Crystal Bases as Tuples of Integer Sequences

Abstract

We describe a set ℛ^∞ consisting of tuples of integer sequences and provide certain explicit maps on it. We show that this defines a semiregular crystal for 𝔰𝔩_𝑛+1 and 𝔰𝔭_2𝑛, respectively. Furthermore, we define for any dominant integral weight 𝜆 a connected subcrystal ℛ(𝜆) in ℛ^∞, such that this crystal is isomorphic to the crystal graph 𝐵(𝜆). Finally, we provide an explicit description of these connected crystals ℛ(𝜆).

1. Introduction

Let 𝔤 be a symmetrizable Kac-Moody algebra and let 𝔘_𝑞(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 𝑞=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_𝑞(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 [2–5]. 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 𝜆 can be generated by an algorithm using the straight path connecting 0 and 𝜆.

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 𝜆 is isomorphic to the crystal graph 𝐵(𝜆) obtained from Kashiwara's crystal bases theory. For special highest weight monomials these connected components are explicitly characterized for 𝔰𝔩_𝑛+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 𝜆 and those in [7, 8] is provided in [9] for the types 𝐴 and 𝐶 and in [10] for the types 𝐵 and 𝐷.

In this paper we introduce a set ℛ^∞ consisting of tuples of integer sequences; that is, a typical element in ℛ^∞ is given by 𝐱=𝑥₁,𝑥₂,…∈ℛ^∞, (1)where each component 𝑥_𝑗 consists of certain ordered pairs of integers, 𝑥_𝑗=(𝑖₁,𝑖^′₁)⋯(𝑖_𝑠,𝑖^′_𝑠) (see Definition 3). Furthermore, the number of nonzero components is finite. We provide certain maps on ℛ^∞, the Kashiwara operators 𝑒_𝑙̃, 𝑓_𝑙, and maps 𝜖_𝑙, 𝜑_𝑙 for all 𝑙=1,…,𝑛 and prove that ℛ^∞ is a semiregular crystal if 𝔤 is 𝔰𝔩_𝑛+1 or 𝔰𝔭_2𝑛 (see Definition 3 and Proposition 9).

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

Theorem 1. Let 𝜆 be a dominant integral weight, then there exists a crystal isomorphism ℛ(𝜆)⟶𝐵(𝜆), (2) mapping 𝑟_𝜆 to the highest weight element 𝑏_𝜆∈𝐵(𝜆).

Therefore, similar to the setting of Nakajima monomials, a natural question arises; namely, can one characterize for each dominant integral weight 𝜆 explicitly the sequences appearing in ℛ(𝜆)? 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 𝜆 the subcrystals ℛ(𝜆) and describe them explicitly. Finally, in Section 6 we prove that they are isomorphic to 𝐵(𝜆).

2. Notations and a Review of Crystal Theory

Let 𝔤 be a complex simple Lie algebra of rank 𝑛 with index set 𝐼={1,…,𝑛} 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={𝛼₁,…,𝛼_𝑛} the corresponding basis of Φ and the basis of the dual root system Φ^∨⊆𝔥 is denoted by Π^∨={𝛼^∨₁,…,𝛼^∨_𝑛}. Let 𝔤=𝔫⁺⊕𝔥⊕𝔫⁻ be a Cartan decomposition and for a given root 𝛼∈Φ let 𝔤_𝛼 be the corresponding root space. For a dominant integral weight 𝜆 we denote by 𝑉(𝜆) the irreducible 𝔤-module with a highest weight 𝜆. Fix a highest weight vector 𝑣_𝜆∈𝑉(𝜆), then 𝑉(𝜆)=𝔘(𝔫⁻)𝑣_𝜆, where 𝔘(𝔫⁻) denotes the universal enveloping algebra of 𝔫⁻. For an indetermined element 𝑞 we denote by 𝔘_𝑞(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 𝑀 in the category 𝒪^𝑞 as follows: ch𝑀=_𝜇♯𝐵_𝜇𝑒^𝜇, (3) whereby (𝐿,𝐵) is the crystal bases of 𝑀 (see [11]). The crystal graph associated with the irreducible module of highest weight 𝜆 is denoted by 𝐵(𝜆). So finding expressions for the characters can be achieved by finding explicit combinatorial description of crystal bases. For some examples we refer to [2–4].

From now on we assume that 𝔤 is a classical Lie algebra of type 𝐴_𝑛 or 𝐶_𝑛. Note that the positive roots are all of the following form: Type𝐴_𝑛∶𝛼_𝑖,𝑗=𝛼_𝑖+𝛼_𝑖+1+⋯+𝛼_𝑗,for1≤𝑖≤𝑗≤𝑛,Type𝐶_𝑛∶𝛼_𝑖,𝑗=𝛼_𝑖+𝛼_𝑖+1+⋯+𝛼_𝑗,𝛼for1≤𝑖≤𝑗≤𝑛,_𝑖,𝑗=𝛼_𝑖+𝛼_𝑖+1+⋯+𝛼_𝑛+𝛼_𝑛−1+⋯+𝛼_𝑗,for1≤𝑖≤𝑗≤𝑛. (4) Furthermore, let 𝑃=⨁_𝑖∈𝐼ℤ𝜔_𝑖 be the set of classical integral weights and 𝑃⁺=⨁_𝑖∈𝐼ℤ₊𝜔_𝑖 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.

2.1. Abstract Crystals

Crystal bases of integrable 𝔘_𝑞(𝔤)-modules in the category 𝒪^𝑞 are characterized by certain maps satisfying some properties. One can define the abstract notion of crystals associated with a Cartan datum as follows.

Definition 2. Let 𝐼 be a finite index set and let 𝐴=(𝑎_𝑖,𝑗)_{𝑖,𝑗∈𝐼} be a generalized Cartan matrix with the Cartan datum (𝐴,Π,Π^∨,𝑃,𝑃^∨). A crystal associated with the Cartan datum (𝐴,Π,Π^∨,𝑃,𝑃^∨) is a set 𝐵 together with maps 𝑤𝑡∶𝐵→𝑃, 𝑒_𝑙̃,𝑓_𝑙∶𝐵→𝐵∪{0}, and 𝜖_𝑙,𝜑_𝑙∶𝐵→ℤ∪{−∞} satisfying the following properties for all 𝑙∈𝐼:(1)𝜑_𝑙(𝑏)=𝜖_𝑙(𝑏)+⟨𝛼^∨_𝑙,𝑤𝑡(𝑏)⟩, (2)𝑤𝑡(𝑒_𝑙̃𝑏)=𝑤𝑡(𝑏)+𝛼_𝑙 if 𝑒_𝑙̃𝑏∈𝐵, (3)𝑤𝑡(𝑓_𝑙𝑏)=𝑤𝑡(𝑏)−𝛼_𝑙 if 𝑓_𝑙𝑏∈𝐵, (4)𝜖_𝑙(𝑒_𝑙̃𝑏)=𝜖_𝑙(𝑏)−1, 𝜑_𝑙(𝑒_𝑙̃𝑏)=𝜑_𝑙(𝑏)+1 if 𝑒_𝑙̃𝑏∈𝐵, (5)𝜖_𝑙(𝑓_𝑙𝑏)=𝜖_𝑙(𝑏)+1, 𝜑_𝑙(𝑓_𝑙𝑏)=𝜑_𝑙(𝑏)−1 if 𝑓_𝑙𝑏∈𝐵, (6)𝑓_𝑙𝑏=𝑏^′ if and only if 𝑒_𝑙̃𝑏^′=𝑏 for 𝑏,𝑏^′∈𝐵, (7)if 𝜑_𝑙(𝑏)=−∞ for 𝑏∈𝐵, then 𝑓_𝑙𝑏=𝑒_𝑙̃𝑏 = 0. Furthermore, a crystal 𝐵 is said to be semiregular if the equalities 𝜖_𝑙(𝑏)=max𝑘≥0∣𝑒^𝑘_𝑙̃,𝜑𝑏≠0_𝑙(𝑏)=max𝑘≥0∣𝑓^𝑘_𝑙𝑏≠0 (5) hold.

The maps 𝑒_𝑙̃ and 𝑓_𝑙 are called Kashiwara's crystal operators and the map 𝑤𝑡 is called the weight function. So, on the one hand, one can associate with any integrable 𝔘_𝑞(𝔤)-module a set 𝐵 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 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 𝐵₂-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 𝐴_𝑛 and 𝐶_𝑛. 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.

3. 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 𝜆 certain subcrystals ℛ(𝜆); that is, ⋃_𝜆∈𝑃+ℛ(𝜆)⊆ℛ^∞, such that ℛ(𝜆) has a strong connection to the crystal graph 𝐵(𝜆) (see Corollary 21).

3.1. Set of Tuples of Integer Sequences

In order to define ℛ^∞ we consider a total order on 𝐈={1,…,𝑛} if 𝔤 is of type 𝐴_𝑛 and a total order on 𝐈={1,…,𝑛,𝑛−1,…,1} if 𝔤 is of type 𝐶_𝑛, namely, 1<2<⋯<𝑛,1<2<⋯<𝑛<𝑛−1<⋯<1, (6) respectively. Furthermore, especially in Section 5, we need for type 𝐶_𝑛 the following bijective map: ⁻∶𝐈⟶𝐈,𝑛↦𝑛,𝑖↦𝑖,𝑖↦𝑖,for𝑖∈{1,…,𝑛−1}. (7) For 1≤𝑖≤𝑛,𝑠∈ℤ_≥0 we set ℛ^𝑖_𝑠 to be the set of all sequences (𝑖₁,𝑖^′₁)⋯(𝑖_𝑠,𝑖^′_𝑠) with 𝑖_𝑗,𝑖^′_𝑗∈𝐈, such that 1≤𝑖_𝑠<𝑖_𝑠−1<⋯<𝑖₁≤𝑖≤𝑖^′₁<𝑖^′₂<⋯<𝑖^′_𝑠𝑖≤max𝐈,_′𝑗≤𝑖_𝑗,𝑗=1,…,𝑠, (8) where max𝐈 denotes the maximal element in 𝐈 with respect to <. We denote by ∅_𝑖 the unique element in ℛ^𝑖₀.

Definition 3. We define ℛ^∞ to be the set of all infinite sequences 𝐱=(𝑥₁,𝑥₂,𝑥₃,…) where each component 𝑥_𝑗 is contained in ℛ=⋃_𝑠,𝑖ℛ^𝑖_𝑠∪{0} and only finitely many components are nonzero. We identify ℛ^𝑘 with the sequences of the form (𝑥₁,…,𝑥_𝑘,0,0,…).

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 𝑥=(𝑖₁,𝑖^′₁)⋯(𝑖_𝑠,𝑖^′_𝑠)∈ℛ^𝑖_𝑠 be an arbitrary element and fix 𝑙∈𝐼:(a)𝑙∉𝑖₁,…,𝑖_𝑠,𝑖^′₁,…,𝑖^′_𝑠,𝑙+1∈𝑖₁,…,𝑖_𝑠𝑖,if𝑙<𝑖,𝑙−1∈^′₁,…,𝑖^′_𝑠,if𝑙>𝑖,/,if𝑙=𝑖,𝑙+1∈𝑖^′₁,…,𝑖^′_𝑠∨𝑙∉𝑖^′₁,…,𝑖^′_𝑠, (9)(a^′) replace in (a) ∨ by ∧, (b)𝑙∉𝑖^′₁,…,𝑖^′_𝑠,𝑙+1∈𝑖^′₁,…,𝑖^′_𝑠,𝑙+1∉𝑖₁,…,𝑖_𝑠∨𝑙∈𝑖₁,…,𝑖_𝑠𝑖,if𝑙<𝑖,𝑙−1∉^′₁,…,𝑖^′_𝑠∨𝑙∈𝑖^′₁,…,𝑖^′_𝑠,if𝑙>𝑖,𝑙=𝑖₁∨𝑙=𝑖^′₁,if𝑙=𝑖, (10)(c)𝑙∈𝑖₁,…,𝑖_𝑠,𝑖^′₁,…,𝑖^′_𝑠,𝑙+1∉𝑖₁,…,𝑖_𝑠𝑖,if𝑙<𝑖,𝑙−1∉^′₁,…,𝑖^′_𝑠𝑖,if𝑙>𝑖,₁,𝑖^′₁=(𝑙,𝑙),if𝑙=𝑖,𝑙+1∈𝑖^′₁,…,𝑖^′_𝑠∨𝑙∉𝑖^′₁,…,𝑖^′_𝑠, (11)(d)𝑙∈𝑖^′₁,…,𝑖^′_𝑠,𝑙+1∉𝑖^′₁,…,𝑖^′_𝑠,𝑙+1∉𝑖₁,…,𝑖_𝑠∨𝑙∈𝑖₁,…,𝑖_𝑠𝑖,if𝑙<𝑖,𝑙−1∉^′₁,…,𝑖^′_𝑠∨𝑙∈𝑖^′₁,…,𝑖^′_𝑠,if𝑙>𝑖,𝑙=𝑖₁∨𝑙=𝑖^′₁,if𝑙=𝑖, (12)(d^′) replace everywhere in (d) ∨ by ∧.

Let us consider an example.

