Assembling crystals of type A

Regular $A_n$-crystals are certain edge-colored directed graphs which are related to representations of the quantized universal enveloping algebra $U_q(\mathfrak{sl}_{n+1})$. For such a crystal $K$ with colors $1,2,...,n$, we consider its maximal connected subcrystals with colors $1,...,n-1$ and with colors $2,...,n$ and characterize the interlacing structure for all pairs of these subcrystals. This is used to give a recursive description of the combinatorial structure of $K$ and develop an efficient procedure of assembling $K$.


Introduction
Crystals are certain "exotic" edge-colored graphs. This graphtheoretic abstraction, introduced by Kashiwara [1,2], has proved its usefulness in the theory of representations of Lie algebras and their quantum analogues. In general, a finite crystal is a finite directed graph such that the edges are partitioned into subsets, or color classes, labeled 1, . . . , , each connected monochromatic subgraph of is a simple directed path, and there are certain interrelations between the lengths of such paths, which depend on the × Cartan matrix = ( ) related to a given Lie algebra g. Of most interest are crystals of representations, or regular crystals. They are associated to elements of a certain basis of the highest weight integrable modules (representations) over the quantized universal enveloping algebra (g).
This paper continues our combinatorial study of crystals begun in [3,4] and considers -colored regular crystals of type A, where the number of colors is arbitrary. Recall that type A concerns g = sl +1 ; in this case the Cartan matrix is viewed as = −1 if | − | = 1, = 0 if | − | > 1, and = 2. We will refer to a regular -colored crystal of type A as an -crystal and omit the term when the number of colors is not specified. Since we are going to deal with finite regular crystals only, the adjectives "finite" and "regular" will usually be omitted. Also we assume that any crystal in question is (weakly) connected; that is, it is not the disjoint union of two nonempty graphs (which does not lead to loss of generality).
It is known that any A-crystal possesses the following properties. (i) is acyclic (i.e., has no directed cycles) and has exactly one zero-indegree vertex, called the source, and exactly one zero-outdegree vertex, called the sink of . (ii) For any ⊆ {1, . . . , }, each (inclusion-wise) maximal connected subgraph of whose edges have colors from is a crystal related to the corresponding × submatrix of the Cartan matrix for . Throughout, speaking of a subcrystal of , we will mean a subgraph of this sort.
Two-colored subcrystals are of especial importance, due to the result in [5] that for a crystal (of any type) with exactly one zero-indegree vertex, the regularity of all two-colored subcrystals implies the regularity of the whole crystal. Let be a two-colored subcrystal with colors , in an A-crystal . Then is the Cartesian product of a path with color and a path with color (forming an 1 × 1 -crystal) when | − | > 1, and an 2 -crystal when | − | = 1. (The A-crystals belong to the group of simply-laced crystals, which are characterized by the property that each two-colored subcrystal is of type Algebra Another important fact is that for any -tuple = ( 1 , . . . , ) of nonnegative integers, there exists exactly onecrystal such that each is equal to the length of the maximal path with color beginning at the source (for short combinatorial explanations, see [4,Section 2]. We denote the crystal determined by in this way by ( ) and refer to as the parameter of this crystal.
There have been known several ways to define A-crystals; in particular, via Gelfand-Tsetlin pattern model, semistandard Young tableaux, and Littelmann's path model; see [7][8][9][10]. In the last decade there appeared additional, more enlightening, descriptions. A short list of "local" defining axioms for A-crystals is pointed out in [6], and an explicit construction for 2 -crystals is given in [3]. According to that construction, any 2 -crystal can be obtained from an 1 × 1 -crystal by replacing each monochromatic path of the latter by a graph viewed as a triangle-shaped half of a directed square grid.
