Zero Action on Perfect Crystals for U_q(G_2^{(1)})

The actions of 0-Kashiwara operators on the $U'_q(G_2^{(1)})$-crystal $B_l$ in [Yamane S., J. Algebra 210 (1998), 440-486] are made explicit by using a similarity technique from that of a $U'_q(D_4^{(3)})$-crystal. It is shown that $\{B_l\}_{l\ge1}$ forms a coherent family of perfect crystals.


Introduction
Let g be a symmetrizable Kac-Moody algebra. Let I be its index set for simple roots, P the weight lattice, α i ∈ P a simple root (i ∈ I), and h i ∈ P * (= Hom(P, Z)) a simple coroot (i ∈ I).
To each i ∈ I we associate a positive integer m i and setα i = m i α i ,h i = h i /m i . Suppose ( h i ,α j ) i,j∈I is a generalized Cartan matrix for another symmetrizable Kac-Moody algebrag. Then the subsetP of P consisting of λ ∈ P such that h i , λ is an integer for any i ∈ I can be considered as the weight lattice ofg. For a dominant integral weight λ let B g (λ) be the highest weight crystal with highest weight λ over U q (g). Then, in [5] Kashiwara showed the following. (The theorem in [5] is more general.) Theorem 1. Let λ be a dominant integral weight inP . Then, there exists a unique injective map S : Bg(λ) → B g (λ) such that . In this paper, we use this theorem to examine the so-called Kirillov-Reshetikhin crystal. Let g be the affine algebra of type D 4 . The generalized Cartan matrix ( h i , α j ) i,j∈I (I = {0, 1, 2}) is given by Set (m 0 , m 1 , m 2 ) = (3, 3, 1). Then,g defined above turns out to be the affine algebra of type G 2 . Their Dynkin diagrams are depicted as follows D 4 : 2 a family of perfect crystals {B l } l≥1 was constructed in [7]. However, the crystal elements there were realized in terms of tableaux given in [2], and it was not easy to calculate the action of 0-Kashiwara operators on these tableaux. On the other hand, an explicit action of these operators was given on perfect crystals in [6]. Hence, it is a natural idea to use Theorem 1 to obtain the explicit action of e 0 , f 0 on B l from that onB l ′ with suitable l ′ . We remark that Kirillov-Reshetikhin crystals are parametrized by a node of the Dynkin diagram except 0 and a positive integer. Both B l andB l correspond to the pair (1, l).
Our strategy to do this is as follows. We define V l as an appropriate subset ofB 3l that is closed under the action ofê m i i ,f m i i whereê i ,f i stand for the Kashiwara operators onB 3l . Hence, we can regard V l as a U ′ q G (1) 2 -crystal. We next show that as a U q G )-crystal, V l has the same decomposition as B l . Then, we can conclude from Theorem 6.1 of The paper is organized as follows. In Section 2 we review the U ′ q D (3) 4 -crystalB l . We then with the aid of Theorem 1 and see it coincides with B l given in [7] in Section 3. Minimal elements of B l are found and {B l } l≥1 is shown to form a coherent family of perfect crystals in Section 4. The crystal graphs of B 1 and B 2 are included in Section 5.

Review on
In this section we recall the perfect crystal for U ′ q D (3) 4 constructed in [6]. Since we also -crystals later, we denote it byB l . Kashiwara operators e i , f i and ε i , ϕ i onB l are denoted byê i ,f i andε i ,φ i . Readers are warned that the coordinates x i ,x i and steps by Kashiwara operators in [6] are divided by 3 here, since it is more convenient for our purpose. As a set In order to define the actions of Kashiwara operatorsê i andf i for i = 0, 1, 2, we introduce some notations and conditions. Set (x) + = max(x, 0). For b = (x 1 , x 2 , x 3 ,x 3 ,x 2 ,x 1 ) ∈B l we set and Now we define conditions (E 1 )-(E 6 ) and (F 1 )-(F 6 ) as follows The conditions (F 1 )-(F 6 ) are disjoint and they exhaust all cases. . (2.5) does not belong toB l , namely, if x j orx j for some j becomes negative or s(b) exceeds l/3, we should understand it to be 0. Forgetting the 0-arrows, -crystal of highest weight λ and G † 2 stands for the simple Lie algebra G 2 with the reverse labeling of the indices of the simple roots (α 1 is the short root). Forgetting 2-arrows, In this section we define a subset V l ofB 3l and see it is isomorphic to the U ′ q G (1) 2 -crystal B l . The set V l is defined as a subset ofB 3l satisfying the following conditions: (3.1) Proof . We first count the number of elements (x 2 , x 3 ,x 3 ,x 2 ) satisfying the conditions of coordinates as an element of V l and x 2 + (x 3 +x 3 )/2 +x 2 = m (m = 0, 1, . . . , k). According to (a, b, c, d) (a, d ∈ {0, 1/3, 2/3}, b, c ∈ {0, 1/3, 2/3, 1, 4/3, 5/3}) such that x 2 ∈ Z + a, x 3 ∈ 2Z + b, x 3 ∈ 2Z + c,x 2 ∈ Z + d, we divide the cases into the following 18: . Since there is one case with e = 0 (i) and e = 3 (xviii) and 8 cases with e = 1 and e = 2, the number of (x 2 , x 3 ,x 3 ,x 2 ) such that x 2 +(x 3 +x 3 )/2+x 2 = m is given by For each (x 2 , x 3 ,x 3 ,x 2 ) such that x 2 +(x 3 +x 3 )/2+x 2 = m (m = 0, 1, . . . , k) there are (k−m+1) cases for (x 1 ,x 1 ), so the number of b ∈ V l such that s(b) = k is given by k m=0 1 2 (2m + 1)(3m 2 + 3m + 2)(k − m + 1).
A direct calculation leads to the desired result.
Zero Action on Perfect Crystals for U q G (1) 2

5
We define the action of operators e i , f i (i = 0, 1, 2) on V l as follows.
For (2) we only treat f i . To prove the i = 0 case consider the following table This table signifies the difference (z j forf 0 b) − (z j for b) when b belongs to the case (F i ). Note that the left hand sides of the inequalities of each (F i ) (2.3) always decrease by 1/3. Since z 1 , z 2 , z 3 ∈ Z, z 4 ∈ Z/3 for b ∈ V l , we see that if b belongs to (F i ),f 0 b andf 2 0 b also belong to (F i ) except two cases: (a) b ∈ (F 4 ) and z 4 = 1/3, and (b) b ∈ (F 4 ) and z 4 = 2/3. If (a) occurs, we havef 0 b,f 2 0 b ∈ (F 3 ). Hence, we obtain f 0 =f 3 0 in this case. If (b) occurs, we havef 0 b ∈ (F 4 ), . Therefore, we obtain f 0 =f 3 0 in this case as well. In the i = 1 case, if b belongs to one of the 3 cases,f 1 b andf 2 1 b also belong to the same case. Hence, we obtain f 1 =f 3 1 . For i = 2 there is nothing to do.
Proposition 1, together with Theorem 1, shows that V l can be regarded as a U ′ q G (1) 2 -crystal with operators e i , f i (i = 0, 1, 2).
However, this is equal to ♯{b ∈ V l | s(b) = k} by Lemma 1. Therefore, ⊂ in (3.2) should be =, and the proof is completed.
Proof . For integers i, j 0 , j 1 such that Here we have set y a = (l − i − j a )/3 for a = 0, 1. From (2.5) one notices that b i,j 0 , s(b i,j 0 ,j 1 ) = l and max A = 2z 1 + z 2 + z 3 + 3z 4 = j 0 . By setting g =g = A 2 , (m 0 , m 1 ) = (3,3) in Theorem 1, the connected component generated from b i,j 0 ,j 1 by f i =f 3 i (i = 0, 1) is isomorphic to B A 2 (j 0 Λ 0 + j 1 Λ 1 ). Hence, by Proposition 1 (1) we have However, from Proposition 2 one knows that Therefore, the proof is completed. Theorem 6.1 in [6] shows that if two U ′ q G (1) 2 -crystals decompose into 0≤k≤l B G 2 (kΛ 1 ) as U q (G 2 )-crystals, then they are isomorphic to each other. Therefore, we now have [7]. The values of ε i , ϕ i with our representation are given by

Minimal elements and a coherent family
The notion of perfect crystals was introduced in [3] to construct the path realization of a highest weight crystal of a quantum affine algebra. The crystal B l was shown to be perfect of level l in [7]. In this section we obtain all the minimal elements of B l in the coordinate representation and also show {B l } l≥1 forms a coherent family of perfect crystals. For the notations such as P cl , (P + cl ) l , see [3].

Minimal elements
where z j (1 ≤ j ≤ 4) are given in (2.2) and A is given in (2.4). The following lemma was proven in [6], although Z is replaced with Z/3 here.

Coherent family of perfect crystals
The notion of a coherent family of perfect crystals was introduced in [1]. Let {B l } l≥1 be a family of perfect crystals B l of level l and (B l ) min be the subset of minimal elements of B l . Set Let σ denote the isomorphism of (P + cl ) l defined by σ = ε•ϕ −1 . For λ ∈ P cl , T λ denotes a crystal with a unique element t λ defined in [4]. For our purpose the following facts are sufficient. For any P cl -weighted crystal B and λ, µ ∈ P cl consider the crystal The definition of T λ and the tensor product rule of crystals implỹ Definition 1. A crystal B ∞ with an element b ∞ is called a limit of {B l } l≥1 if it satisfies the following conditions: • wt b ∞ = 0, ε(b ∞ ) = ϕ(b ∞ ) = 0, • for any (l, b) ∈ J, there exists an embedding of crystals • B ∞ = (l,b)∈J Im f (l,b) .
If a limit exists for the family {B l }, we say that {B l } is a coherent family of perfect crystals.
For ε i , ϕ i with i = 1, 2 we adopt the formulas in Section 3. For ε 0 , ϕ 0 we define