Example 4. (1) Let 𝔤=𝐴₅, 𝑙=2, and 𝑥₁=(3,4)(1,5)∈ℛ³₂,𝑥₂=(2,3)∈ℛ²₁, (13) then 𝑥₁ satisfies (a) while 𝑥₂ violates (a).
(2) Let 𝔤=𝐶₃, 𝑙=1, and 𝑥=1,2∈ℛ²₁, (14) then 𝑥 satisfies (c).

Remark 5. If 𝑥 satisfies (a^′) and (d^′), respectively, then it satisfies also (a) and (d), respectively. If 𝔤 is further of type 𝐴_𝑛, 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=(𝑥₁,𝑥₂,𝑥₃,…) 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 𝑤𝑡(𝐱)=^𝑛_𝑖=1𝑐_𝑖𝜔_𝑖−_𝑗𝑤𝑡𝑥_𝑗, (15) where 𝑤𝑡𝑥_𝑗=^𝑠_𝑗∑₌₁𝛼_{𝑖𝑗,𝑖′𝑗}if𝑥_𝑗=𝑖₁,𝑖^′₁⋯𝑖_𝑠,𝑖^′_𝑠,0if𝑥_𝑗=0or𝑥_𝑗∈∅₁,…,∅_𝑛 (16) and 𝑐_𝑖=♯{𝑥_𝑗≠0∣𝑥_𝑗∈⋃_𝑠ℛ^𝑖_𝑠}. Suppose that the nonzero components in 𝐱 are given by 𝑥_𝑞1∈ℛ^𝑗1_𝑠1,…,𝑥_𝑞𝑘∈ℛ^𝑗𝑘_𝑠𝑘. For fixed 𝑙∈𝐼 we define the following maps.

(i) For  2≤𝑗≤𝑘+1, let 𝜎^𝑗_𝑙∶ℛ^∞→ℤ_≥0 be the map given by 𝜎^𝑗_𝑙(𝐱)=𝑎^𝑗_𝑙(𝐱)+𝑏^𝑗_𝑙(𝐱), (17) where 𝑎^𝑗_𝑙(𝐱)=♯𝑥_𝑞𝑝∣1≤𝑝≤𝑗−1,𝑥_𝑞𝑝(a)(b,𝑏satisfiesor)_𝑗𝑙(𝐱)=♯𝑥_𝑞𝑝∣1≤𝑝≤𝑗−1,𝑥_𝑞𝑝satisfiesa^′. (18) Furthermore, we define 𝜃_𝑙(𝑥_𝑞𝑝) to be the sequence which arises out of 𝑥_𝑞𝑝 by replacing𝑙+1by𝑙,if𝑙<𝑗_𝑝,replacing𝑙−1by𝑙,if𝑙>𝑗_𝑝,adding(𝑙,𝑙),if𝑙=𝑗_𝑝, (19) if 𝑥_𝑞𝑝 satisfies (a). If 𝑥_𝑞𝑝 satisfies (b), let 𝜃_𝑙(𝑥_𝑞𝑝) be the sequence which arises out of 𝑥_𝑞𝑝 by replacing 𝑙+1 by 𝑙. If neither (a) nor (b) is fulfilled, we set 𝜃_𝑙(𝑥_𝑞𝑝)=0.

(ii) For 2≤𝑗≤𝑘, let 𝜏^𝑗_𝑙∶ℛ^∞→ℤ_≥0 be the map given by 𝜏^𝑗_𝑙(𝐱)=𝑐^𝑗_𝑙(𝐱)+𝑑^𝑗_𝑙(𝐱), (20) where 𝑐^𝑗_𝑙(𝐱)=♯𝑥_𝑞𝑝∣2≤𝑝≤𝑗,𝑥_𝑞𝑝(c)(d,𝑑satisfiesor)_𝑗𝑙(𝐱)=♯𝑥_𝑞𝑝∣2≤𝑝≤𝑗,𝑥_𝑞𝑝satisfiesd^′. (21) We define 𝜌_𝑙(𝑥_𝑞𝑝) to be the sequence which arises out of 𝑥_𝑞𝑝 by replacing𝑙by𝑙+1,if𝑙<𝑗_𝑝,replacing𝑙by𝑙−1,if𝑙>𝑗_𝑝,erasing(𝑙,𝑙),if𝑙=𝑗_𝑝, (22) if 𝑥_𝑞𝑝 satisfies (c). If 𝑥_𝑞𝑝 satisfies (d), let 𝜌_𝑙(𝑥_𝑞𝑝) be the sequence which arises out of 𝑥_𝑞𝑝 by replacing 𝑙 by 𝑙+1. If neither (c) nor (d) is fulfilled, we set 𝜌_𝑙(𝑥_𝑞𝑝)=0.

Remark 6. (1) For 𝑗=1, we set 𝜎^𝑗_𝑙=𝜏^𝑗_𝑙=0.
(2) Note that the image of 𝑥∈⋃_𝑠ℛ^𝑖_𝑠 under the maps  𝜃_𝑙 and 𝜌_𝑙, respectively, is contained in ⋃_𝑠ℛ^𝑖_𝑠∪{0}, that is, 𝜃_𝑙∶_𝑠ℛ^𝑖_𝑠⟶_𝑠ℛ^𝑖_𝑠∪{0},𝜌_𝑙∶_𝑠ℛ^𝑖_𝑠⟶_𝑠ℛ^𝑖_𝑠∪{0}. (23)

One important fact about these maps is described in the next lemma.

Lemma 7. Let 𝑥,𝑥 be nonzero sequences as in (8), then one has 𝜃_𝑙(𝑥)=𝑥iff𝜌_𝑙(𝑥)=𝑥. (24)

Proof. One can easily show that 𝑥 satisfies (a) if and only if 𝑥 satisfies (c). Hence, we can suppose that 𝑥 does not fulfill all properties enumerated in (a). By observing the action we see that 𝑥 arises from 𝑥 by replacing 𝑙+1 by 𝑙, which means that (c) is violated. In particular 𝑙+1 does not appear in 𝑥 and 𝑙 appears in 𝑥, which means that the properties in (d) hold. Hence, 𝜌(𝑥)=𝑥. The arguments for the reverse direction are the same.

Let 𝑓_𝑙(𝐱)∶=max1≤𝑝≤𝑘∣𝜎^𝑝_𝑙(𝐱)−𝜏^𝑝_𝑙(𝐱)=min𝜎^𝑗_𝑙(𝐱)−𝜏^𝑗_𝑙(𝐱),𝑒∣1≤𝑗≤𝑘_𝑙(𝐱)∶=min1≤𝑝≤𝑘∣𝜎^𝑝_𝑙(𝐱)−𝜏^𝑝_𝑙(𝐱)=min𝜎^𝑗_𝑙(𝐱)−𝜏^𝑗_𝑙(𝐱)∣1≤𝑗≤𝑘. (25)(26) Now we are able to define the Kashiwara operators, 𝑓_𝑙𝐱=0,if𝜃_𝑙𝑥_{𝑞𝑓𝑙(𝐱)}=0,…,𝜃_𝑙𝑥_{𝑞𝑓𝑙(𝐱)}𝑒̃,…,else,_𝑙𝐱=0,if𝜌_𝑙𝑥_{𝑞𝑒𝑙(𝐱)}=0,…,𝜌_𝑙𝑥_{𝑞𝑒𝑙(𝐱)},…,else. (27) Let us consider an example.

Example 8. (1) Let 𝔤=𝐴₄ and 𝐱=(𝑥₁,𝑥₂,𝑥₃,0,0,…) with 𝑥₁=(2,2)(1,3)∈ℛ²₂,𝑥₂=(2,4)∈ℛ⁴₁,𝑥₃=(3,3)(2,4)∈ℛ³₂. (28) For 𝑙=2, we get 𝜎¹₂(𝐱)=𝜎²₂(𝐱)=𝜎³₂(𝐱)=𝜏¹₂(𝐱)=0,𝜏²₂(𝐱)=1,𝜏³₂(𝐱)=2 and hence 𝑓₂𝐱=0. (29)
(2)  Let 𝔤=𝐶₃ and 𝐱=(𝑥₁,𝑥₂,0,0,…) with 𝑥₁=1,2∈ℛ²₁,𝑥₂=(3,3)1,2∈ℛ³₂, (30) then 𝜎²₃(𝐱)=0,𝜏²₃(𝐱)=1, and 𝑒₃̃𝐱=𝑥₁,𝑥₁,0,0⋯. (31)

It remains to define the maps 𝜑_𝑙 and 𝜖_𝑙. These maps are given by the next formula: 𝜖_𝑙(𝐱)=𝜏²_𝑙∅₁,𝑥_𝑞1𝜎,0,0⋯−min^𝑗_𝑙(𝐱)−𝜏^𝑗_𝑙(𝐱),𝜑∣1≤𝑗≤𝑘_𝑙(𝐱)=𝜎_𝑙^𝑘+1(𝐱)−𝜏^𝑘_𝑙(𝐱)−min𝜎^𝑗_𝑙(𝐱)−𝜏^𝑗_𝑙(𝐱)∣1≤𝑗≤𝑘. (32)

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, 𝑥_𝑞1∈ℛ^𝑗1_𝑠1,…,𝑥_𝑞𝑘∈ℛ^𝑗𝑘_𝑠𝑘. Then we order these components in a way such that the first components are contained in ⋃_𝑠⋃_𝑚<𝑙ℛ^𝑚_𝑠 followed by components in ⋃_𝑠⋃_𝑚>𝑙ℛ^𝑚_𝑠 and the last ones are contained in ⋃_𝑠ℛ^𝑙_𝑠. So we can write the set of nonzero components of 𝐱 as a disjoint union of three subsets 𝐴_<𝑙∪𝐴_>𝑙∪𝐴_=𝑙. Further let 𝐱₁ be the element in ℛ^∞ obtained from 𝐱 by replacing all components not belonging to 𝐴_<𝑙 by 0. And 𝐱₂ and 𝐱₃, respectively, are similarly defined using 𝐴_>𝑙 and 𝐴_=𝑙, respectively. Then we get 𝜑_𝑙(𝐱)−𝜖_𝑙(𝐱)=𝜎_𝑙^𝑘+1(𝐱)−𝜏^𝑘_𝑙(𝐱)−𝜏²_𝑙∅₁,𝑥_𝑞1,0,0⋯=𝜎^|𝐴<𝑙_𝑙^|+1𝐱₁−𝜏^|𝐴<𝑙|_𝑙𝐱₁+𝜎^|𝐴>𝑙_𝑙^|+1𝐱₂−𝜏^|𝐴>𝑙|_𝑙𝐱₂+𝜎^|𝐴=𝑙_𝑙^|+1𝐱₃−𝜏^|𝐴=𝑙|_𝑙𝐱₃−³_𝑖=1𝜏²_𝑙∅₁,𝑦_𝑖,0,0⋯, (33) where the sequence 𝑦_𝑖 is the first nonzero element in 𝐱_𝑖. A short calculation shows 𝜎^|𝐴<𝑙_𝑙^|+1𝐱₁−𝜏^|𝐴<𝑙|_𝑙𝐱₁−𝜏²_𝑙∅₁,𝑦₁,0,0⋯=−_{1≤𝑖≤𝑘∶𝑥𝑞𝑖∈𝐴<𝑙}𝛼^∨_𝑙,𝑤𝑡𝑥_𝑞𝑖,𝜎^|𝐴>𝑙_𝑙^|+1𝐱₂−𝜏^|𝐴>𝑙|_𝑙𝐱₂−𝜏²_𝑙∅₁,𝑦₂,0,0⋯=−_{1≤𝑖≤𝑘∶𝑥𝑞𝑖∈𝐴>𝑙}𝛼^∨_𝑙,𝑤𝑡𝑥_𝑞𝑖,𝜎^|𝐴=𝑙_𝑙^|+1𝐱₃−𝜏^|𝐴=𝑙|_𝑙𝐱₃−𝜏²_𝑙∅₁,𝑦₃,0,0⋯=𝑐_𝑙−_{1≤𝑖≤𝑘∶𝑥𝑞𝑖∈𝐴=𝑙}𝛼^∨_𝑙,𝑤𝑡𝑥_𝑞𝑖. (34) Now we proceed to prove (6). By the definition of the Kashiwara operators and Lemma 7, it is enough to show that  𝑓_𝑙(𝑒_𝑙̃𝐱)=𝑒_𝑙(𝐱) and 𝑒_𝑙(𝑓_𝑙𝐱)=𝑓_𝑙(𝐱). Since the proofs are similar, we prove only the latter equation. Assume that 𝑓_𝑙𝐱≠0 and 𝑓_𝑙(𝐱)≠1, then 𝜎^𝑗_𝑙𝑓_𝑙𝐱=𝜎^𝑗_𝑙(𝐱),if𝑗≤𝑓_𝑙(𝐱),𝜎^𝑗_𝑙(𝐱)−1,if𝑗>𝑓_𝑙(𝐱),𝜏^𝑗_𝑙𝑓_𝑙𝐱=𝜏^𝑗_𝑙(𝐱),if𝑗≤𝑓_𝑙(𝐱)𝜏−1,_𝑗𝑙(𝐱)+1,if𝑗≥𝑓_𝑙(𝐱). (35) Subsequently we have 𝑒_𝑙𝑓_𝑙𝐱=min1≤𝑝≤𝑘∣𝜎^𝑝_𝑙𝑓_𝑙𝐱−𝜏^𝑝_𝑙𝑓_𝑙𝐱=min𝜎^𝑗_𝑙𝑓_𝑙𝐱−𝜏^𝑗_𝑙𝑓_𝑙𝐱∣1≤𝑗≤𝑘=min1≤𝑝≤𝑘∣𝜎^𝑝_𝑙𝑓_𝑙𝐱−𝜏^𝑝_𝑙𝑓_𝑙𝐱=𝜎^𝑓𝑙_𝑙^(𝐱)(𝐱)−𝜏^𝑓𝑙_𝑙^(𝐱)(𝐱)−1=𝑓_𝑙(𝐱). (36) In the case where 𝑓_𝑙(𝐱)=1, we obtain 𝑒_𝑙𝑓_𝑙𝐱=min1≤𝑝≤𝑘∣𝜎^𝑝_𝑙𝑓_𝑙𝐱−𝜏^𝑝_𝑙𝑓_𝑙𝐱=0=1. (37)

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 𝜑_𝑙 and 𝜖_𝑙, respectively, determine how often one can act with 𝑓_𝑙 and 𝑒_𝑙̃, respectively. The semiregularity is a necessary condition of a crystal 𝐵, if one wants to identify it with the crystal graph 𝐵(𝜆).

Lemma 10. Let 𝜖_𝑙 and 𝜑_𝑙  be as in (32). For a given element 𝐱∈ℛ^∞ one obtains 𝜖_𝑙(𝐱)=max𝑘≥0∣𝑒^𝑘_𝑙̃,𝜑𝐱≠0_𝑙(𝐱)=max𝑘≥0∣𝑓^𝑘_𝑙𝐱≠0. (38)

Proof. We proof the statement by induction on 𝑧∶=max{𝑘≥0∣𝑒^𝑘_𝑙̃𝐱≠0}; so let 𝑧=0 and suppose first that 𝑒_𝑙(𝐱)≠1. By the definition of 𝑒_𝑙(𝐱) and 𝜌_𝑙 we have 𝜎^𝑒𝑙_𝑙^(𝐱)(𝐱)−𝜏^𝑒𝑙_𝑙^(𝐱)(𝐱)<𝜎^𝑒𝑙_𝑙^(𝐱)−1(𝐱)−𝜏^𝑒𝑙_𝑙^(𝐱)−1(𝐱)⟺𝜎^𝑒𝑙_𝑙^(𝐱)(𝐱)<𝜎^𝑒𝑙_𝑙^(𝐱)−1(𝐱), (39) which is a contradiction to 𝑒_𝑙(𝐱)≠1. Hence, we have 𝑒_𝑙(𝐱)=1 and as a consequence we obtain 𝜖_𝑙(𝐱)=𝜏²_𝑙((∅₁,𝑥_𝑞1,0,0,⋯))=0, which proves the initial step. Now assume that 𝑧>0 and consider the element 𝑒_𝑙̃𝐱, where we again presume initially 𝑒_𝑙(𝐱)≠1. By applying the induction hypothesis and using 𝜎^𝑗_𝑙𝑒_𝑙̃𝐱=𝜎^𝑗_𝑙(𝐱),if𝑗≤𝑒_𝑙(𝐱),𝜎^𝑗_𝑙(𝐱)+1,if𝑗>𝑒_𝑙(𝐱),𝜏^𝑗_𝑙𝑒_𝑙̃𝐱=𝜏^𝑗_𝑙(𝐱),if𝑗≤𝑒_𝑙(𝐱)𝜏−1,_𝑗𝑙(𝐱)−1,if𝑗≥𝑒_𝑙(𝐱), (40) we arrive at max𝑘≥0∣𝑒̃_𝑙^𝑘+1𝐱≠0=𝜖_𝑙𝑒_𝑙̃𝐱=𝜏²_𝑙∅₁,𝑥_𝑞1,0,0⋯−𝜎_𝑙^𝑒(𝐱)(𝐱)−𝜏_𝑙^𝑒(𝐱)(𝐱)+1=𝜖_𝑙(𝐱)−1. (41) If 𝑒_𝑙(𝐱)=1, 𝜖_𝑙𝑒_𝑙̃(𝐱)=𝜏²_𝑙∅₁,𝜌_𝑙𝑥_𝑞1,0,0⋯+𝜎_𝑙^𝑒(𝐱)(𝐱)−𝜏_𝑙^𝑒(𝐱)(𝐱)=𝜖_𝑙(𝐱)−1. (42) The proof of the remaining equality 𝜑_𝑙(𝐱)=max{𝑘≥0∣𝑓^𝑘_𝑙𝐱≠0} is quite parallel.

Before we introduce the subcrystals ℛ(𝜆) 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_𝑞(𝔤)-modules.

4. 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 𝐵(𝜆) from our approach, we want to introduce the set of all Nakajima monomials, such that we can think of 𝐵(𝜆) in terms of certain monomials. This theory is discovered by Nakajima [14] and generalized by Kashiwara [6].

4.1. Tensor Product of Crystals

Suppose that we have two abstract crystals 𝐵₁, 𝐵₂ in the sense of Definition 3, then we can construct a new crystal which is as a set 𝐵₁×𝐵₂. This crystal is denoted by 𝐵₁⊗𝐵₂ and the Kashiwara operators are given as follows: 𝑓_𝑙𝑏₁⊗𝑏₂=𝑓_𝑙𝑏₁⊗𝑏₂,if𝜑_𝑙𝑏₁>𝜖_𝑙𝑏₂,𝑏₁⊗𝑓_𝑙𝑏₂,if𝜑_𝑙𝑏₁≤𝜖_𝑙𝑏₂,𝑒_𝑙̃𝑏₁⊗𝑏₂=𝑒_𝑙̃𝑏₁⊗𝑏₂,if𝜑_𝑙𝑏₁≥𝜖_𝑙𝑏₂,𝑏₁⊗𝑒_𝑙̃𝑏₂,if𝜑_𝑙𝑏₁<𝜖_𝑙𝑏₂. (43) Furthermore, one can describe explicitly the maps 𝑤𝑡, 𝜑_𝑙, and 𝜖_𝑙 on 𝐵₁⊗𝐵₂, namely, 𝑤𝑡𝑏₁⊗𝑏₂=𝑤𝑡𝑏₁+𝑤𝑡𝑏₂,𝜑_𝑙𝑏₁⊗𝑏₂=max𝜑_𝑙𝑏₂,𝜑_𝑙𝑏₁+𝜑_𝑙𝑏₂−𝜖_𝑙𝑏₂,𝜖_𝑙𝑏₁⊗𝑏₂=max𝜖_𝑙𝑏₁,𝜖_𝑙𝑏₁+𝜖_𝑙𝑏₂−𝜑_𝑙𝑏₁. (44) One of the most important interpretations of the tensor product rule is the following theorem (for more details see [11]).

Theorem 11. Let 𝑀_𝑗 be an integrable module in the category 𝒪^𝑞 and let (𝐿_𝑗,𝐵_𝑗)  be a crystal  bases of 𝑀_𝑗,(𝑗=1,2). Set 𝐿=𝐿₁⊗𝐿₂ and 𝐵=𝐵₁⊗𝐵₂. Then (𝐿,𝐵)  is a crystal  bases of 𝑀₁⊗𝑀₂.

4.2. Nakajima Monomials

For 𝑖∈𝐼 and 𝑛∈ℤ, we consider monomials in the variables 𝑌_𝑖(𝑛); that is, we obtain the set of Nakajima monomials ℳ as follows: ℳ∶=_{𝑖∈𝐼,𝑛∈ℤ}𝑌_𝑖(𝑛)^{𝑦𝑖(𝑛)}∣𝑦_𝑖(𝑛)(.∈ℤvanishexceptforfinitelymany𝑖,𝑛) (45) With the goal of defining a crystal structure on ℳ, we take some integers 𝑐=(𝑐_𝑖,𝑗)_𝑖≠𝑗 such that 𝑐_𝑖,𝑗+𝑐_𝑗,𝑖=1. Let now 𝑀=∏_{𝑖∈𝐼,𝑛∈ℤ}𝑌_𝑖(𝑛)^{𝑦𝑖(𝑛)} be an arbitrary monomial in ℳ and 𝑙∈𝐼; then we set. 𝑤𝑡(𝑀)=_𝑖𝑛𝑦_𝑖(𝑛)𝜔_𝑖,𝜑_𝑙(𝑀)=max_𝑘≤𝑛𝑦_𝑙(𝑘),𝜖∣𝑛∈ℤ_𝑙(𝑀)=max−_𝑘>𝑛𝑦_𝑙(𝑘),𝑛∣𝑛∈ℤ_𝑙𝑓=min𝑛∣𝜑_𝑙(𝑀)=_𝑘≤𝑛𝑦_𝑙(𝑘,𝑛_𝑙𝑒)=max𝑛∣𝜖_𝑙(𝑀)=−_𝑘>𝑛𝑦_𝑙(𝑘). (46) The Kashiwara operators are defined as follows: 𝑓_𝑙𝐴𝑀=^𝑙𝑛^𝑙_𝑓⁻¹𝑀,if𝜑_𝑙(𝑀)>0,0,if𝜑_𝑙(𝑀)𝑒̃=0,_𝑙𝐴𝑀=^𝑙𝑛^𝑙_𝑒𝑀,if𝜖_𝑙(𝑀)>0,0,if𝜖_𝑙(𝑀)=0, (47) whereby 𝐴_𝑙(𝑛)∶=𝑌_𝑙(𝑛)𝑌_𝑙(𝑛+1)_𝑖≠𝑙𝑌_𝑖𝑛+𝑐_𝑖,𝑙^{⟨𝛼∨𝑖,𝛼𝑙⟩}. (48) The following two results are shown by Kashiwara [6].

Proposition 12. With the maps 𝑤𝑡, 𝜑_𝑙, 𝜖_𝑙, 𝑓_𝑙, 𝑒_𝑙̃, 𝑙∈𝐼, the set ℳ becomes a semiregular crystal.

Remark 13. A priori the crystal structure depends on 𝑐; hence, we will denote this crystal by ℳ_𝑐. But it is easy to see that the isomorphism class of ℳ_𝑐 does not depend on this choice. In the literature 𝑐 is often chosen as 𝑐_𝑖,𝑗=0,if𝑖>𝑗,1,elseor𝑐_𝑖,𝑗=0,if𝑖<𝑗,1,else. (49)

Proposition 14. Let 𝑀 be a monomial in ℳ, such that 𝑒_𝑙̃𝑀=0 for all 𝑙∈𝐼. Then the connected component of ℳ containing 𝑀 is isomorphic to 𝐵(𝑤𝑡(𝑀)).

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 𝐴_𝑛⁽¹⁾ in [15]. We recall the results here only for type 𝐶_𝑛 stated originally in [8].

Proposition 15. Let 𝜆=∑^𝑛_𝑖=1𝑚_𝑖𝜔_𝑖 be a dominant integral weight and consider the highest weight monomial 𝑀=𝑌₁(1)^𝑚1⋯𝑌_𝑛(1)^𝑚𝑛. Then the connected component of ℳ containing 𝑀 is characterized as the set of monomials of the form 𝑋_𝑡1,1(1)⋯𝑋_𝑡11,𝛼(1)⋯𝑋_𝑡𝑛,1(𝑛)⋯𝑋_{𝑡𝑛𝑛,𝛼}(𝑛),𝑡_𝑟,𝑠∈1<⋯<𝑛<𝑛<⋯<1, (50) satisfying (1)𝛼_𝑗=𝑚_𝑗+⋯+𝑚_𝑛  for all 𝑗=1,…,𝑛,(2)𝑡_𝑗,1≥⋯≥𝑡_{𝑗,𝛼𝑗}  for all 𝑗=1,…,𝑛,(3)𝑡_{𝑗−1,𝑘}>𝑡_𝑗,𝑘  for all 𝑗=2,…,𝑛 and 𝑘=1,…,𝛼_𝑗, where 𝑋_𝑖(𝑚)=𝑌_𝑖−1(𝑚+1)⁻¹𝑌_𝑖(𝑚),𝑋_𝑖(𝑚)=𝑌_𝑖−1(𝑚+(𝑛−𝑖+1))𝑌_𝑖(𝑚+(𝑛−𝑖+1))⁻¹. (51)

Summarized, we have a semiregular crystal ℳand for each dominant integral weight 𝜆 certain connected subcrystals contained in ℳ. These are isomorphic to 𝐵(𝜆) and an explicit description of these components is worked out for the classical simple Lie algebras and 𝐴_𝑛⁽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^∞.

5. Explicit Description of the Connected Components

In this section we define for the dominant integral weights 𝜆=∑^𝑛_𝑖=1𝑚_𝑖𝜔_𝑖 certain connected subcrystals ℛ(𝜆)⊆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(𝜆).

Definition 16. For a dominant integral weight 𝜆=∑^𝑛_𝑖=1𝑚_𝑖𝜔_𝑖, let ℛ(𝜆) be the connected component of ℛ^∞ containing 𝑟_𝜆=∅₁,…,∅_1𝑚1,…,∅_𝑛,…,∅_𝑛𝑚𝑛,0,0⋯. (52)

Note that the weight of 𝑟_𝜆 is precisely 𝜆. Furthermore, by definition, ℛ(𝜆) is connected and for 𝜆=𝜔_𝑖 we can immediately provide a description of ℛ(𝜔_𝑖). To be more accurate we prove as a first step the following proposition:

Proposition 17. Let 𝜆=𝜔_𝑖; then one obtains ℛ(𝜆)=_𝑠𝑅^𝑖_𝑠,0,0,…=_𝑠𝑅^𝑖_𝑠. (53)

Proof. Since (⋃_𝑠𝑅^𝑖_𝑠,0,0,…) is stable under the Kashiwara operators 𝑓_𝑙 and 𝑒_𝑙̃ (Remark 6), it is enough to prove that ∅_𝑖 is the unique highest weight element in ⋃_𝑠𝑅^𝑖_𝑠, that is, 𝑒_𝑙̃()∅𝑥,0,0,…=0∀𝑙∈𝐼⟹𝑥=_𝑖,0,0⋯. (54) Assume 𝑥=(𝑖₁,𝑖^′₁)⋯(𝑖_𝑠,𝑖^′_𝑠) and 𝑒_𝑙̃(𝑥,0,0,…)=0forall𝑙∈𝐼. If 𝑖₁<𝑖, we have 𝑒̃_𝑖1(𝑥,0,0,…)≠0 and if 𝑖<𝑖^′₁≤𝑛, we have 𝑒̃_𝑖′1(𝑥,0,0,…)≠0. So let 𝑛−1≤𝑖^′₁ (this case cannot appear if 𝔤=𝐴_𝑛); then since 𝑖^′₁∈{𝑖^′₁,…,𝑖^′_𝑠}, 𝑖^′₁+1∉{𝑖^′₁,…,𝑖^′_𝑠} and 𝑖^′₁≤𝑖₁=𝑖, we obtain that 𝑖^′₁∈{𝑖,…,𝑛−1}. Thus, 𝑥 satisfies (d) and hence 𝑒̃_𝑖′1(𝑥,0,0,…)≠0. According to these calculations, the pair (𝑖,𝑖) must appear in 𝑥. This implies 𝑒_𝑖̃(𝑥,0,0,…)≠0.
For general λ’s, we can describe the connected components and refer to the following two subsections.

5.1. Explicit Description of ℛ(𝜆) in Type 𝐴_𝑛

In this subsection we give an explicit characterization of ℛ(𝜆) if 𝔤 is the special linear Lie algebra; that is, we give conditions whether a sequence 𝐱 is contained in ℛ(𝜆) or not (Theorem 18). Initially we note that ℛ(𝜆) lives in ℛ^{𝑘=∑𝑖𝑚𝑖} and the first 𝑘 components are nonzero. For simplicity we set 𝑒_𝑙(𝐱)=𝑒(𝐱), 𝑓_𝑙(𝐱)=𝑓(𝐱), 𝜃_𝑙=𝜃, and 𝜌_𝑙=𝜌.

Theorem 18. The crystal ℛ(𝜆) consists of all sequences 𝐱=𝑥₁,…,𝑥_𝑘,0,0,…, (55) such that (1) for all pairs (𝑥_𝑞≠∅_𝑖,𝑥_𝑞+1≠∅_𝑗), say,  𝑥_𝑞=(𝑖₁,𝑖^′₁)⋯(𝑖_𝑠,𝑖^′_𝑠)∈ℛ^𝑖_𝑠 and 𝑥_𝑞+1=(𝑗₁,𝑗^′₁)⋯(𝑗_𝑡,𝑗^′_𝑡)∈ℛ^𝑗_𝑡 one has(i)♯{𝑖_𝑘∣1≤𝑖_𝑘≤𝑗_𝑡−𝑝}≥𝑝+1, for all 0≤𝑝≤𝑢∶=max{0≤𝑟≤𝑡−1∣𝑗_𝑡−𝑟≤𝑖}, (ii)𝑗_𝑡−𝑝≤𝑖^′_{𝑗𝑡−𝑝−𝑝+𝑠−𝑖}, for all 𝑝∈{𝑢+1≤𝑟≤𝑡−1∣𝑗_𝑡−𝑟≤𝑟+𝑖}, (iii)𝑗^′_𝑝≤𝑖^′_{𝑗−𝑖+𝑠−𝑡+𝑝}, for all 1≤𝑝≤𝑖−𝑗+𝑡, (2) there is no pair (𝑥_𝑞,𝑥_𝑞+1) of the form (∅_𝑖,𝑥_𝑞+1≠∅_𝑗) with 𝑖≥𝑗_𝑡.

Proof. First we note that the element 𝑟_𝜆 is contained in ℛ(𝜆) and is a highest weight element. Furthermore, we claim that 𝑟_𝜆 is the unique highest weight element. So suppose that we have another element 𝐱=(𝑥₁,…,𝑥_𝑘,0,0,…) satisfying 𝑒_𝑙̃𝐱=0 for all 𝑙∈𝐼. Let 𝑧 be the lowest integer which appears in one of the sequences 𝑥₁,…,𝑥_𝑘 and let 𝑝 be the minimal integer such that 𝑧 appears in 𝑥_𝑝, say,𝑥_𝑝=(𝑗₁,𝑗^′₁)⋯(𝑗_𝑡=𝑧,𝑗^′_𝑡)∈ℛ^𝑗_𝑡. In the case where such a 𝑧 does not exist, we have 𝐱=𝑟_𝜆. We remark that 𝑒_𝑙̃𝑥_𝑝,0,0⋯=0∀𝑙≥𝑧 (56) would imply the claim, whereby the reason is the following: assume that (56) holds and let 𝑟=max{1≤𝑟≤𝑡∣𝑗_𝑟−1≠𝑗_𝑟+1}. If 𝑗₁=𝑗 and 𝑟=1, we set 𝑖=𝑗^′₁+1 and 𝑖=𝑗_𝑟+1else. In either case we obtain 𝜖_𝑖−1(𝑥_𝑝,0,0,…)≠0, which is a contradiction to (56). So if suffices to show (56).
Primarily we claim that 𝑥₁,…,𝑥_𝑝−1∈{∅₁,…,∅_𝑛}. Assume that the element 𝑥_𝑝−1=(𝑖₁,𝑖^′₁)⋯(𝑖_𝑠,𝑖^′_𝑠)∈ℛ^𝑖_𝑠 is not in the aforementioned set; then the property (1)(i) is violated by the choice of 𝑧 and 𝑝 because 0=♯𝑖_𝑘∣1≤𝑖_𝑘≤𝑧≥1. (57) Hence, there exists a pair (∅_𝑗,𝑥_𝑝) which forces 𝑗<𝑧 and thus, again by the choice of 𝑧, the remaining sequences are again contained in the set {∅₁,…,∅_𝑛}. As a consequence, a short calculation by using Lemma 10 shows the uniqueness of the a highest weight element, namely, 𝜖_𝑙(𝐱)=0∀𝑙∈𝐼⟹𝜖_𝑙𝑥_𝑝,…,𝑥_𝑘,0,0⋯=0∀𝑙≥𝑧⟹𝜖_𝑙𝑥_𝑝,0,0⋯=0∀𝑙≥𝑧. (58) In order to obtain a connected crystal it remains to show that 𝑒_𝑙̃𝐱,𝑓_𝑙𝐱∈ℛ(𝜆)∪{0}. Assume that 𝑓_𝑙𝐱≠0, say,𝑓_𝑙𝐱=…,𝑥_{𝑓(𝐱)−1},𝜃𝑥_𝑓(𝐱),𝑥_𝑓(𝐱)+1,…, (59) where we set for simplicity 𝑓(𝐱)=𝑞. Our goal here is to show that the properties (1) and (2) hold for 𝑓_𝑙𝐱, 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:(i)(𝑥_𝑞−1,𝑥_𝑞)=(∅_𝑖,∅_𝑖)and𝑙=𝑖,(ii)𝑥_𝑞=(𝑖₁,𝑖^′₁)⋯(𝑖_𝑠,𝑖^′_𝑠)∈ℛ^𝑖_𝑠,𝑥_𝑞−1=∅_𝑖𝑠−1and𝑙=𝑖_𝑠−1<𝑖. In either case we obtain 𝜎_𝑙^𝑞−1(𝐱)−𝜏_𝑙^𝑞−1(𝐱)<𝜎^𝑞_𝑙(𝐱)−𝜏^𝑞_𝑙(𝐱), which is a contradiction to the choice of 𝑞. The proof of the fact that (1) holds will proceed in several cases. In the remaining parts we denote the entries of 𝜃(𝑥_𝑞) by 𝜃(𝑖_𝑘),𝜃(𝑖^′_𝑘), that is, 𝜃𝑥_𝑞=𝜃𝑖₁,𝜃𝑖^′₁⋯𝜃𝑖_{𝑠+𝛿𝑙,𝑖},𝜃𝑖^′_{𝑠+𝛿𝑙,𝑖}. (60)
Case 1.1 ((1) is violated for the pair (𝜃(𝑥_𝑞),𝑥_𝑞+1)). Let 𝑥_𝑞=(𝑖₁,𝑖^′₁)⋯(𝑖_𝑠,𝑖^′_𝑠)∈ℛ^𝑖_𝑠 and 𝑥_𝑞+1=(𝑗₁,𝑗^′₁)⋯(𝑗_𝑡,𝑗^′_𝑡)∈ℛ^𝑗_𝑡. We first consider the case 𝑙<𝑖, which means that we replace 𝑙+1 by 𝑙. Since the entries 𝑖^′_𝑝 stay unchanged, only property (1)(i) can be violated. However, property (1)(i) is still fulfilled because ♯𝑖_𝑘∣1≤𝑖_𝑘≤𝑗_𝑡−𝑝≤♯𝜃𝑖_𝑘∣1≤𝜃𝑖_𝑘≤𝑗_𝑡−𝑝,∀0≤𝑝≤𝑢. (61) If 𝑙>𝑖 we replace 𝑙−1 by 𝑙 and hence (1) is obviously fulfilled. So suppose that 𝑙=𝑖, which means that we add the entry (𝑙,𝑙). The equality ♯𝑖_𝑘∣1≤𝑖_𝑘≤𝑗_𝑡−𝑝=♯𝜃𝑖_𝑘∣1≤𝜃𝑖_𝑘≤𝑗_𝑡−𝑝,∀0≤𝑝≤𝑢−1 (62) and inequality ♯𝑖_𝑘∣1≤𝑖_𝑘≤𝑗_𝑡−𝑢≤♯𝜃𝑖_𝑘∣1≤𝜃𝑖_𝑘≤𝑗_𝑡−𝑢 (63) imply that property (1)(i) is still fulfilled. Furthermore, since 𝜃(𝑖^′_𝑘)=𝑖^′_𝑘−1 for 𝑘>1, 𝜃(𝑖^′₁)=𝑖, and 𝜃(𝑥_𝑞)∈ℛ^𝑖_𝑠+1, we obtain 𝑗_𝑡−𝑝≤𝑖^′_{𝑗𝑡−𝑝−𝑝+𝑠−𝑖}=𝜃𝑖^′_{𝑗𝑡−𝑝−𝑝+𝑠+1−𝑖},∀𝑝∈𝑢+1≤𝑟≤𝑡−1∣𝑗_𝑡−𝑟,𝑗≤𝑟+𝑖_′𝑝≤𝑖^′_{𝑗−𝑖+𝑠−𝑡+𝑝}=𝜃𝑖^′_{𝑗−𝑖+𝑠+1−𝑡+𝑝},∀1≤𝑝≤𝑖−𝑗+𝑡, (64) which particularly means that Case 1.1 can never appear.
Case 1.2 ((1)(i) is violated for the pair (𝑥_𝑞−1,𝜃(𝑥_𝑞))). For simplicity, we set 𝑥_𝑞−1=(𝑖₁,𝑖^′₁)⋯(𝑖_𝑠,𝑖^′_𝑠)∈ℛ^𝑖_𝑠 and 𝑥_𝑞=(𝑗₁,𝑗^′₁)⋯(𝑗_𝑡,𝑗^′_𝑡)∈ℛ^𝑗_𝑡. In that case there exists at least one 0≤𝑝≤𝜃(𝑢)=max{𝑟∣𝜃(𝑗_𝑡−𝑟)≤𝑖}, such that ♯𝑖_𝑘∣1≤𝑖_𝑘≤𝜃𝑗_𝑡−𝑝<𝑝+1. (65) Necessarily we must have 𝑙≤𝑖 and in the case where 𝑙<𝑗 (65) implies ♯𝑖_𝑘∣1≤𝑖_𝑘≤𝑙+1>♯𝑖_𝑘∣1≤𝑖_𝑘,♯𝑖≤𝑙_𝑘∣1≤𝑖_𝑘≤𝑙−1=♯𝑖_𝑘∣1≤𝑖_𝑘≤𝑙. (66) As a consequence we get that 𝑙+1 appears in 𝑥_𝑞−1 while 𝑙 does not appear and thus 𝜎_𝑙^𝑞−1(𝐱)−𝜏_𝑙^𝑞−1(𝐱)<𝜎^𝑞_𝑙(𝐱)−𝜏^𝑞_𝑙(𝐱), which is a contradiction to the choice of 𝑞.
Eventually if 𝑙=𝑗=𝑖, we obtain in a similar way 𝑡≥♯𝑖_𝑘∣1≤𝑖_𝑘≤𝜃𝑗₁𝑖=𝑗=𝑠≥♯_𝑘∣1≤𝑖_𝑘≤𝑗₁≥𝑡, (67) which forces on the one hand 𝑠=𝑡 and on the other hand that 𝑗 does not appear in 𝑥_𝑞−1. To be more precise, we can conclude the latter statement with the help of (1)(iii) and (1)(i), namely, 𝑗<𝑗^′₁≤𝑖^′_{𝑠−𝑡+1}=𝑖^′₁,𝑠≥♯𝑖_𝑘∣1≤𝑖_𝑘<𝑖≥♯𝑖_𝑘∣1≤𝑖_𝑘≤𝑗₁≥𝑡=𝑠. (68) Thus, we get again 𝜎_𝑙^𝑞−1(𝐱)−𝜏_𝑙^𝑞−1(𝐱)<𝜎^𝑞_𝑙(𝐱)−𝜏^𝑞_𝑙(𝐱), which is once more a contradiction to the choice of 𝑞.
Case 1.3 ((1)(ii) is violated for the pair (𝑥_𝑞−1,𝜃(𝑥_𝑞))). Here we have 𝑙≤𝑗 and if 𝑙=𝑗, we must have 𝑗>𝑖 because otherwise (1)(ii) would not be violated. We first consider the case where 𝑙=𝑗 and notice that the only possible violation is given by the following inequality: 𝑗>𝑖^′_{𝑗−𝑡+𝑠−𝑖}. (69) We can conclude that 𝑗 does not occur in 𝑥_𝑞−1 because either 𝑗=𝑡+𝑖 and hence 𝑖^′_𝑠<𝑗 (70) or 𝑗+1≤𝑡+𝑖 and (1)(iii) is applicable, which yields 𝑗<𝑗^′₁≤𝑖^′_{𝑗−𝑡+𝑠−𝑖+1}. To obtain a contradiction we have to show that 𝑗−1 appears in 𝑥_𝑞−1. If 𝑗₁>𝑖, this follows by the subsequent calculation: 𝑖^′_{𝑗−𝑡+𝑠−𝑖}≤𝑗−1≤𝑗₁+𝑗−1−𝑗₁≤𝑖^′_{𝑗1−𝑡+1+𝑠−𝑖}+𝑗−1−𝑗₁≤𝑖^′_{𝑗−𝑡+𝑠−𝑖}. (71) If 𝑗₁≤𝑖, we obtain with property (1)(i) that 𝑠≥{𝑖_𝑘∣1≤𝑖_𝑘≤𝑗₁}≥𝑡. In particular we actually have 𝑠=𝑡 because otherwise we would get 𝑗≤𝑗−𝑡+𝑠−1≤𝑖^′_{𝑗−𝑡+𝑠−𝑖}. (72) Eventually we can conclude again that 𝑗−1 must appear in 𝑥_𝑞−1𝑖^′_𝑗−𝑖≤𝑗−1≤𝑖^′_𝑗−𝑖. (73) Now we suppose 𝑙<𝑗. Then an easy consideration shows that (1)(ii) can only be violated if 𝑙>𝑖 and thus we obtain similarly as before that the only violation which can occur is the following: 𝑙>𝑖^′_{𝑙−𝑟+𝑠−𝑖}, (74) where we expect 𝑗_𝑡−𝑟=𝑙+1. We would like to show as before that 𝑙 does not appear in 𝑥_𝑞−1 while 𝑙−1 appears. We either have 𝑙=𝑟+𝑖 and thus 𝑙>𝑖^′_𝑠 or 𝑙+1≤𝑟+𝑖. In the latter case we apply property (1)(ii) and obtain 𝑙<𝑖^′_{𝑙−𝑟+𝑠−𝑖+1}. (75) In either case we notice that 𝑙 does not occur in 𝑥_𝑞−1.
In order to prove the remaining part we consider the element 𝑗_{𝑡−(𝑟−1)} which is considerable, since 𝑙−𝑟+𝑠−1≤𝑖^′_{𝑙−𝑟+𝑠−𝑖}≤𝑙−1⟹1≤𝑠≤𝑟. (76) In the case where 𝑗_{𝑡−(𝑟−1)} is greater than 𝑖, we obtain 𝑖^′_{𝑙−𝑟+𝑠−𝑖}≤𝑙−1≤𝑗_{𝑡−(𝑟−1)}+𝑙−1−𝑗_{𝑡−(𝑟−1)}≤𝑖^′_{𝑗𝑡−(𝑟−1)−𝑟+1+𝑠−𝑖}+𝑙−1−𝑗_{𝑡−(𝑟−1)}≤𝑖^′_{𝑙−𝑟+𝑠−𝑖} (77) and otherwise by using property (1)(i) we get 𝑟=𝑠. Thus, 𝑙−1≥𝑖^′_𝑙−𝑖≥𝑙−1. (78)
Case 1.4 ((1)(iii) is violated for the pair (𝑥_𝑞−1,𝜃(𝑥_𝑞))). We suppose that (1)(iii) is violated, which forces 𝑙≥𝑗. In the case where 𝑙>𝑗, we have ∃𝑝∶𝑗^′_𝑝=𝑙−1=𝑖^′_{𝑗−𝑖+𝑠−𝑡+𝑝}. (79) It follows that 𝑙−1 appears and 𝑙 does not appear in 𝑥_𝑞−1 because 𝑙=𝑖^′_{𝑗−𝑖+𝑠−𝑡+𝑝+1} would imply 𝑙=𝑖^′_{𝑗−𝑖+𝑠−𝑡+𝑝+1}≥𝑗^′_𝑝+1>𝑗^′_𝑝=𝑙−1⟹𝑗^′_𝑝+1=𝑙. (80) Consequently, 𝜎_𝑙^𝑞−1(𝐱)−𝜏_𝑙^𝑞−1(𝐱)<𝜎^𝑞_𝑙(𝐱)−𝜏^𝑞_𝑙(𝐱) and we obtain as usual a contradiction to the choice of 𝑞.
If the remaining case 𝑙=𝑗 occurs, then one of the inequalities 𝜃𝑗^′_𝑝≤𝑖^′_{𝑗−𝑖+𝑠−𝑡−1+𝑝},1≤𝑝≤𝑖−𝑗+𝑡+1 (81) must be violated. Clearly, the only possibility is that 𝑗≤𝑖^′_{𝑗−𝑖+𝑠−𝑡} does not hold. If 𝑡=𝑗−𝑖, we get 𝑗>𝑖^′_𝑠 (82) and else we can assume 𝑗−𝑖+1≤𝑡 so that (1)(iii) is applicable, which yields 𝑖≤𝑖^′_{𝑗−𝑖+𝑠−𝑡}<𝑗<𝑗^′₁≤𝑖^′_{𝑗−𝑖+𝑠−𝑡+1}. (83) Therefore, 𝑗 does not appear in 𝑥_𝑞−1. In what follows, we finish our proof by showing that 𝑗−1 appears in 𝑥_𝑞−1. If 𝑗₁>𝑖, we can apply (1)(ii) and get 𝑖^′_{𝑗−𝑖+𝑠−𝑡}≤𝑗₁+𝑗−1−𝑗₁≤𝑖^′_{𝑗1−𝑡+1+𝑠−𝑖}+𝑗−1−𝑗₁≤𝑖^′_{𝑗−𝑖+𝑠−𝑡}. (84) If 𝑗₁≤𝑖, we can verify with (1)(i) that 𝑠≥𝑡, but the assumption 𝑠−1≥𝑡  yields in a contradiction, namely, 𝑖^′_{𝑗−𝑖+𝑠−𝑡}≥𝑗+𝑠−𝑡−1≥𝑗. (85) So 𝑠=𝑡 and we obtain the required equality 𝑗−1≥𝑖^′_𝑗−𝑖≥𝑗−1.
The proof of 𝑒_𝑙̃𝐱∈ℛ(𝜆)∪{0} is similar, which completes the proof.