When > 2, the combinatorial structure of -crystals becomes much more complicated, even for = 3. Attempting to learn more about this structure, we elaborated in [4] a new combinatorial construction, the so-called crossing model (which is a refinement of the Gelfand-Tsetlin pattern model). This powerful tool has helped us to reveal more structural features of an -crystal = ( ). In particular, has the so-called principal lattice, a vertex subset Π with the following nice properties: (P3) the set K (− ) of ( − 1)-colored subcrystals of having colors 1, . . . , − 1 is bijective to Π; more precisely, ∩ Π consists of exactly one vertex, called the heart of w.r.t. , and similarly for the set K (−1) of subcrystals of with colors 2, . . . , .
Note that a sort of "principal lattice" satisfying (P1) and (P2) can be introduced for crystals of types B and C as well, and probably for the other classical types (see [11,Section 8]); for more about 2 -crystals, see also [12]). However, (P3) does not remain true in general for those types. Property (P3) is crucial in our study of A-crystals in this paper.
For ∈ B( ), let ↑ [ ] (resp., ↓ [ ]) denote the subcrystal in K (− ) (resp., in K (−1) ) that contains the principal vertex[ ]; we call it the upper (resp., lower) subcrystal at . It is shown in [4] that the parameter of this subcrystal is expressed by a linear function of and , and that the total amount of upper (lower) subcrystals with a fixed parameter is expressed by a piecewise linear function of and .
In this paper, we further essentially use the crossing model, aiming to obtain a refined description of the structure of an -crystal . We study the intersections of subcrystals ↑ [ ] and ↓ [ ] for any , ∈ B( ). This intersection may be empty or consist of one or more subcrystals with colors 2, . . . , − 1, called middle subcrystals of . Each of these middle subcrystals̃is therefore a lower subcrystal of ↑ [ ] and an upper subcrystal of ↓ [ ]; sõhas a unique vertex in the principal lattice Π ↑ of the former, and a unique vertex in the principal lattice Π ↓ of the latter. Our main structural results-Theorems 7 and 8-give explicit expressions showing how the "loci" and in Π, the "deviation" of (the heart of)̃from in Π ↑ , and the "deviation" of̃from in Π ↓ are interrelated.
This gives rise to a recursive procedure of assembling of the -crystal ( ). More precisely, suppose that the ( − 1)colored crystals ↑ [ ] and ↓ [ ] for all , ∈ B( ) are already constructed. Then we can combine these subcrystals to obtain the desired crystal ( ), by properly identifying the corresponding middle subcrystals (if any) for each pair This recursive method is implemented as an efficient algorithm which, given a parameter ∈ Z + , outputs the crystal ( ). The running time of the algorithm and the needed space are bounded by 2 | ( )|, where is a constant and | ( )| is the size of ( ). (It may be of practical use for small and ; in general, an -crystal has "dimension" ( + 1)/2 and its size grows sharply by increasing .) This paper is organized as follows. Section 2 contains basic definitions and backgrounds. Here we recall "local" axioms and the crossing model for A-crystals and review the needed results on the principal lattice Π of an -crystal and relations between Π and the ( − 1)-colored subcrystals from [4]. Section 3 states Theorems 7 and 8 and gives a recursive description of the structure of an -crystal and the algorithm of assembling . These theorems are proved in Section 4. Section 5 illustrates our assembling method for two special cases of A-crystals: for an arbitrary 2 -crystal (in which case the method can be compared with the explicit combinatorial construction in [3]) and for the particular 3crystal (1, 1, 1).
It should be noted that the obtained structural results on A-crystals can also be applied to give a direct combinatorial proof of the known fact that any regular -crystal (crystal) can be extracted, in a certain way, from a symmetric 2 −1 -crystal (resp., 2 -crystal); this is discussed in detail in ( [11], . Here an -crystal with parameter ( 1 , . . . , ) is called symmetric if = +1− .

Preliminaries
In this section we recall "local" axioms defining A-crystals, explain the construction of crossing model, and review facts about the principal lattice and subcrystals established in [4] that will be needed later.

A-Crystals.
