,
i i+1
(3.15) ei(ǫ1,··· ,ǫn) = (········· ,n+−1,+n) if ǫ =−,ǫ =−, i=n,
 0 othenr−w1ise. n
Then the module Vs(p+,n) (resp. Vs(p−,n)) is isomorphic to V(Λn) (resp. V(Λn−1)) as a Dn-module.
Remark. SimilartothecaseB ,inthiscasewecanassociatethecrystalstructureonthesetB(+,n)
n sp
(resp. B(−,n)) by setting f˜ = f and e˜ = e in (3.14) and (3.15) respectively, which is also denoted
sp i i i i
by B(±,n) and is isomorphic to B(Λ ) (resp. B(Λ )).
sp n n−1
4. Decorated geometric crystals
The basic reference for this section is [1, 2, 15].
4.1. Definitions. LetA=(a ) beanindecomposableCartanmatrix. Letg=g(A)=ht,e ,f (i∈
ij i,j∈I i i
I)ibethesimpleLiealgebraassociatedwithAoverCasaboveand∆=∆ ⊔∆ betherootsystem
+ −
associated with g. Define the simple reflections s ∈ Aut(t) (i ∈ I) by s (h):=h−α (h)h , which
i i i i
generate the Weyl groupW. Let G be the simply connected simple algebraicgroupoverC whose Lie
algebra is g = n ⊕t⊕n , which is the usual triangular decomposition. Let U :=expg (α ∈ ∆)
+ − α α
be the one-parameter subgroup of G. The group U± are generated by {U |α ∈ ∆ }. Here U± is a
α ±
unipotent radical of G and Lie(U±)=n . For any i∈I, there exists a unique group homomorphism
±
φ : SL (C)→G such that
i 2
1 t 1 0
φ =exp(te ), φ =exp(tf ) (t∈C).
i 0 1 i i t 1 i
(cid:18)(cid:18) (cid:19)(cid:19) (cid:18)(cid:18) (cid:19)(cid:19)
Set α∨(c):=φ c 0 , x (t):=exp(te ), y (t):=exp(tf ), G :=φ (SL (C)), T :=α∨(C×) and
i i 0c−1 i i i i i i 2 i i
N :=N (T ). Let T be a maximal torus of G which has P as its weight lattice and Lie(T) = t.
Leit B±(G⊃i Ti,U±(cid:0)(cid:0)) be t(cid:1)h(cid:1)e Borel subgroup of G. We have the isomorphism φ : W−∼→N/T defined
by φ(s ) = N T/T. An element s := x (−1)y (1)x (−1) is in N (T), which is a representative of
i i i i i i G
s ∈W =N (T)/T.
i G
Definition 4.1. Let X be anaffine algebraicvarietyoverC, γ , ε ,f (i∈I)rationalfunctions onX,
i i
ande :C××X →XaunitalrationalC×-action(i∈I). A5-tupleχ=(X,{e } ,{γ ,} ,{ε } ,f)
i i i∈I i i∈I i i∈I
is a G (or g)-decorated geometric crystal if
(i) ({1}×X)∩dom(e ) is open dense in {1}×X for any i∈I, where dom(e ) is the domain of
i i
definition of e : C××X →X.
i
(ii) The rational functions {γi}i∈I satisfy γj(eci(x))=caijγj(x) for any i,j ∈I.
(iii) The function f satisfies
(4.1) f(ec(x))=f(x)+(c−1)ϕ (x)+(c−1−1)ε (x),
i i i
for any i∈I and x∈X, where ϕ :=ε ·γ .
i i i
8 TOSHIKINAKASHIMA
(iv) e and e satisfy the following relations:
i j
ec1ec2 =ec2ec1 if a =a =0,
i j j i ij ji
ec1ec1c2ec2 =ec2ec1c2ec1 if a =a =−1,
i j i j i j ij ji
ec1ec21c2ec1c2ec2 =ec2ec1c2ec21c2ec1 if a =−2, a =−1,
i j i j j i j i ij ji
ec1ec31c2ec21c2ec31c22ec1c2ec2 =ec2ec1c2ec31c22ec21c2ec31c2ec1 if a =−3, a =−1.
i j i j i j j i j i j i ij ji
(v) The rational functions {ε } satisfy ε (ec(x)) = c−1ε (x) and ε (ec(x)) = ε (x) if a =
i i∈I i i i i j i i,j
a =0.
j,i
Wecallthefunctionf in(iii)thedecorationofχandtherelationsin(iv)arecalledVermarelations.
If χ=(X,{e }, {γ },{ε }) satisfies the conditions (i), (ii), (iv) and (v), we call χ a geometric crystal.
i i i
Remark. The definitions of ε and ϕ are different from the ones in e.g., [2] since we adopt the
i i
definitions following [11, 12]. Indeed, if we flip ε →ε−1 and ϕ →ϕ−1, they coincide with ours.
i i
4.2. Characters. Let U := Hom(U,C) be the set of additive characters of U. The elementary
character χ ∈U and the standard regular character χst ∈U are defined as follows:
i
b
χ (x (c))=δ ·c (c∈C, i∈I), χst = χ .
b i j i,j b i
i∈I
X
We also define the anti-automorphism η :G→G by
η(x (c))=x (c), η(y (c))=y (c), η(t)=t−1 (c∈C, t∈T),
i i i i
which is called the positive inverse([2]).
The rational function f on G is defined by
B
(4.2) f (g)=χst(π+(w−1g))+χst(π+(w−1η(g))),
B 0 0
for g ∈Bw B, where π+ :B−U →U is the projection defined by π+(bu)=u.
0
For a split algebraic torus T over C, let us denote its lattice of (multiplicative )characters(resp.
co-characters)by X∗(T) (resp. X (T)). By the usual way,we identify X∗(T)(resp. X (T)) with the
∗ ∗
weight lattice P (resp. the dual weight lattice P∗).
4.3. Positive structure and ultra-discretization. Letusreviewthenotionpositivestructureand
the ultra-discretization.
Definition 4.2. Let T,T′ be split algebraic tori over C.
(i) A regular function f = c ·µ on T is positive if all coefficients c are non-negative
µ∈X∗(T) µ µ
numbers. ArationalfunctiononT issaidtobepositiveifthereexistpositiveregularfunctions
g,h such that f = g (hP6=0).
h
(ii) Letf :T →T′ be a rationalmapbetween T andT′. Then we saythat f is positive if for any
ξ ∈X∗(T′) we have that ξ◦f is positive in the above sense.
Note that if f,g are positive rational functions on T, then f ·g, f/g and f +g are all positive.
Definition 4.3. Let χ = (X,{e } ,{wt } ,{ε } ,f) be a decorated geometric crystal, T′ an
i i∈I i i∈I i i∈I
algebraic torus and θ :T′ →X a birationalmap. The birational map θ is called positive structure on
χ if it satisfies:
(i) For any i ∈ I the rational functions γ ◦θ,ε ◦θ,f ◦θ : T′ → C are all positive in the above
i i
sense.
(ii) For any i ∈ I, the rational map e : C× ×T′ → T′ defined by e (c,t) := θ−1◦ec◦θ(t) is
i,θ i,θ i
positive.
DECORATIONS ON GEOMETRIC CRYSTALS AND MONOMIAL REALIZATIONS OF CRYSTAL BASES 9
Let v :C(c)\→Z be a mapdefined by v(f(c)):=deg(f(c−1)), which is different fromthat in e.g.,
[11, 12, 15, 17].
Let f: T → T′ be a positive rational mapping of algebraic tori T and T′. We define the map
f: X (T)→X (T′) by
∗ ∗
hχ,f(ξ)i=v(χ◦f ◦ξ),
wbhere χ∈X∗(T′) and ξ ∈X (T).
∗
Let T be the category whose objects arbe algebraic tori over C and whose morphisms are positive
+
rational maps. Then, we obtain the “ultra-discretization” functor
UD : T −→ Set
+
T 7→ X (T)
∗
(f :T →T′) 7→ (f :X (T)→X (T′)).
∗ ∗
Note that this definition of the functor UD is called tropicalization in [1] and much simpler than the
b
one in [2].
Letθ :T′ →X beapositivestructureonadecoratedgeometriccrystalχ=(X,{e } ,{wt } ,{ε } ,f).
i i∈I i i∈I i i∈I
Applying the functor UD to positive rationalmorphisms e :C××T′ →T′ and f ◦Θ,γ ◦θ,ε ◦Θ:
i,θ i i
T′ →C, we obtain
e˜ := UD(e ):Z×X (T′)→X (T′)
i i,θ ∗ ∗
wt := UD(γ ◦θ):X (T′)→Z,
i i ∗
ε := UD(ε ◦θ):X (T′)→Z,
i i ∗
f := UD(f ◦θ):X (T′)→Z.
∗
e
Now,forgivenpositivestructureθ :T′ →X onageometriccrystalχ=(X,{e } ,{wt } ,{ε } ),
i i∈I i i∈I i i∈I
we associate the quadruple (Xe(T′),{e˜} ,{wt } ,{ε } ) with a free pre-crystal structure (see
∗ i i∈I i i∈I i i∈I
[1, 2.2]) and denote it by UDθ,T′(χ). We have the following theorem:
Theorem 4.4 ([1, 2, 15]). For any geometric crystal χe= (X,{e } ,{γ } ,{ε } ) and positive
i i∈I i i∈I i i∈I
structure θ : T′ → X, the associated pre-crystal UDθ,T′(χ) = (X∗(T′),{ei}i∈I,{wti}i∈I,{εi}i∈I) is a
Langlands dual Kashiwara’s crystal.
Remark. The definition of ε is different from the one in [2, 6.1.] since our definition eof ε corre-
i i
sponds to ε−1 in [2].
i
For a positive decorated geoemetric crystal X =((X,{ei}i∈I,{γi}i∈I,{εi}i∈I,f),θ,T′), set
(4.3) Bfe:={x∈X∗(T′)(=Zdim(T′))|f(x)≥0},
and define Bf,θ :=(Bfe,wti|Befee,εi|Befe,eei|Befe)i∈I. ee
Proposition4.5([2]). ForapositivedecoratedgeometriccrystalX =((X,{e } ,{γ } ,{ε } ,f),θ,T′),
e i i∈I i i∈I i i∈I
the quadruple B is a normal crystal.
f,θ
4.4. Decorated geometric crystal on B . For a Weyl group element w ∈ W, define B− :=
w w
B−∩UwU and set B :=TB−. Let γ :B →C be the rational function defined by
w w i w
(4.4) γ :B ֒→ B− −∼→ T ×U− −pr→oj T −α→∨i C.
i w
For any i∈I, there exists the natural projectionpr :B− →B−∩φ(SL ). Hence, for any x∈B
i 2 w
b 0
there exists unique v = 11 ∈ SL such that pr (x) = φ (v). Using this fact, we define the
b b 2 i i
21 22
(cid:18) (cid:19)
rational function ε on B as in [18]:
i w
b
(4.5) ε (x)= 22 (x∈B ).
i w
b
21
10 TOSHIKINAKASHIMA
The rational C×-action e on B is defined by
i w
(4.6) ec(x):=x ((c−1)ϕ (x))·x·x (c−1−1)ε (x) (c∈C×, x∈B ),
i i i i i w
if εi(x) is well-defined, that is, b21 6=0, and defi(cid:0)ne eci(x)=x if(cid:1)b21 =0.
Remark. Thedefinition(4.5)isdifferentfromtheonein[2]. Indeed,ifin(4.5)wetakeε (x)=b /b ,
i 21 22
then it coincides with the one in [2].
Proposition 4.6 ([2]). For any w ∈ W, the 5-tuple χ := (B ,{e } ,{γ } ,{ε } ,f ) is a decorated
w i i i i i i B
geometric crystal, where f is in (4.2), γ is in (4.4), ε is in (4.5) and e is in (4.6).
B i i i
For the longest Weyl group element w0 ∈ W, let i0 = i1...iN be one of its reduced expressions
and define the positive structure on B− Θ− :(C×)N −→B− by
w0 i0 w0
Θ−(c ,··· ,c ):=yyy (c )···yyy (c ),
i0 1 N i1 1 iN N
where yyy (c)= y (c)α∨(c−1), which is different from Y (c) in [15, 14, 11, 12]. Indeed, Y (c) =yyy (c−1).
i i i i i
We alsodefine the positive structure onB as TΘ− :T ×(C×)N −→ B by TΘ−(t,c ,··· ,c )=
w0 i0 w0 i0 1 N
tΘ−(c ,··· ,c ).
i0 1 N
Now, for this positive structure, we describe the geometric crystal structure on B = TB−
w0 w0
explicitly.
Proposition 4.7 ([18]). The action ec on tΘ−(c ,··· ,c ) is given by
i i0 1 N
ec(tΘ−(c ,··· ,c ))=tΘ−(c′,··· ,c′ )
i i0 1 N i0 1 N
where
c·cai1,i···caim−1,ic + cai1,i···caim−1,ic
1 m−1 m 1 m−1 m
(4.7) c′j :=cj · 1≤m<Xj,im=ic·cai1,i···caim−1,ic +j≤m≤XN,im=icai1,i···caim−1,ic .
1 m−1 m 1 m−1 m
1≤m≤Xj,im=i j<m≤XN,im=i
The explicit forms of rational functions ε and γ are:
i i
−1
1 α (t)
(4.8) εi(tΘ−i0(c))= c caim+1,i···caiN,i , γi(tΘ−i0(c))= cai1,i·i··caiN,i.
1≤m≤XN,im=i m m+1 N 1 N
 
5. Monomial Realization of Crystals
5.1. Definitions of Monomial Realization of Crystals. Following [9, 13], we shallintroduce the
monomial realization of crystals. For doubly-indexed variables {Y |i∈I,m∈Z.}, define the set of
m,i
monomials
Y :={Y = Ylm,i|l ∈Z\{0} except for finitely many (m,i)}.
m,i m,i
m∈Z,i∈I
Y
Fix a set of integers p = (p ) such that p +p = 1, which we call a sign. Take a sign
i,j i,j∈I,i6=j i,j j,i
p:=(p ) and a Cartan matrix (a ) . For m∈Z, i∈I define the monomial
i,j i,j∈I,i6=j i,j i,j∈I
a
A =Y Y Y j,i .
m,i m,i m+1,i m+pj,i,j
j6=i
Y
(p)
Here, when we emphasize the sign p, we shall denote the monomial A by A . For any cyclic
m,i m,i
sequence of the indices ι = ···(i i ···i )(i i ···i )··· s.t. {i ,··· ,i } = I, we can associate the
1 2 n 1 2 n 1 n