5.2. Explicit Description of ℛ(𝜆) in Type 𝐶_𝑛

In this subsection we would like to give an explicit characterization of ℛ(𝜆) if 𝔤 is a symplectic Lie algebra. In order to state the main theorem we fix some notation.

For an arbitrary subset 𝐴⊆𝐈, 𝑛∈{0,1}, and 𝑦∈𝐈, we set 𝛿_𝑦,𝐴=1,if𝑦∈𝐴,0,if𝑦∉𝐴,𝑛𝐴=𝐴,if𝑛=1,∅,if𝑛=0. (86) The analogue result to Theorem 18 for type 𝐶_𝑛 is the following.

Theorem 19. The crystal ℛ(𝜆) consists of all sequences 𝐱=𝑥₁,…,𝑥_𝑘,0,0,…, (87) such that (1)for all pairs (𝑥_𝑞≠∅_𝑖,𝑥_𝑞+1≠∅_𝑗), say,   𝑥_𝑞=(𝑖₁,𝑖^′₁)⋯(𝑖_𝑠,𝑖^′_𝑠)∈ℛ^𝑖_𝑠 and 𝑥_𝑞+1=(𝑗₁,𝑗^′₁)⋯(𝑗_𝑡,𝑗^′_𝑡)∈ℛ^𝑗_𝑡 one has(i)♯{𝑖_𝑘∣1≤𝑖_𝑘≤𝑗_𝑡−𝑝}≥𝑝+1, for all 0≤𝑝≤𝑢∶=max{0≤𝑟≤𝑡−1∣𝑗_𝑡−𝑟≤𝑖}, (ii)𝑗_𝑡−𝑝≤𝑖^′_{𝑗𝑡−𝑝−𝑝+𝑠−𝑖}, for all 𝑝∈{𝑢+1≤𝑟≤𝑡−1∣𝑗_𝑡−𝑟≤𝑟+𝑖}, (iii)𝑗^′_𝑝≤𝑖^′_{𝑗−𝑖+𝑠−𝑡+𝑝}, for all 1≤𝑝≤𝑖−𝑗+𝑡, (2)there is no pair (𝑥_𝑞,𝑥_𝑞+1) of the form (∅_𝑖,𝑥_𝑞+1≠∅_𝑗) with 𝑖≥𝑗_𝑡, (3)there is no pair (𝑥_𝑞≠∅_𝑖,𝑥_𝑞+1) with the following property: there exists 𝑖^′_𝑝≥𝑖^′_𝑟≥𝑛 with(a)𝑖^′_𝑟−𝛿_{𝑖′𝑟,{𝑖+1,…,𝑛}}∉(1−𝛿_{𝑖′𝑟,{𝑖+1,…,𝑛}}){𝑖₁,…,𝑖_𝑠}∪𝛿_{𝑖′𝑟,{𝑖+1,…,𝑛}}(𝐈−{𝑖^′₁,…,𝑖^′_𝑠}), (b)𝑖^′_𝑝−𝛿_{𝑖′𝑝,{𝑗+1,…,𝑛}}∉(1−𝛿_{𝑖′𝑝,{𝑗+1,…,𝑛}}){𝑗₁,…,𝑗_𝑡}∪𝛿_{𝑖′𝑝,{𝑗+1,…,𝑛}}(𝐈−{𝑗^′₁,…,𝑗^′_𝑡}), (c)(𝑝−𝑟)+♯{𝑗_𝑘∣𝑗_𝑘<𝑖^′_𝑝}−♯{𝑖_𝑘∣𝑖_𝑘<𝑖^′_𝑟}+♯{𝑖^′_𝑘∣𝑖^′_𝑘<𝑖^′_𝑟}−♯{𝑗^′_𝑘∣𝑗^′_𝑘<𝑖^′_𝑝} ≥ max{0,𝑖^′_𝑟−𝑖}−max{0,𝑖^′_𝑝−𝑗},(4)there is no pair (𝑥_𝑞≠∅_𝑖,𝑥_𝑞+1≠∅_𝑗) with the following property: there exists 𝑖^′_𝑝≥𝑗^′_𝑟≥𝑛 with(a)𝑖^′_𝑝−𝛿_{𝑖′𝑝,{𝑗+1,…,𝑛}}∉(1−𝛿_{𝑖′𝑝,{𝑗+1,…,𝑛}}){𝑗₁,…,𝑗_𝑡}∪𝛿_{𝑖′𝑝,{𝑗+1,…,𝑛}}(𝐈−{𝑗^′₁,…,𝑗^′_𝑡}), (b)𝑗^′_𝑟−𝛿_{𝑗′𝑟,{𝑗+1,…,𝑛}}∉(1−𝛿_{𝑗′𝑟,{𝑗+1,…,𝑛}}){𝑗₁,…,𝑗_𝑡}∪𝛿_{𝑗′𝑟,{𝑗+1,…,𝑛}}(𝐈−{𝑗^′₁,…,𝑗^′_𝑡}), (c)(𝑖−𝑗)+(𝑡−𝑠)+(𝑝−𝑟)+♯{𝑗^′_𝑘∣𝑖^′_𝑝≤𝑗^′_𝑘<𝑗^′_𝑟}−♯{𝑗_𝑘∣𝑖^′_𝑝≤𝑗_𝑘<𝑗^′_𝑟} ≥ max{0,𝑗^′_𝑟−𝑗}−max{0,𝑖^′_𝑝−𝑗}.

Proof. First of all we note as in the 𝐴_𝑛 case that the element 𝑟_𝜆 is contained in ℛ(𝜆) and is a highest weight element. In order to prove that 𝑟_𝜆 is the unique highest weight element in ℛ(𝜆) we assume 𝐱=(𝑥₁,…,𝑥_𝑘,0,0,…) to be another one. Let 𝑧 be the lowest integer which appears in one of the sequences 𝑥₁,…,𝑥_𝑘 and let 𝑝 be the minimal integer such that 𝑧 appears in 𝑥_𝑝, say,𝑥_𝑝=(𝑖₁,𝑖^′₁)⋯(𝑖_𝑠=𝑧,𝑖^′_𝑠)∈ℛ^𝑖_𝑠. We can prove similar to Theorem 18 that the elements 𝑥₁,…,𝑥_𝑝 are contained in {∅₁,…,∅_𝑛}. Accordingly we get once more with Lemma 10 (similar to the 𝐴_𝑛 case) 𝑒_𝑙̃𝐱=0∀𝑙∈𝐼⟹𝑒_𝑙̃𝑥_𝑝,…,𝑥_𝑘𝑒̃,0,0⋯=0∀𝑙≥𝑧⟹_𝑙𝑥_𝑝,0,0⋯=0∀𝑙≥𝑧. (88) Our aim is again to prove the impossibility of 𝑒_𝑙̃((𝑥_𝑝,0,0⋯))=0forall𝑙≥𝑧. Let 𝑟=max{1≤𝑟≤𝑠∣𝑖_𝑟−1≠𝑖_𝑟+1}. If 𝑖₁=𝑖, 𝑟=1, and 𝑖^′₁∉{𝑛−1,…,1}, we set 𝑗=𝑖^′₁+1 and if 𝑖₁=𝑖,𝑟=1 is not satisfied, we set 𝑗=𝑖_𝑟+1 and obtain similar to Theorem 18 that 𝜖_𝑗−1(𝑥_𝑝,0,0,…)≠0. Thus, the only remaining case which can appear is when 𝑖₁=𝑖, 𝑟=1, and 𝑖^′₁∈{𝑛−1,…,1}. In this particular case we set 𝑗=𝑖^′₁ and claim 𝜖_𝑗(𝑥_𝑝,0,0,…)≠0. The latter claim is true because on the one hand we have 𝑖^′₁∈𝑖^′₁,…,𝑖^′_𝑠,𝑖^′₁+1∉𝑖^′₁,…,𝑖^′_𝑠 (89) and on the other hand we got 𝑗=𝑖^′₁≤𝑖₁=𝑖⇒𝑗≥𝑖, which verifies the properties listed in (d). To be more precise, if 𝑗>𝑖, we have 𝑗−1∉{𝑖^′₁,…,𝑖^′_𝑠} and if 𝑗=𝑖, we have 𝑗=𝑖=𝑖₁.
In order to finish the theorem it remains to show that ℛ(𝜆) is stable under the Kashiwara operators, that is, 𝑒_𝑙̃𝐱,𝑓_𝑙𝐱∈ℛ(𝜆)∪{0}. Assume that 𝑓_𝑙𝐱≠0, say,𝑓_𝑙𝐱=…,𝑥_𝑞−1,𝜃𝑥_𝑞,𝑥_𝑞+1,…. (90) Our aim here is to prove that the properties (1)--(4) hold for 𝑓_𝑙𝐱, whereby the verification of the first and the second properties proceeds almost similar to Theorem 18. Nevertheless we will demonstrate some parts of it in Case 1. In the remaining parts of our proof, we set 𝑥_𝑞=(𝑖₁,𝑖^′₁)⋯(𝑖_𝑠,𝑖^′_𝑠)∈ℛ^𝑖_𝑠 and 𝑥_𝑞+1=(𝑗₁,𝑗^′₁)⋯(𝑗_𝑡,𝑗^′_𝑡)∈ℛ^𝑗_𝑡, whenever they are contained in ℛ−{∅₁,…,∅_𝑛}. We will divide our proof into several cases.
Case  1 (𝑙±1↦𝑙 or +(𝑙,𝑙)). Here we assume that the action of the Kashiwara operator on 𝐱 is given in a way such that 𝑥_𝑞 satisfies property (a). It means that we either replace 𝑙±1 by 𝑙 or add the pair (𝑙,𝑙) as described in Section 3. Here we consider again several cases, where each case assumes that a condition described in Theorem 19 is violated.
Subcase 1.1 ((1)(i) is violated for the pair (𝑥_𝑞−1,𝜃(𝑥_𝑞)). Since (1)(i) is violated, there must exist an element 𝜃(𝑗_{𝑡+𝛿𝑙,𝑗−𝑟})=𝑙 such that ♯𝑖_𝑘∣1≤𝑖_𝑘≤𝑙≤𝑟. (91) Further, as in Subcase 1.2 of Theorem 18, one can verify that 𝑙 does not appear in 𝑥_𝑞−1 and if in addition 𝑙≤𝑖<𝑗 holds, then 𝑙+1 occurs in 𝑥_𝑞−1. Consequently 𝑙 must be contained in {𝑖^′₁,…,𝑖^′_𝑠} because otherwise we would obtain a contradiction to the choice of 𝑞. Hence, if we take 𝑖^′_𝑝=𝑖^′_𝑟=𝑙, it follows immediately that the properties (a), (b), and (c) in (3) hold for the pair (𝑥_𝑞−1,𝑥_𝑞), which is impossible. For instance (c) is fulfilled with (91), since ♯𝑖_𝑘∣1≤𝑖_𝑘≤𝑙=♯𝑖_𝑘∣𝑖_𝑘<𝑙≤𝑟≤♯𝑗_𝑘∣𝑗_𝑘<𝑙. (92) 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.
Subcase 1.2 ((3) is violated for the pair (𝜃(𝑥_𝑞),𝑥_𝑞+1)). A simple case-by-case observation shows that this case can only occur if there exists 𝜃(𝑖^′_𝑝)≥𝜃(𝑖^′_𝑟)=𝑙+1, such that (a), (b) and (c) are satisfied. Suppose that there exists an element 𝑚 in the set {𝜃(𝑖^′_𝑟+1),…,𝜃(𝑖^′_𝑝)}(𝑚=𝜃(𝑖^′_𝑟+𝑢)), such that 𝑚−𝛿_{𝑚,{𝑖+1,…,𝑛}}∉1−𝛿_{𝑚,{𝑖+1,…,𝑛}}𝑖₁,…,𝑖_𝑠∪𝛿_{𝑚,{𝑖+1,…,𝑛}}𝐈−𝑖^′₁,…,𝑖^′_𝑠 (93) and 𝑚 is minimal with this property. In the case where such an element does not exist we set 𝑚=𝜃(𝑖^′_𝑝). We claim the following.
Claim. Let 𝑚 be as described before; then 𝑢+min{𝑙,𝑖}−min{𝑚,𝑖}−♯𝑖_𝑘∣𝑚≤𝑖_𝑘𝑖<𝑙+♯^′_𝑘∣𝑚≤𝑖^′_𝑘<𝑙≤𝑙+1−𝑚. (94)
Proof of the Claim. We consider again various cases, starting with (i)𝑙≤𝑖: the minimality of 𝑚 implies 𝜃(𝑖^′_𝑟+1),…,𝜃(𝑖^′_{𝑟+𝑢−1})∈{𝑖₁,…,𝑖_𝑠}, which yields ♯𝑖^′_𝑘∣𝜃𝑖^′_𝑟<𝑖^′_𝑘<𝑚_=𝑢−1≤♯𝑖_𝑘∣𝑚≤𝑖_𝑘𝑖<𝑙⟺♯^′_𝑘∣𝑚≤𝑖^′_𝑘<𝑙−♯𝑖_𝑘∣𝑚≤𝑖_𝑘<𝑙≤−𝑢+1, (95)(ii)𝑙>𝑖 and 𝑚≤𝑖: using the minimality of 𝑚 we obtain similarly 𝜃(𝑖^′_{𝑟+𝑢−1}),…,𝜃(𝑖^′_𝑟+𝑥)∈{𝑖₁,…,𝑖_𝑠} and 𝜃(𝑖^′_𝑟+1)−1,…,𝜃(𝑖^′_{𝑟+𝑥−1})−1∉{𝑖^′₁,…,𝑖^′_𝑠}, for some integer 𝑥. Therefore, the following calculation implies (94): 𝑖−♯𝑖_𝑘∣𝑚≤𝑖_𝑘+♯𝑖^′_𝑘∣𝑖^′_𝑘()𝑖<𝑙≤𝑙−𝑥−1−♯_𝑘∣𝑚≤𝑖_𝑘≤𝑙−♯𝑖^′_𝑘∣𝜃𝑖^′_𝑟<𝑖^′_𝑘<𝑚, (96)(iii)𝑚>𝑖: the minimality of 𝑚 provides 𝜃(𝑖^′_𝑟+1)−1,…,𝜃(𝑖^′_{𝑟+𝑢−1})−1∉{𝑖^′₁,…,𝑖^′_𝑠}. Hence, ♯𝑖^′_𝑘∣𝑚≤𝑖^′_𝑘<𝑙≤𝑙−𝑚−♯𝑖^′_𝑘∣𝜃𝑖^′_𝑟<𝑖^′_𝑘<𝑚, (97) which finishes the proof of the claim.
As a consequence of (94) we obtain ♯𝑗_𝑘∣𝑗_𝑘<𝜃𝑖^′_𝑝−♯𝜃𝑖_𝑘∣𝜃𝑖_𝑘<𝜃𝑖^′_𝑟+♯𝜃𝑖^′_𝑘∣𝜃𝑖^′_𝑘<𝜃𝑖^′_𝑟−♯𝑗^′_𝑘∣𝑗^′_𝑘<𝜃𝑖^′_𝑝≤♯𝑗_𝑘∣𝑗_𝑘<𝑖^′_{𝑝−𝛿𝑙,𝑖}−♯𝑖_𝑘∣𝑚≤𝑖_𝑘𝑖<𝑙−♯_𝑘∣𝑖_𝑘𝑖<𝑚+♯^′_𝑘∣𝑚≤𝑖^′_𝑘<𝑙+♯𝑖^′_𝑘∣𝑖^′_𝑘𝑗<𝑚−♯^′_𝑘∣𝑗^′_𝑘<𝑖^′_{𝑝−𝛿𝑙,𝑖}<𝑙+1−𝑚−min{𝑙,𝑖}+min{𝑚,𝑖}+max{0,𝑚−𝑖}−max0,𝑖^′_{𝑝−𝛿𝑙,𝑖}−{}−𝑗𝑝−𝑟=𝑙+1−min𝑙,𝑖−max0,𝑖^′_{𝑝−𝛿𝑙,𝑖}−{}−𝑗𝑝−𝑟≤max0,𝑙+1−𝑖−max0,𝑖^′_{𝑝−𝛿𝑙,𝑖}−𝑗−𝑝−𝑟+𝛿_{𝑙,{1,…,𝑖−1}}, (98) where the first estimation is strict if 𝑙<𝑖. Hence, we have a contradiction to the assumption that (c) holds.
Subcase 1.3  ((3) is violated for the pair (𝑥_𝑞−1,𝜃(𝑥_𝑞))). In that case there exists 𝑖^′_𝑝=𝑙+1≥𝑖^′_𝑟, such that (a), (b), and (c) is satisfied. Therefore, we must have 𝑙=𝑖^′_𝑝+1 because otherwise we obtain a contradiction to the choice of 𝑞. It follows that 𝑝−𝑟+♯𝜃𝑗_𝑘∣𝜃𝑗_𝑘<𝑖^′_𝑝−♯𝑖_𝑘∣𝑖_𝑘<𝑖^′_𝑟+♯𝑖^′_𝑘∣𝑖^′_𝑘<𝑖^′_𝑟−♯𝜃𝑗^′_𝑘∣𝜃𝑗^′_𝑘<𝑖^′_𝑝≤𝑝+1−𝑟+♯𝑗_𝑘∣𝑗_𝑘<𝑖^′_𝑝+1−♯𝑖_𝑘∣𝑖_𝑘<𝑖^′_𝑟+♯𝑖^′_𝑘∣𝑖^′_𝑘<𝑖^′_𝑟−♯𝑗^′_𝑘∣𝑗^′_𝑘<𝑖^′_𝑝+1<max0,𝑖^′_𝑟−i−max0,𝑙−𝑗≤max0,𝑖^′_𝑟−𝑖−max0,𝑙+1−𝑗+𝛿_{𝑙,{𝑗,…,𝑛}}, (99) where the first estimation is strict whenever 𝑙≥𝑗 and thus provides a contradiction to (c).
Subcase 1.4 ((4) is violated for the pair (𝜃(𝑥_𝑞),𝑥_𝑞+1)). It is easy to see that this case can never appear.
Subcase 1.5 ((4) is violated for the pair (𝑥_𝑞−1,𝜃(𝑥_𝑞))). In that case we have two possibilities, where we start by supposing that there is 𝑖^′_𝑝=𝑙+1≥𝜃(𝑗^′_𝑟), such that the properties (a), (b), and (c) are fulfilled. Similar to Subcase 1.3, we must have an element 𝑙=𝑖^′_𝑝+1 because otherwise we would obtain a contradiction to the choice of 𝑞. Subsequently we get 𝑖−𝑗+𝑡+𝛿_𝑙,𝑗+𝜃𝑗−𝑠𝑝−𝑟+♯^′_𝑘∣𝑖^′_𝑝≤𝜃𝑗^′_𝑘<𝜃𝑗^′_𝑟−♯𝜃𝑗_𝑘∣𝑖^′_𝑝≤𝜃𝑗_𝑘<𝜃𝑗^′_𝑟≤𝑖−𝑗+𝑡+𝛿_𝑙,𝑗+𝑗−𝑠𝑝+1−𝑟+♯^′_𝑘∣𝑙<𝑗^′_𝑘≤𝜃𝑗^′_𝑟−♯𝑗_𝑘∣𝑙≤𝑗_𝑘<𝜃𝑗^′_𝑟<max0,𝜃𝑗^′_𝑟−𝑗−max0,𝑙−𝑗≤max0,𝜃𝑗^′_𝑟−𝑗−max0,𝑙+1−𝑗+𝛿_{𝑙,{𝑗,…,𝑛}}, (100) where the first estimation is strict provided that𝑙≥𝑗 meaning that this calculation contradicts once more property (c).
The last and second possibility which can occur is that there exists 𝑖^′_𝑝≥𝜃(𝑗^′_𝑟)=𝑙+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 𝑚∈𝜃𝑗^′_𝑟+1,…,𝜃𝑗^′_{𝑡+𝛿𝑙,𝑗},say,𝑚=𝜃𝑗^′_𝑢=𝑗^′_{𝑢−𝛿𝑙,𝑗}, (101) such that 𝑚−𝛿_{𝑚,{𝑗+1,…,𝑛}}∉1−𝛿_{𝑚,{𝑗+1,…,𝑛}}𝑗₁,…,𝑗_𝑡∪𝛿_{𝑚,{𝑗+1,…,𝑛}}𝐈−𝑗^′₁,…,𝑗^′_𝑡 (102) and 𝑚≥𝑖^′_𝑝. If such an element exists we choose 𝑚 maximal with this property and otherwise we set 𝑚=𝑖^′_𝑝. Using the maximality, we can verify similar to (94) the correctness of (𝑢−𝑟)−♯𝑗_𝑘∣𝑚≤𝑗_𝑘<𝑙+♯𝑗^′_𝑘∣𝑚≤𝑗^′_𝑘<𝑙+min𝑙,𝑗−min𝑚,𝑗≤𝑙+1−𝑚. (103) As a corollary we obtain as usual a contradiction to property (c) (recall that 𝜃(𝑥_𝑞)∈ℛ^𝑗_{𝑡+𝛿𝑙,𝑗}) 𝑖−𝑗+𝑡+𝛿_𝑙,𝑗+𝜃𝑗−𝑠𝑝−𝑟+♯^′_𝑘∣𝑖^′_𝑝≤𝜃𝑗^′_𝑘<𝜃𝑗^′_𝑟−♯𝜃𝑗_𝑘∣𝑖^′_𝑝≤𝜃𝑗_𝑘<𝜃𝑗^′_𝑟≤𝑖−𝑗+𝑡+𝛿_𝑙,𝑗++()𝑗−𝑠𝑝−𝑢𝑢−𝑟+♯^′_𝑘∣𝑖^′_𝑝≤𝑗^′_𝑘𝑗<𝑚+♯^′_𝑘∣𝑚≤𝑗^′_𝑘<𝑙−♯𝑗_𝑘∣𝑖^′_𝑝≤𝑗_𝑘𝑗<𝑚−♯_𝑘∣𝑚≤𝑗_𝑘<𝑙<max0,𝑚−𝑗−max0,𝑖^′_𝑝−𝑗+𝑙+1−𝑚−min𝑙,𝑗+min𝑚,𝑗≤𝑙+1−min𝑙,𝑗−max0,𝑖^′_𝑝−𝑗≤max0,𝑙+1−𝑗−max0,𝑖^′_𝑝−𝑗+𝛿_{𝑙,{1,…,𝑗−1}}, (104) where the first estimation is strict whenever 𝑙<𝑗 is satisfied.
Case 2 (𝑙+1↦𝑙). Now we assume that the action of the Kashiwara operator on 𝐱 is given in a way such that 𝑥_𝑞 satisfies property (b) while (a) is violated, which in particular means that we replace the entry 𝑙+1 by 𝑙. 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 (𝑥_𝑞−1,𝜃(𝑥_𝑞))). It is easy to see that this case can never appear.
Subcase 2.2 ((3) is violated for the pair (𝜃(𝑥_𝑞),𝑥_𝑞+1)). The first possibility which can occur is that there exists 𝜃(𝑖^′_𝑝)=𝑖^′_𝑝≥𝜃(𝑖^′_𝑟)=𝑙, such that the properties (a), (b), and (c) are fulfilled. Because of (a) and 𝑖^′_𝑟=𝑙+1, we must have 𝑙+1−𝛿_{𝑙+1,{𝑖+1,…,𝑛}}∉1−𝛿_{𝑙+1,{𝑖+1,…,𝑛}}𝑖₁,…,𝑖_𝑠∪𝛿_{𝑙+1,{𝑖+1,…,𝑛}}𝐈−𝑖^′₁,…,𝑖^′_𝑠, (105) since otherwise we would obtain that 𝑥_𝑞 satisfies (a) (from Section 3) and thus the Kashiwara operator would act as in Case 1. Accordingly we can apply our assumptions to 𝑖^′_𝑝≥𝑖^′_𝑟=𝑙+1 and obtain a contradiction to (c), namely, ♯𝑗_𝑘∣𝑗_𝑘<𝑖^′_𝑝−♯𝜃𝑖_𝑘∣𝜃𝑖_𝑘𝜃𝑖<𝑙+♯^′_𝑘∣𝜃𝑖^′_𝑘<𝑙−♯𝑗^′_𝑘∣𝑗^′_𝑘<𝑖^′_𝑝=♯𝑗_𝑘∣𝑗_𝑘<𝑖^′_𝑝{}{}𝑖+min𝑙,𝑖−min𝑙,𝑖−♯_𝑘∣𝑖_𝑘<𝑙+♯𝑖^′_𝑘∣𝑖^′_𝑘<𝑙−♯𝑗^′_𝑘∣𝑗^′_𝑘<𝑖^′_𝑝<♯𝑗_𝑘∣𝑗_𝑘<𝑖^′_𝑝{}{}𝑖+min𝑙+1,𝑖−min𝑙,𝑖−♯_𝑘∣𝑖_𝑘𝑖<𝑙+1+♯^′_𝑘∣𝑖^′_𝑘<𝑙+1−♯𝑗^′_𝑘∣𝑗^′_𝑘<𝑖^′_𝑝<min{𝑙+1,𝑖}−min{𝑙,𝑖}+max{0,𝑙+1−𝑖}−max0,𝑖^′_𝑝−{}−𝑗𝑝−𝑟=1+max0,𝑙−𝑖−max0,𝑖^′_𝑝−𝑗−𝑝−𝑟. (106) The second and last possibility which can occur in that case isthat there exists 𝜃(𝑖^′_𝑝)=𝑙≥𝜃(𝑖^′_𝑟)=𝑖^′_𝑟, such that the properties (a), (b), and (c) are fulfilled. As before we can assume 𝑖^′_𝑝=𝑙+1. Suppose that there exists 𝑚∈{𝑖^′_𝑟,…,𝑖^′_𝑝} (𝑚=𝑖^′_𝑟+𝑢), such that 𝑚−𝛿_{𝑚,{𝑗+1,…,𝑛}}∉1−𝛿_{𝑚,{𝑗+1,…,𝑛}}𝑗₁,…,𝑗_𝑡∪𝛿_{𝑚,{𝑗+1,…,𝑛}}𝐈−𝑗^′₁,…,𝑗^′_𝑡 (107) and 𝑚 is minimal with this property. If such an element does not exist, we set 𝑚=𝑖^′_𝑟. Similar to (94) one gets ()𝑗𝑝−𝑟+𝑢−min𝑙,𝑗+min𝑚,𝑗−♯_𝑘∣𝑙≤𝑗_𝑘<𝑚+♯𝑗^′_𝑘∣𝑙≤𝑗^′_𝑘<𝑚≤𝑚−𝑙. (108) Using this inequality we arrive once more at a contradiction, namely, ♯𝑗_𝑘∣𝑗_𝑘<𝑙−♯𝜃𝑖_𝑘∣𝜃𝑖_𝑘<𝑖^′_𝑟+♯𝜃𝑖^′_𝑘∣𝜃𝑖^′_𝑘<𝑖^′_𝑟−♯𝑗^′_𝑘∣𝑗^′_𝑘𝑗<𝑙=♯_𝑘∣𝑗_𝑘<𝑚−♯𝑗_𝑘∣𝑙≤𝑗_𝑘𝑖<𝑚−♯_𝑘∣𝑖_𝑘<𝑖^′_𝑟+♯𝑖^′_𝑘∣𝑖^′_𝑘<𝑖^′_𝑟−♯𝑗^′_𝑘∣𝑗^′_𝑘<𝑚+♯𝑗^′_𝑘∣𝑙≤𝑗^′_𝑘<𝑚<max0,𝑖^′_𝑟−𝑖−max0,𝑚−𝑗−𝑝−𝑟−min𝑚,𝑗+min𝑙,𝑗+𝑚−𝑙=max0,𝑖^′_𝑟−𝑖−max0,𝑙−𝑗−𝑝−𝑟. (109) The proof of 𝑒_𝑙̃𝐱∈ℛ(𝜆)∪{0} is similar, which completes the proof.

6. Crystal Bases as Tuples of Integer Sequences

In this section we will verify with Theorem 20 that the crystal ℛ(𝜆) can be identified with the crystal graph 𝐵(𝜆) obtained from Kashiwara's crystal bases theory. Our strategy here is to show that there exists an isomorphism of ℛ(𝜆) onto the connected component of ⨂^𝑛_𝑖=1𝐵(𝜔_𝑖)^⊗𝑚𝑖 containing ⨂^𝑛_𝑖=1𝑏^⊗𝑚𝑖_𝑖, where 𝑏_𝑖 denotes the highest weight element in 𝐵(𝜔_𝑖). For the proofs in type 𝐴_𝑛 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 𝐵(𝜔_𝑖). In type 𝐶_𝑛 we will use a short induction argument to prove our results.

Theorem 20. Let 𝜆 be an arbitrary dominant integral weight and set 𝑘=∑_𝑖𝑚_𝑖 as before. Thenone has the following.(1)If 𝑘=1, one has an isomorphism of crystals ℛ(𝜆)−→^∼𝐵(𝜆).(2)If 𝑘>1 and 𝑗 is the maximal integer such that 𝑚_𝑗≠0, one obtains a strict crystal morphism 𝜙∶ℛ(𝜆)⟶ℛ𝜆−𝜔_𝑗⊗ℛ𝜔_𝑗 (110) mapping 𝐱=(𝑥₁,…,𝑥_𝑘,0,0⋯) to the tensor product 𝐱_𝑘−1⊗𝑥_𝑘, where 𝐱_𝑘−1=(𝑥₁,…,𝑥_𝑘−1,0,0⋯).

Proof. With the help of Proposition 17, the crystal ℛ(𝜔_𝑖) is characterized as ⋃_𝑠ℛ^𝑖_𝑠. In the case where 𝔤 is of type 𝐴_𝑛, we will consider the map 𝑥=𝑖₁,𝑖^′₁⋯𝑖_𝑠,𝑖^′_𝑠↦𝑋, (111) whereby 𝑋 is a pattern as in [16, Definition 2], with filling 𝑎_𝑝,𝑞=1,if𝑝,𝑞=𝑖_𝑟,𝑖^′_𝑟forsome𝑟0,else. (112) By an inspection of the crystal structure on the KR-crystal 𝐵^1,𝑖≅𝐵(𝜔_𝑖) ([16, Section 3.2]), it is easy to see that this map becomes an isomorphism of crystals.
If 𝔤 is of type 𝐶_𝑛, the proof of part (1) will proceed by upward induction on 𝑖. An observation of the crystal graph of 𝑅(𝜔₁)∅⟶₁¹(1,1)⟶²⋯⟶^𝑛−1(1,𝑛−1)⟶^𝑛(1,𝑛_𝑛−1)⟶1,𝑛−1⟶^𝑛−2⋯⟶¹1,1 (113) proves the initial step. Now we assume the correctness of the claim for all integers less than 𝑖, especially we have ℛ(𝜔_𝑖−1)≅𝐵(𝜔_𝑖−1). For the purpose of completing the induction we consider the injective map 𝜂∶ℛ𝜔_𝑖⟶ℛ𝜔_𝑖−1⊗ℛ𝜔₁ (114) given by 𝜂(𝑥)=∅_𝑖−1⊗()∅1,𝑖−1,if𝑠=0_𝑖−1⊗1,𝑖^′₁,if𝑠=1,𝑖₁𝑖=𝑖₁,𝑖−1^{(1−𝛿𝑖1,𝑖)}𝑖₂,𝑖^′₁⋯𝑖_𝑠,𝑖^′_𝑠−1⊗1,𝑖^′_𝑠,else. (115) If 𝜂 would be a strict crystal morphism, we would get ℛ(𝜔_𝑖)≅ℐ𝑚(𝜂)≅𝐵(𝜔_𝑖), which finishes the induction. Therefore, we prove that 𝜂 is a strict crystal morphism, where we consider the cases 𝑠=0 and 𝑠=1,𝑖₁=𝑖 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 𝑠=0, we obtain ∅⟶_𝑖^𝑖(),∅𝑖,𝑖_𝑖−1⊗(1,𝑖−1)=𝜂∅⟶_𝑖^𝑖∅_𝑖−1⊗(1,𝑖)=𝜂((𝑖,𝑖)) (116) and if 𝑠=1,𝑖₁=𝑖, we get(1)if 𝑖^′₁≤𝑛−1  (see Figure 1),(2)if 𝑖^′₁≥𝑛 and 𝑖^′₁≠𝑖  (see Figure 2),(3)if 𝑖^′₁≥𝑛 and 𝑖^′₁=𝑖  (see Figure 3).Thus, from now on we can presume that 𝑠>1 or 𝑠=1,𝑖₁≠𝑖. Let 𝑙∈𝐼 be an arbitrary integer. We set for convenience 𝜂(𝑥)=𝑥₁⊗𝑥₂; then 𝜑_𝑙𝜂(𝑥)=max𝜑_𝑙𝑥₂,𝜑_𝑙𝑥₁+𝜑_𝑙𝑥₂−𝜖_𝑙𝑥₂=𝜑_𝑙𝑥₁+𝜑_𝑙𝑥₂,if𝑙≠𝑖^′_𝑠,𝑖^′_𝑠max0,𝜑_𝑙𝑥₁−1,else. (117) The proof of 𝜑_𝑙(𝜂(𝑥))=𝜑_𝑙(𝑥) is an intensive investigation of the properties (a)–(d) listed in Section 3. To avoid confusion with indices we consider only the following case:(i)𝑙=𝑖^′_𝑠 or 𝑙=𝑖^′_𝑠: the case 𝑙=𝑖^′_𝑠 is very simple because 𝑥 satisfies neither (a) nor (b), which yields 𝜑_𝑙(𝑥)=0. Furthermore, since 𝑙+1=𝑖^′_𝑠+1∉{𝑖^′₁,…,𝑖^′_𝑠}, we get 𝜑_𝑙(𝑥₁)≤1. Consequently 𝜑_𝑙(𝜂(𝑥))=0. So it remains to consider the case 𝑙=𝑖^′_𝑠. Here we claim the following: let (𝑎) be the property which arises from (𝑎) by erasing the condition 𝑙∉{𝑖^′₁,…,𝑖^′_𝑠}, then 𝑥 satisfies (𝑎) if and only if 𝑥₁ satisfies (𝑎).
First we observe that the property 𝑙+1∉{𝑖^′₁,…,𝑖^′_𝑠} is equivalent to 𝑙+1∉{𝑖^′₁,…,𝑖^′_𝑠−1}, so that we can ignore it. We start the proof by assuming that 𝑥₁ satisfies (𝑎) and 𝑥 does not satisfy (𝑎), for example, 𝑙∈{𝑖₁,…,𝑖_𝑠,𝑖^′₁,…,𝑖^′_𝑠}. This case is only possible if 𝑙=𝑖₁=𝑖 and thus 𝑙−1=𝑖−1∈{𝑖^′₁,…,𝑖^′_𝑠−1}, which is a contradiction. Since 𝑙−1∈{𝑖^′₁,…,𝑖^′_𝑠} is always fulfilled if 𝑙>𝑖>𝑖−1, we can assume that 𝑙<𝑖 and additionally 𝑙+1∉{𝑖₁,…,𝑖_𝑠}. Without loss of generality we set 𝑙=𝑖−1 because else we have 𝑙+1∈{𝑖^{(1−𝛿𝑖1,𝑖)}₁,𝑖₂,…,𝑖_𝑠}. As a consequence we get 𝑙+1=𝑖∉𝑖₁,…,𝑖_𝑠⟹𝑖₁≤𝑖−1⟹𝑙≠𝑖−1, (118) which is again a contradiction.
According to these calculations, 𝑥 must satisfy (𝑎). To show the other direction, let 𝑥₁ violate one of the properties in (𝑎), for example, 𝑙∈{(𝑖₁,𝑖−1)^{(1−𝛿𝑖1,𝑖)},𝑖₂,…,𝑖_𝑠,𝑖^′₁,…,𝑖^′_𝑠−1}. Then we must have 𝑙=𝑖−1 and 𝑖₁<𝑖 which violates 𝑙+1∈{𝑖₁,…,𝑖_𝑠}. The only additional possibility which can occur, such that 𝑥₁ does not satisfy (𝑎), is 𝑙<𝑖−1 and 𝑙+1∉{𝑖^{(1−𝛿𝑖1,𝑖)}₁,𝑖₂,…,𝑖_𝑠}. Then we get 𝑙+1=𝑖₁=𝑖, which is a contradiction to 𝑙<𝑖−1 and so the reverse direction is also completed.
According to this we have 𝜑_𝑙(𝑥)=1⇒𝜑_𝑙(𝑥₁)=2 and 𝜑_𝑙(𝑥)=0⇒𝜑_𝑙(𝑥₁)≤1.
To show the existence of a morphism of crystals we have to show (among others) the weight invariance of 𝜂, which is proven by the following calculation: 𝑤𝑡𝜂(𝑥)=𝜔_𝑖−1+𝜔₁−^𝑠−1_𝑗=1𝛼_{𝑖𝑗+1,𝑖′𝑗}−𝛼_{1,𝑖′𝑠}−1−𝛿_𝑖1,𝑖𝛼_{𝑖1,𝑖−1}=𝜔_𝑖−1+𝜔₁−𝛼_1,𝑖−1−^𝑠_𝑗=1𝛼_{𝑖𝑗,𝑖′𝑗}=𝜔_𝑖−^𝑠_𝑗=1𝛼_{𝑖𝑗,𝑖′𝑗}. (119) The verification of 𝜖_𝑙(𝜂(𝑥))=𝜖_𝑙(𝑥) is therefore proven with Definition 3(1) and (119). Suppose now that 𝑓_𝑙𝜂(𝑥)≠0 and 𝑓_𝑙 acts on the second tensor. A short investigation of the crystal graph (113) yields 𝑖𝑙=^′𝑠+1,if𝑖^′_𝑠≤𝑛−1𝑖^′_𝑠−1,else. (120) Since 𝜑_𝑙(𝜂(𝑥))=𝜑_𝑙(𝑥)=1≠0, we obtain that 𝑓_𝑙𝑖𝑥=¹,𝑖^′₁⋯𝑖_𝑠,𝑖^′_𝑠+1,if𝑙=𝑖^′_𝑠𝑖+1₁,𝑖^′₁⋯𝑖_𝑠,𝑖^′_𝑠−1,𝑙=𝑖^′_𝑠−1 (121) because if 𝑙=𝑖^′_𝑠−1 and 𝑥 would satisfy (a), it would automatically follow that 𝑥 satisfies (a^′) which is impossible since 𝜑_𝑙(𝑥)=1≠2. Thus, 𝑓_𝑙𝜂(𝑥)=𝜂(𝑓_𝑙𝑥).
If 𝑓_𝑙𝜂(𝑥)≠0 and 𝑓_𝑙 acts on the first tensor, we get with the tensor product rule 𝜑_𝑙(𝑥₁)>𝜖_𝑙(𝑥₂). Note that the operation with the Kashiwara operator 𝑓_𝑙 would change the entry 𝑖^′_𝑠 in 𝑥 if and only if either 𝑙=𝑖^′_𝑠+1 or 𝑙=𝑖^′_𝑠−1 and 𝑥 is not subject to (a). Our goal is to show here that 𝑓_𝑙 has no effect on 𝑖^′_𝑠. If 𝑙=𝑖^′_𝑠+1, we would get 𝜑_𝑙(𝑥₁)=0 and thus a contradiction to 𝜑_𝑙(𝑥₁)>𝜖_𝑙(𝑥₂). In the case where 𝑙=𝑖^′_𝑠−1, we have that (b) is not fulfilled for 𝑥₁. Therefore, 𝑥₁ must be subject to (a). Consequently we obtain that 𝑥 must fulfill (a) as well because otherwise we would end in a contradiction; namely, the only property in (a) which can be violated is 𝑙∉{𝑖₁,…,𝑖_𝑠,𝑖^′₁,…,𝑖^′_𝑠}. So if 𝑙 is contained in the aforementioned set, we get by the definition of 𝜂 that 𝑙=𝑖₁=𝑖. Hence, 𝑙−1=𝑖−1 must be contained in the set {𝑖≤𝑖^′₁,…,𝑖^′_𝑠−1}, which is impossible.
Thus, the entry 𝑖^′_𝑠 stays unchanged in 𝑓_𝑙𝑥 which provides 𝑓_𝑙𝜂(𝑥)=𝜂(𝑓_𝑙𝑥). The proof of 𝑒_𝑙̃𝜂(𝑥)=𝜂(𝑒_𝑙̃𝑥) is similar, which completes the proof of (1).
In order to prove (2) we will check as in (1) step for step the properties of a morphism of crystals. We get 𝜑_𝑙𝐱_𝑘−1⊗𝑥_𝑘=max𝜑_𝑙𝑥_𝑘,𝜑_𝑙𝐱_𝑘−1+𝜑_𝑙𝑥_𝑘−𝜖_𝑙𝑥_𝑘=max𝜑_𝑙𝑥_𝑘,𝜎^𝑘_𝑙𝐱_𝑘−1−𝜏_𝑙^𝑘−1𝐱_𝑘−1−min𝜎^𝑗_𝑙𝐱_𝑘−1−𝜏^𝑗_𝑙𝐱_𝑘−1∣1≤𝑗≤𝑘−1+𝜑_𝑙𝑥_𝑘−𝜖_𝑙𝑥_𝑘=max𝜑_𝑙𝑥_𝑘,𝜑_𝑙𝑥_𝑘+𝜎^𝑘_𝑙(𝐱)−𝜏^𝑘_𝑙(𝐱)−min𝜎^𝑗_𝑙(𝐱)−𝜏^𝑗_𝑙(𝐱)∣1≤𝑗≤𝑘−1=𝜎_𝑙^𝑘+1(𝐱)−𝜏^𝑘_𝑙(𝐱)−min𝜎^𝑗_𝑙(𝐱)−𝜏^𝑗_𝑙(𝐱)∣1≤𝑗≤𝑘=𝜑_𝑙(𝐱). (122) The same is trivially fulfilled for 𝜖_𝑙 because of Definition 3(1) and the weight invariance of 𝜙. Now suppose that 𝑓_𝑙(𝐱)=𝑝 is as in (25). If we apply the Kashiwara operator 𝑓_𝑙 to the tensor product 𝐱_𝑘−1⊗𝑥_𝑘, we obtain with the above calculations𝑓_𝑙𝐱_𝑘−1⊗𝑥_𝑘=𝑓_𝑙𝐱_𝑘−1⊗𝑥_𝑘,if𝜑_𝑙𝐱_𝑘−1>𝜖_𝑙𝑥_𝑘𝐱_𝑘−1⊗𝑓_𝑙𝑥_𝑘,if𝜑_𝑙𝐱_𝑘−1≤𝜖_𝑙𝑥_𝑘=𝑓_𝑙𝐱_𝑘−1⊗𝑥_𝑘,if𝜎^𝑘_𝑙(𝐱)−𝜏^𝑘_𝑙(𝐱)>𝜎^𝑝_𝑙(𝐱)−𝜏^𝑝_𝑙(𝐱)𝐱_𝑘−1⊗𝑓_𝑙𝑥_𝑘,if𝜎^𝑘_𝑙(𝐱)−𝜏^𝑘_𝑙(𝐱)≤𝜎^𝑝_𝑙(𝐱)−𝜏^𝑝_𝑙(𝐱). (123) According to this we get that 𝑝∈{1,…,𝑘−1} if 𝑓_𝑙 acts on the first tensor and 𝑝=𝑘 if 𝑓_𝑙 acts on the second tensor. The proof for the Kashiwara operator 𝑒_𝑙̃ works similarly.

Figure 1 

Figure 2 

Figure 3 

Corollary 21. One has an isomorphism of crystals ℛ(𝜆)≅𝐵(𝜆). (124)

Proof. The proof will proceed by induction on 𝑘, where the initial step is exactly part (1) of Theorem 20. If 𝑘>1 and 𝑗 is the maximal integer where 𝑚_𝑗 is non zero, we can assume with the induction hypothesis that ℛ(𝜆−𝜔_𝑗)≅𝐵(𝜆−𝜔_𝑗) and ℛ(𝜔_𝑗)≅𝐵(𝜔_𝑗). The rest of the proof is done with part (2) of Theorem 20, since the map 𝜙 is injective and the image is a connected crystal containing the highest weight element 𝑟_{𝜆−𝜔𝑗}⊗𝑟_𝜔𝑗.

Acknowledgment

The author was sponsored by the SFB/TR 12-Symmetries and Universality in Mesoscopic Systems.