Stembridge [6] pointed out a list of "local" graph-theoretic axioms for the regular simply laced crystals. The (regular) A-crystals form a subclass of those and are defined by axioms (A1)-(A5) below; these axioms are given in a slightly different, but equivalent, form compared with [6]. = ( ( ), ( )) be a directed graph whose edge set is partitioned into subsets 1 , . . . , , denoted as = ( ( ), 1 ⊔ ⋅ ⋅ ⋅ ⊔ ). We assume to be (weakly) connected. We say that an edge ∈ has color or is an -edge.
The first axiom concerns the structure of monochromatic subgraphs of .
So each vertex of has at most one incoming -edge and at most one outgoing -edge, and therefore one can associate to the set a partial invertible operator acting on vertices: ( , ) is an -edge if and only if acts at and ( ) = (or = −1 ( ), where −1 is the partial operator inverse to ). Since is connected, one can use the operator notation to express any vertex via another one. For example, the expression −1 1 2 3 2 ( ) determines the vertex obtained from a vertex by traversing 2-edge ( , ), followed by traversing 3 edges ( , ) and ( , ), followed by traversing 1-edge ( , ) in backward direction. Emphasize that every time we use such an operator expression in what follows, this automatically says that all corresponding edges do exist in .
We refer to a monochromatic path with color on the edges as an -path. So each maximal -path is an 1 -subcrystal with color in . The maximal -path passing a given vertex (possibly consisting of the only vertex ) is denoted by ( ), its part from the first vertex to by in ( ), and its part from to the last vertex by out ( ) (the tail and head parts of w.r.t. ). The lengths (i.e., the numbers of edges) of in ( ) and out ( ) are denoted by ( ) and ℎ ( ), respectively. Axioms (A2)-(A5) concern interrelations of different colors , . They say that each component of the two-colored graph ( ( ), ⊔ ) forms an 2 -crystal when colors , are neighboring, which means that | − | = 1, and forms an 1 × 1 -crystal otherwise.
When we traverse an edge of color , the head and tail part lengths of maximal paths of another color behave as follows.
These constants are just the off-diagonal entries of the Cartan × matrix related to the crystal type A and the number of colors.
It follows that for neighboring colors , , each maximal -path contains a unique vertex such that when traversing any edge of before (i.e., ∈ in ( )), the tail length decreases by 1 while the head length ℎ does not change, and when traversing any edge of after , does not change while ℎ increases by 1. This is called the critical vertex for , , .
One easily shows that if four vertices are connected by two -edges , and two -edges̃,̃(forming a "square"), then ℓ ( ) = ℓ ( ) ̸ = ℓ (̃) = ℓ (̃) (as illustrated in the picture). Another important consequence of (A3) is that for neighboring colors , , if is the critical vertex on a maximal -path w.r.t. color , then is also the critical vertex on the maximal -path passing w.r.t. color ; that is, we can speak of common critical vertices for the pair { , }.
The fourth axiom points out situations when, for neighboring , , the operators , and their inverse ones Again, one shows that the label w.r.t. , of each of the eight involved edges is determined uniquely, just as indicated in the above picture (where the bigger circles indicate critical vertices).
The final axiom concerns nonneighboring colors. (A5) Let | − | ≥ 2. Then for any ∈ { , −1 } and ∈ { , −1 }, the operators , commute at each vertex where both act. This is equivalent to saying that each component of the two-colored subgraph ( ( ), ⊔ ) is the Cartesian product of an -path and a -path , or that each subcrystal of with nonneighboring colors , is an 1 × 1 -crystal. One shows that any -crystal is finite and has exactly one zero-indegree vertex and one zero-outdegree vertex , called the source and sink of , respectively. Furthermore, the -crystals admit a nice parameterization: the lengths ℎ 1 ( ), . . . , ℎ ( ) of monochromatic paths starting at the source determine , and for each tuple = ( 1 , . . . , ) of nonnegative integers, there exists a (unique) -crystal such that = ℎ ( ) for = 1, . . . , . (See [4,6].) We call the parameter of and denote by ( ).

The Crossing Model for -Crystals.
Following [4], the crossing model M ( ) generating the -crystal = ( ) with a parameter = ( 1 , . . . , ) ∈ Z + consists of three ingredients: (i) a directed graph = = ( ( ), ( )) depending on , called the supporting graph of the model; To explain the construction of the supporting graph , we first introduce another directed graph G = G that we call the protograph of . Its node set consists of elements ( ) for all , ∈ {1, . . . , } such that ≤ . (To avoid a possible mess, we prefer to use the term "node" for vertices in the crossing model, and the term "vertex" for vertices of crystals.) Its edges are all possible pairs of the form ( ( ), −1 ( )) (ascending edges) or ( ( ), +1 ( + 1)) (descending edges). We say that the nodes (1), . . . , ( ) form th level of G and order them as indicated (by increasing ). We visualize G by drawing it on the plane so that the nodes of the same level lie in a horizontal line, the ascending edges point North-East, and the descending edges point South-East. See the picture where = 4.
The resulting is the disjoint union of directed graphs 1 , . . . , , where each contains all vertices of the form ( ). Also is isomorphic to the Cartesian product of two paths, with the lengths − 1 and − . For example, for = 4, the graph is viewed as (4) (where the multinodes are surrounded by ovals) and its components 1 , 2 , 3 , 4 are viewed as (ii) 0 ≤ ( ) ≤ for each ∈ ( ) , = 1, . . . , ; (iii) each multinode ( ) contains a node with the following property : the edge SE ( ) is tight for each node ∈ ( ) preceding , and SW ( ) is tight for each node ∈ ( ) succeeding .
The first node = ( ) (i.e., with minimum) satisfying the property in (iii) is called the switch-node of the multinode ( ). These nodes play an important role in our transformations of feasible functions in the model.
To describe the rule of transforming ∈ F( ), we first extend each by adding extra nodes and extra edges (following [4] and aiming to slightly simplify the description). In the extended directed graph , the node set consists of elements ( ) for all = 0, . . . , +1 and = 0, . . . , such that ≤ . The edge set of consists of all possible pairs of the form ( ( ), −1 ( )) or ( ( ), +1 ( + 1)). Then all are isomorphic. The disjoint union of these gives the extended supporting graph . The creation of 2 from 2 for = 4 is illustrated in the picture: 2 Each feasible function on ( ) is extended to the extra nodes = ( ) as follows: ( ) := if there is a path from to a node of , and ( ) := 0 otherwise (one may say that lies on the left of in the former case and on the right of in the latter case; in the above picture, such nodes are marked by white and black circles, resp.).
is extended to the extra edges accordingly. In particular, each edge of not incident with a node of is tight; that is, ( ) = 0. For a node = ( ) with 1 ≤ ≤ ≤ , define the value ( ) = ( ) by where := ( − 1). For a multinode ( ), define the numbers We call ( ), ( ), and̃( ) the slack at a node , the total slack at a multinode ( ), and the reduced slack at ( ), respectively. (Note that ,̃are defined in (8), (9), and (10) in a slightly different way than in [4], which, however, does not affect the choice of active multinodes and switch-nodes below.) Now we are ready to define the transformations of (or the moves from ). At most transformations 1 , . . . , are possible. Each changes within level and is applicable when this level contains a multinode ( ) with̃( ) > 0. In this case we take the multinode ( ) such that referring to it as the active multinode for the given and . We increase by 1 at the switch-node in ( ), preserving on the other nodes of . It is shown [4] that the resulting function ( ) is again feasible. As a result, the model generates -colored directed graph K( ) = (F, E 1 ⊔⋅ ⋅ ⋅⊔E ), where each color class E is formed by the edges ( , ( )) for all feasible functions to which the operator is applicable. This graph is just an -crystal.

Principal Lattice and ( −1)-Colored Subcrystals of an -Crystal.
Based on the crossing model, [4] reveals some important ingredients and relations for an -crystal = ( ). One of them is the so-called principal lattice, which is defined as follows.
Let ∈ Z + and ≤ . One easily checks that the function on the vertices of the supporting graph that takes the constant value within each subgraph of , = 1, . . . , , is feasible. We denote this function and the vertex of corresponding to it by [ ] and[ ], respectively, and call them principal ones. So the set of principal vertices is bijective to the integer box B( ) := { ∈ Z : 0 ≤ ≤ }; this set is called the principal lattice of and denoted by Π = Π( ). When it is not confusing, the term "principal lattice" may also be applied to B( ).
The following properties of the principal lattice will be essentially used later.
The principal vertex[ ] is contained in the lower lattice Π ↓ [ ] and its coordinate We call[ ] the heart of ↑ [ ] w.r.t. , and similarly for lower subcrystals.
(One more result given in [4, Remark 5] is a piecewise linear formula to compute, for an ( − 1)-tuple , the number of upper subcrystals of ( ) with the parameter equal to , but we do not need this in what follows.) Remark 6. As is mentioned in the Introduction, the crossing model is, in fact, a refinement of the Gelfand-Tsetlin pattern (or GT-pattern) model [7]. More precisely, for ∈ Z + , form the partition = ( 1 ≥ 2 ≥ ⋅ ⋅ ⋅ ≥ ≥ +1 = 0) by setting

Assembling an -Crystal
As mentioned in the Introduction, the structure of an crystal = ( ) can be described in a recursive manner. The idea is as follows. We know that contains |Π| = ( 1 + 1) × ⋅ ⋅ ⋅ × ( + 1) upper subcrystals (with colors 1, . . . , − 1) and |Π| lower subcrystals (with colors 2, . . . , ). Moreover, the parameters of these subcrystals are expressed explicitly by Algebra 7 (15) and (17). Assume that the set K (− ) of upper subcrystals and the set K (−1) of lower subcrystals are already available. Then in order to assemble , it suffices to point out, in appropriate terms, the intersection ↑ [ ] ∩ ↓ [ ] for all pairs , ∈ B( ) (the intersection may either be empty or consist of one or more ( − 2)-colored subcrystals with colors 2, . . . , − 1 in ). We give an appropriate characterization in Theorems 7 and 8.
Proofs of these theorems will be given in the next section. Based on Theorems 7 and 8, the crystal ( ) is assembled as follows. By recursion we assume that all upper and lower subcrystals are already constructed. We also assume that for each upper subcrystal ↑ [ ] ≃ ( ↑ ), its principal lattice is distinguished by the use of the corresponding injective map : B( ↑ ) → ( ( ↑ )), and similarly for the lower subcrystals. We delete the edges with color 1 in each ↑ [ ] and extract the components of the resulting graphs, forming the set K (−1,− ) (arranged as a list) of all middle subcrystals of ( ). Each ↑↓ ∈ K (−1,− ) is encoded by a corresponding pair ( , Δ), where ∈ B( ) and the deviation Δ in Π ↑ [ ] is determined by the use of as above. Acting similarly for the lower subcrystals ↓ [ ] (by deleting the edges with color ), we obtain the same set of middle subcrystals (arranged as another list), each of which being encoded by a corresponding pair ( , ∇), where ∈ B( ) and ∇ is a deviation in Π ↓ ( ). Relations (21) and (20) indicate how to identify each member of the first list with its counterpart in the second one. Now restoring the deleted edges with colors 1 and , we obtain the desired crystal ( ). The corresponding map B( ) → ( ( )) is constructed easily (e.g., by the use of operator strings as in Proposition 2).
We conclude this section with several remarks.

Remark 10.
A straightforward implementation of the above recursive method of constructing = ( ) takes (2 ( ) ) time and space, where ( ) is a polynomial in and is the number of vertices of . Here the factor 2 ( ) appears because the total number of vertices in the upper and lower subcrystals is 2 (implying that there appear 4 vertices in total on the previous step of the recursion, and so on). Therefore, Algebra such an implementation has polynomial complexity of the size of the output for each fixed , but not in general. However, many intermediate subcrystals arising during the recursive process are repeated, and we can use this fact to improve the implementation. More precisely, the colors occurring in each intermediate subcrystal in the process form an interval of the ordered set (1, . . . , ). We call a subcrystal of this sort a colorinterval subcrystal, or a CI-subcrystal, of . In fact, every CIsubcrystal of appears in the process. Since the number of intervals is ( +1)/2 and the CI-subcrystals concerning equal intervals are pairwise disjoint, the total number of vertices of all CI-subcrystals of is ( 2 ). It is not difficult to implement the recursive process so that each CI-subcrystal be explicitly constructed only once. As a result, we obtain the following.

Proofs of Theorems 7 and 8
Let , Δ, , ∇ be as in the hypotheses of Theorem 7. First we show that Theorem 7 follows from Theorem 8.
Let ≤ − 2. Suppose that = ( ) is a node whose SW-edge = ( , ) exists and is not -tight. This is possible only if ∈ 1 and ∈ 2 . In this case, is determined as = − + 2; that is, is the second node in ( ). We observe that (a) for the first node −1 ( ) of ( ), both ends of its SEedge belong to the piece −1 1 ; and (b) for any node ( ) with > in ( ), both ends of its SW-edge belong either to 2 or to 4 . So, such and are -tight. Therefore, the node satisfies the condition in (6) Now consider an arbitrary Δ satisfying (37). Let be such that Δ < − (if any) and define Δ := Δ +1 and Δ := Δ for ̸ = . Then (Δ) < (Δ ). We assume by induction that assertion (ii) is valid for ,Δ , and our aim is to show validity of (ii) for ,Δ .
In what follows stands for the former function ,Δ . Let be the vertex with the deviation Δ in Π ↑ [ ]. Both and are principal vertices of the subcrystal ↑ [ ], and the coordinate of in Π ↑ [ ] is obtained from the one of by increasing its th entry by 1. According to Proposition 2 (with is replaced by − 1), is obtained from by applying the operator string where −1, , = ⋅ ⋅ ⋅ + −1 (cf. (12)). In light of this, we have to show that when (the sequence of moves corresponding to) −1, is applied to , the resulting feasible function is exactly For convenience, th term + −1 in the substring −1, , will be denoted by ( , ), = 1, . . . , . So −1, , = ( , 1) ( , 2) ⋅ ⋅ ⋅ ( , ).
We distinguish between two cases: Δ ≥ 0 and Δ < 0. Let , denote the current function on ( ) just before the application of ( , ) (when the process starts with = ,Δ ). We assert that for each , the application of ( , ) to , increases the value at the node ( ) by 1, whence (40) will immediately follow. In order to show (41), we first examine tight edges and the slacks ( ) of the nodes in levels < for the initial function . One can see from (36) that for = 1, . . . , , each node of the subgraph has at least one entering edge (i.e., SW ( ) or NW ( )) which is -tight, except, possibly, for the nodes (1) , indicated by stars in Figure 1.
Now we are ready to prove (41). When dealing with a current function , and seeking for the node at level + −1 where the operator ( , ) should act to increase , , we can immediately exclude from consideration any node that has at least one tight entering edge (in view of the monotonicity condition (6) (i)).
Due to (42) and (43)(i), for the initial function = 1, , there is only one node in level that has no tight entering edge, namely, (1). So, at the first step of the process, the first operator (1, ) of −1, acts just at (1), as required in (41).
Next consider a step with := , for ( , ) ̸ = (1, ), assuming validity of (41) on the previous steps. are at most two other nodes in level that may have no tight entering edges for (and therefore, for ), namely, (1) and +1 (1). Then (1, ) must act at , as required in (41) (since the nontightness of the SW-edge of implies that none of the nodes (1) in (1) preceding (i.e., with < ) can be the switch-node).
(B) Let > 1. