Crystal bases as tuples of integer sequences

We describe a set $\mathcal{R}^{\infty}$ consisting of tuples of integer sequences and provide certain explicit maps on it. We show that this defines a semiregular crystal for $\mathfrak{sl}_{n+1}$ and $\mathfrak{sp}_{2n}$ respectively. Furthermore we define for any dominant integral weight $\lambda$ a connected subcrystal $\mathcal{R}(\lambda)$ in $\mathcal{R}^{\infty}$, such that this crystal is isomorphic to the crystal graph $B(\lambda)$. Finally we provide an explicit description of these connected crystals $\mathcal{R}(\lambda)$.


Introduction
Let g be a symmetrizable Kac-Moody algebra and let U (g) be the corresponding quantum algebra. For these quantum algebras, Kashiwara developed the crystal bases theory for integrable modules in [1] and thus provided a remarkable combinatorial tool for studying these modules. In particular crystal bases can be viewed as bases at = 0 and they contain structures of edge-colored oriented graphs satisfying a set of axioms, called the crystal graphs. These crystal graphs have certain nice properties; for instance, characters of U (g)-modules can be computed and the decomposition of tensor products of modules into irreducible ones can also be determined from the crystal graphs, to name but a few. It is thus an important problem to have explicit realizations of crystal graphs.
There are many such realizations, combinatorial and geometrical, worked out during the last decades; for instance, we refer to [2][3][4][5]. In [2], the authors give a tableaux realization of crystal graphs for irreducible modules over the quantum algebra for all classical Lie algebras, which is a purely combinatorial model. Another significant combinatorial model for any symmetrizable Kac-Moody algebra is provided in [3], called Littelmann's path model. The underlying set here is a set of piecewise linear maps, and the crystal graph of an irreducible module of any dominant integral highest weight can be generated by an algorithm using the straight path connecting 0 and .
A geometrical realization of crystals is also known and is provided by Nakajima [5] by showing that there exists a crystal structure on the set of irreducible components of a lagrangian subvariety of the quiver variety M. This realization can be translated into a purely combinatorial model, the set of Nakajima monomials, where the action of the Kashiwara operators can be understood as a multiplication with monomials. Moreover, it is shown in [6] that the connected component of any highest weight monomial of highest weight is isomorphic to the crystal graph ( ) obtained from Kashiwara's crystal bases theory. For special highest weight monomials these connected components are explicitly characterized for sl +1 in [7] and for the other classical Lie algebras in [8]. A combinatorial isomorphism from connected components corresponding to arbitrary highest weight monomials of highest weight and those in [7,8] is provided in [9] for the types and and in [10] for the types and .
In this paper we introduce a set R ∞ consisting of tuples of integer sequences; that is, a typical element in R ∞ is given by where each component consists of certain ordered pairs of integers, = ( 1 , 1 ) ⋅ ⋅ ⋅ ( , ) (see Definition 3). Furthermore, the number of nonzero components is finite. We provide certain maps on R ∞ , the Kashiwara operators̃,̃, and maps , for all = 1, . . . , and prove that R ∞ is a semiregular crystal if g is sl +1 or sp 2 (see Definition 3 and Proposition 9).

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Moreover, we introduce for any dominant integral weight a subcrystal R( ) as the connected component of R ∞ containing a highest weight element and prove the following theorem.

Theorem 1. Let be a dominant integral weight, then there exists a crystal isomorphism
mapping to the highest weight element ∈ ( ).
Therefore, similar to the setting of Nakajima monomials, a natural question arises; namely, can one characterize for each dominant integral weight explicitly the sequences appearing in R( )? We answer this question by describing explicitly these connected components (for the special linear Lie algebra in Theorem 18 and the symplectic Lie algebra in Theorem 19).
Our paper is organized as follows: in Section 2 we fix some notations and review briefly the crystal theory. In Section 3 we present the main definitions, especially the definition of R ∞ and we equip our main object with a crystal structure. In Section 4 Nakajima monomials are recalled. In Section 5 we introduce for any dominant integral weight the subcrystals R( ) and describe them explicitly. Finally, in Section 6 we prove that they are isomorphic to ( ).

Notations and a Review of Crystal Theory
Let g be a complex simple Lie algebra of rank with index set = {1, . . . , } and fix a Cartan subalgebra h in g and a Borel subalgebra b ⊇ h. We denote by Φ ⊆ h * the root system of the Lie algebra, and corresponding to the choice of b let Φ + be the subset of positive roots. Further, we denote by Π = { 1 , . . . , } the corresponding basis of Φ and the basis of the dual root system Φ ∨ ⊆ h is denoted by Π ∨ = { ∨ 1 , . . . , ∨ }. Let g = n + ⊕ h ⊕ n − be a Cartan decomposition and for a given root ∈ Φ let g be the corresponding root space. For a dominant integral weight we denote by ( ) the irreducible g-module with a highest weight . Fix a highest weight vector V ∈ ( ), then ( ) = U(n − )V , where U(n − ) denotes the universal enveloping algebra of n − . For an indetermined element we denote by U (g) the corresponding quantum algebra. The theory of studying modules of quantum algebras is quite parallel to that of Kac-Moody algebras and the irreducible modules are classified again in terms of highest weights (see [11]). Using the crystal bases theory, introduced by Kashiwara in [1], we can compute the character of an integrable module in the category O as follows: whereby ( , ) is the crystal bases of (see [11]). The crystal graph associated with the irreducible module of highest weight is denoted by ( ). So finding expressions for the characters can be achieved by finding explicit combinatorial description of crystal bases. For some examples we refer to [2][3][4].
From now on we assume that g is a classical Lie algebra of type or . Note that the positive roots are all of the following form:

Type
: , = + +1 + ⋅ ⋅ ⋅ + , Furthermore, let = ⨁ ∈ Z be the set of classical integral weights and + = ⨁ ∈ Z + be the set of classical dominant integral weights. Before we discuss the crystal bases theory in detail we review first the notion of abstract crystals.

Abstract Crystals.
Crystal bases of integrable U (g)modules in the category O are characterized by certain maps satisfying some properties. One can define the abstract notion of crystals associated with a Cartan datum as follows.
Furthermore, a crystal is said to be semiregular if the equalities The maps̃and̃are called Kashiwara's crystal operators and the map is called the weight function. So, on the one hand, one can associate with any integrable U (g)-module a set satisfying the properties from Definition 3, and, on the other hand, one can study the notion of abstract crystals. A natural question which arises at this point is therefore the following: can one determine whether an abstract crystal is the crystal of a module? Stembridge [12] gave a set of local ISRN Combinatorics 3 axioms to characterize the set of crystals of module in the class of all crystals when g is simply laced and a list of local axioms for 2 -crystals is provided in [13]. In the following sections we define our underlying set and realize the crystal obtained from Kashiwara's crystal bases theory for the types and . We start by equipping our underlying set with an abstract crystal structure and later we prove that this crystal is the crystal of a module.

Tuples of Integer Sequences as Crystals
In this section we introduce a set R ∞ consisting of tuples of integer sequences (see Definition 16) and a crystal structure on it in the sense of Definition 3. Our purpose is to identify for any dominant integral weight certain subcrystals R( ); that is, ⋃ ∈ + R( ) ⊆ R ∞ , such that R( ) has a strong connection to the crystal graph ( ) (see Corollary 21).

Set of Tuples of Integer Sequences.
In order to define R ∞ we consider a total order on I = {1, . . . , } if g is of type and a total order on I = {1, . . . , , − 1, . . . , 1} if g is of type , namely, respectively. Furthermore, especially in Section 5, we need for type the following bijective map: For 1 ≤ ≤ , ∈ Z ≥0 we set R to be the set of all sequences ( 1 , 1 ) ⋅ ⋅ ⋅ ( , ) with , ∈ I, such that where max I denotes the maximal element in I with respect to <. We denote by 0 the unique element in R 0 . Before we mention the crystal structure on R ∞ we will initially introduce a list of properties. We need these to define the Kashiwara operators. Let = ( 1 , 1 ) ⋅ ⋅ ⋅ ( , ) ∈ R be an arbitrary element and fix ∈ : (a ) replace in (a) ∨ by ∧, . . , } , (d ) replace everywhere in (d) ∨ by ∧.
Let us consider an example.

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Remark 5. If satisfies (a ) and (d ), respectively, then it satisfies also (a) and (d), respectively. If g is further of type , these properties can be simplified. In particular, the properties (a ), (b), (d), and (d ) are superfluous.
Henceforth we define a crystal structure on R ∞ , such that the semiregularity holds. For this let x = ( 1 , 2 , 3 , . . .) be such a sequence with finitely many components different from zero; recall that each component is a sequence as in (8). The weight function is given by where and = ♯{ ̸ = 0 | ∈ ⋃ R }. Suppose that the nonzero components in x are given by 1 ∈ R 1 1 , . . . , ∈ R . For fixed ∈ we define the following maps.
(i) For 2 ≤ ≤ + 1, let : R ∞ → Z ≥0 be the map given by where Furthermore, we define ( ) to be the sequence which arises out of by replacing + 1 by , if < , if satisfies (a). If satisfies (b), let ( ) be the sequence which arises out of by replacing + 1 by . If neither (a) nor (b) is fulfilled, we set ( ) = 0.
(ii) For 2 ≤ ≤ , let : R ∞ → Z ≥0 be the map given by where We define ( ) to be the sequence which arises out of by replacing by + 1, if < , if satisfies (c). If satisfies (d), let ( ) be the sequence which arises out of by replacing by + 1. If neither (c) nor (d) is fulfilled, we set ( ) = 0.
(2) Note that the image of ∈ ⋃ R under the maps and , respectively, is contained in ⋃ R ∪ {0}, that is, One important fact about these maps is described in the next lemma.

Lemma 7.
Let ,̃be nonzero sequences as in (8), then one has ( ) =̃iff (̃) = . (24) Proof. One can easily show that satisfies (a) if and only ifs atisfies (c). Hence, we can suppose that does not fulfill all properties enumerated in (a). By observing the action we see that̃arises from by replacing + 1 by , which means that (c) is violated. In particular + 1 does not appear iñand appears iñ, which means that the properties in (d) hold. Hence, (̃) = . The arguments for the reverse direction are the same. Let ISRN Combinatorics 5 Now we are able to define the Kashiwara operators, Let us consider an example. (28) It remains to define the maps and . These maps are given by the next formula: Proposition 9. The set R ∞ becomes a semiregular crystal.
Proof. In Lemma 10 we proof the semiregularity of R ∞ , which ensures that (4) and (5) from Definition 3 hold. So, to verify the proposition, it is sufficient to prove (1), (2), (3), and (6), where (2) and (3) are easily checked with the help of Remark 6. Let us start by proving (1); so let x ∈ R ∞ be arbitrary with finitely many nonzero components, say, Then we order these components in a way such that the first components are contained in ⋃ ⋃ < R followed by components in ⋃ ⋃ > R and the last ones are contained in ⋃ R . So we can write the set of nonzero components of x as a disjoint union of three subsets Further let x 1 be the element in R ∞ obtained from x by replacing all components not belonging to < by 0.
And x 2 and x 3 , respectively, are similarly defined using > and = , respectively. Then we get where the sequence is the first nonzero element in x . A short calculation shows Now we proceed to prove (6). By the definition of the Kashiwara operators and Lemma 7, it is enough to show that (̃x) = (x) and (̃x) = (x). Since the proofs are similar, we prove only the latter equation. Assume that̃x ̸ = 0 and (x) ̸ = 1, then Subsequently we have In the case where (x) = 1, we obtain

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Hence we have shown that the set R ∞ is an abstract crystal provided that the semiregularity is shown. Thus, our aim now is to verify that the maps and , respectively, determine how often one can act with̃and̃, respectively. The semiregularity is a necessary condition of a crystal , if one wants to identify it with the crystal graph ( ). (32). For a given element
Before we introduce the subcrystals R( ) we need some facts about the theory of tensor products of crystals. The tensor product rule is a very nice combinatorial feature and important to realize the crystal bases of a tensor product of two U (g)-modules.

Tensor Products and Nakajima Monomials
In this section, we want to recall tensor products of crystals and investigate the action of Kashiwara operators on tensor products. With the aim of having a different realization of ( ) from our approach, we want to introduce the set of all Nakajima monomials, such that we can think of ( ) in terms of certain monomials. This theory is discovered by Nakajima [14] and generalized by Kashiwara [6].

Tensor Product of Crystals.
Suppose that we have two abstract crystals 1 , 2 in the sense of Definition 3, then we can construct a new crystal which is as a set 1 × 2 . This crystal is denoted by 1 ⊗ 2 and the Kashiwara operators are given as follows: Furthermore, one can describe explicitly the maps , , and on 1 ⊗ 2 , namely, One of the most important interpretations of the tensor product rule is the following theorem (for more details see [11]).

Nakajima Monomials.
For ∈ and ∈ Z, we consider monomials in the variables ( ); that is, we obtain the set of Nakajima monomials M as follows: vanish except for finitely many ( , )}.
With the goal of defining a crystal structure on M, we take some integers = ( , ) ̸ = such that , + , = 1. Let now ISRN Combinatorics 7 = ∏ ∈ , ∈Z ( ) ( ) be an arbitrary monomial in M and ∈ ; then we set.
The Kashiwara operators are defined as follows: whereby The following two results are shown by Kashiwara [6].
Remark 13. A priori the crystal structure depends on ; hence, we will denote this crystal by M . But it is easy to see that the isomorphism class of M does not depend on this choice. In the literature is often chosen as According to the latter proposition, it is of great interest to describe these connected components explicitly. This is worked out for special highest weight monomials for all classical Lie algebras in [7,8] and for the affine Lie algebra (1) in [15]. We recall the results here only for type stated originally in [8].
be a dominant integral weight and consider the highest weight monomial = is characterized as the set of monomials of the form satisfying Summarized, we have a semiregular crystal Mand for each dominant integral weight certain connected subcrystals contained in M. These are isomorphic to ( ) and an explicit description of these components is worked out for the classical simple Lie algebras and (1) . In the remaining parts of this paper we prove a similar result as Propositions 14 and 15, whereby our "big" semiregular crystal is R ∞ .

Explicit Description of the Connected Components
In this section we define for the dominant integral weights = ∑ =1 certain connected subcrystals R( ) ⊆ R ∞ . Furthermore, we provide an explicit description of these crystals in Theorems 18 and 19, respectively; that is, we give a set of conditions describing R( ).
Note that the weight of is precisely . Furthermore, by definition, R( ) is connected and for = we can immediately provide a description of R( ). To be more accurate we prove as a first step the following proposition: such that there is no pair ( , +1 ) of the form (0 , +1 ̸ = 0 ) with ≥ .
Proof. First we note that the element is contained in R( ) and is a highest weight element. Furthermore, we claim that is the unique highest weight element. So suppose that we have another element x = ( 1 , . . . , , 0, 0, . . .) satisfying̃x = 0 for all ∈ . Let be the lowest integer which appears in one of the sequences 1 , . . . , and let be the minimal integer such that appears in , say, In the case where such a does not exist, we have x = .
We remark that̃( In order to obtain a connected crystal it remains to show that x,̃x ∈ R( ) ∪ {0}. Assume that̃x ̸ = 0, say, where we set for simplicity (x) = . Our goal here is to show that the properties (1) and (2) hold for̃x, where we start by proving (2). It is easy to see that (2) can only be violated if one of the following two cases occurs:

In either case we obtain
which is a contradiction to the choice of . The proof of the fact that (1) holds will proceed in several cases. In the remaining parts we denote the entries of ( ) by ( ), ( ), that is, We first consider the case < , which means that we replace + 1 by . Since the entries stay unchanged, only property (1)(i) can be violated. However, property (1)(i) is still fulfilled because If > we replace − 1 by and hence (1) is obviously fulfilled. So suppose that = , which means that we add the entry ( , ).
Necessarily we must have ≤ and in the case where < (65) implies As a consequence we get that +1 appears in −1 while does not appear and thus −1 (x) − −1 (x) < (x) − (x), which is a contradiction to the choice of . Eventually if = = , we obtain in a similar way which forces on the one hand = and on the other hand that does not appear in −1 .
To be more precise, we can conclude the latter statement with the help of (1)(iii) and (1)(i), namely, (68) Thus, we get again −1 (x) − −1 (x) < (x) − (x), which is once more a contradiction to the choice of . ( −1 , ( ))). Here we have ≤ and if = , we must have > because otherwise (1)(ii) would not be violated. We first consider the case where = and notice that the only possible violation is given by the following inequality:

Case 1.3 ((1)(ii) is violated for the pair
We can conclude that does not occur in −1 because either = + and hence < (70) or + 1 ≤ + and (1)(iii) is applicable, which yields < To obtain a contradiction we have to show that − 1 appears in −1 . If 1 > , this follows by the subsequent calculation: If 1 ≤ , we obtain with property (1)(i) that ≥ { | 1 ≤ ≤ 1 } ≥ . In particular we actually have = because otherwise we would get Eventually we can conclude again that − 1 must appear in Now we suppose < . Then an easy consideration shows that (1)(ii) can only be violated if > and thus we obtain similarly as before that the only violation which can occur is the following: where we expect − = + 1. We would like to show as before that does not appear in −1 while − 1 appears. We either have = + and thus > or + 1 ≤ + . In the latter case we apply property (1)(ii) and obtain In either case we notice that does not occur in −1 .
In order to prove the remaining part we consider the element −( −1) which is considerable, since In the case where −( −1) is greater than , we obtain and otherwise by using property (1)(i) we get = . Thus, Case 1.4 ((1)(iii) is violated for the pair ( −1 , ( ))). We suppose that (1)(iii) is violated, which forces ≥ . In the case where > , we have It follows that − 1 appears and does not appear in −1 because = − + − + +1 would imply Consequently, and else we can assume − +1 ≤ so that (1)(iii) is applicable, which yields Therefore, does not appear in −1 . In what follows, we finish our proof by showing that − 1 appears in −1 . If 1 > , we can apply (1)(ii) and get If 1 ≤ , we can verify with (1)(i) that ≥ , but the assumption − 1 ≥ yields in a contradiction, namely, So = and we obtain the required equality −1 ≥ − ≥ −1.

Explicit Description of R( ) in Type .
In this subsection we would like to give an explicit characterization of R( ) if g is a symplectic Lie algebra. In order to state the main theorem we fix some notation.
and on the other hand we got = 1 ≤ 1 = ⇒ ≥ , which verifies the properties listed in (d).
Case 1 ( ± 1 → or +( , )). Here we assume that the action of the Kashiwara operator on x is given in a way such that satisfies property (a). It means that we either replace ± 1 by or add the pair ( , ) as described in Section 3. Here we consider again several cases, where each case assumes that a condition described in Theorem 19 is violated.
Further, as in Subcase 1.2 of Theorem 18, one can verify that does not appear in −1 and if in addition ≤ < holds, then +1 occurs in −1 . Consequently must be contained in { 1 , . . . , } because otherwise we would obtain a contradiction to the choice of . Hence, if we take = = , it follows immediately that the properties (a), (b), and (c) in (3) hold for the pair ( −1 , ), which is impossible. For instance (c) is fulfilled with (91), since The other violations of the properties in (1) or (2) can be proven similarly, so that we consider as a next step the following case.
and is minimal with this property. In the case where such an element does not exist we set = ( ). We claim the following.

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As a consequence of (94) we obtain where the first estimation is strict if < . Hence, we have a contradiction to the assumption that (c) holds. satisfied. Therefore, we must have = +1 because otherwise we obtain a contradiction to the choice of . It follows that where the first estimation is strict whenever ≥ and thus provides a contradiction to (c).
Subcase 1.4 ((4) is violated for the pair ( ( ), +1 )). It is easy to see that this case can never appear.
where the first estimation is strict provided that ≥ meaning that this calculation contradicts once more property (c). The last and second possibility which can occur is that there exists ≥ ( ) = + 1, such that the properties (a), (b), and (c) are fulfilled.
Then we make a similar construction as in Subcase 1.2; namely, we suppose that where the first estimation is strict whenever < is satisfied.
Case 2 ( + 1 → ). Now we assume that the action of the Kashiwara operator on x is given in a way such that satisfies property (b) while (a) is violated, which in particular means that we replace the entry + 1 by . Since the proofs are similar to Case 1, we do not give them in full details. We only consider the case where we presume that property (3) is violated, that is, one has the following.
Subcase 2.1 ( (3) is violated for the pair ( −1 , ( ))). It is easy to see that this case can never appear.
since otherwise we would obtain that satisfies (a) (from Section 3) and thus the Kashiwara operator would act as in Case 1. Accordingly we can apply our assumptions to ≥ = + 1 and obtain a contradiction to (c), namely, The Using this inequality we arrive once more at a contradiction, namely,

Crystal Bases as Tuples of Integer Sequences
In this section we will verify with Theorem 20 that the crystal R( ) can be identified with the crystal graph ( ) obtained from Kashiwara's crystal bases theory. Our strategy here is to show that there exists an isomorphism of R( ) onto the connected component of ⨂ =1 ( ) ⊗ containing ⨂ =1 ⊗ , where denotes the highest weight element in ( ). For the proofs in type we will need a result stated in [16], where the affine type A Kirillov-Reshetikhin crystals are realized via polytopes. We will need the realization of level 1 KR-crystals especially, since they are as classical crystals isomorphic to ( ). In type we will use a short induction argument to prove our results.
Theorem 20. Let be an arbitrary dominant integral weight and set = ∑ as before. Thenone has the following.
(2) If > 1 and is the maximal integer such that ̸ = 0, one obtains a strict crystal morphism Proof. With the help of Proposition 17, the crystal R( ) is characterized as ⋃ R . In the case where g is of type , we will consider the map whereby is a pattern as in [16,Definition 2], with filling By an inspection of the crystal structure on the KR-crystal 1, ≅ ( ) ([16, Section 3.2]), it is easy to see that this map becomes an isomorphism of crystals.
If g is of type , the proof of part (1) will proceed by upward induction on . An observation of the crystal graph of ( 1 ) proves the initial step. Now we assume the correctness of the claim for all integers less than , especially we have R( −1 ) ≅ ( −1 ). For the purpose of completing the induction we consider the injective map given by If would be a strict crystal morphism, we would get R( ) ≅ I ( ) ≅ ( ), which finishes the induction. Therefore, we prove that is a strict crystal morphism, where we consider the cases = 0 and = 1, 1 = separately. In the separated cases we draw a part of the crystal graph in order to see that the properties of a crystal morphism hold. If = 0, we obtain 0 → ( , ) , and if = 1, 1 = , we get (1) if 1 ≤ − 1 (see Figure 1), (2) if 1 ≥ and 1 ̸ = (see Figure 2), (3) if 1 ≥ and 1 = (see Figure 3).
To show the existence of a morphism of crystals we have to show (among others) the weight invariance of , which is proven by the following calculation: The verification of ( ( )) = ( ) is therefore proven with

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If̃( ) ̸ = 0 and̃acts on the first tensor, we get with the tensor product rule ( 1 ) > ( 2 ). Note that the operation with the Kashiwara operator̃would change the entry in if and only if either = + 1 or = − 1 and is not subject to (a). Our goal is to show here that̃has no effect on . If = + 1, we would get ( 1 ) = 0 and thus a contradiction to ( 1 ) > ( 2 ). In the case where = − 1, we have that (b) is not fulfilled for 1 . Therefore, 1 must be subject to (a). Consequently we obtain that must fulfill (a) as well because otherwise we would end in a contradiction; namely, the only property in (a) which can be violated is ∉ { 1 , . . . , , 1 , . . . , }. So if is contained in the aforementioned set, we get by the definition of that = 1 = . Hence, − 1 = − 1 must be contained in the set { ≤ 1 , . . . , −1 }, which is impossible.
In order to prove (2) we will check as in (1) step for step the properties of a morphism of crystals. We get The same is trivially fulfilled for because of Definition 3 (1) and the weight invariance of . Now suppose that (x) = is as in (25). If we apply the Kashiwara operator̃to the tensor product x −1 ⊗ , we obtain with the above calculations According to this we get that ∈ {1, . . . , − 1} if̃acts on the first tensor and = if̃acts on the second tensor. The proof for the Kashiwara operator̃works similarly.

Corollary 21. One has an isomorphism of crystals
Proof. The proof will proceed by induction on , where the initial step is exactly part (1) of Theorem 20. If > 1 and is the maximal integer where is non zero, we can assume with the induction hypothesis that R( − ) ≅ ( − ) and R( ) ≅ ( ). The rest of the proof is done with part (2) of Theorem 20, since the map is injective and the image is a connected crystal containing the highest weight element − ⊗ .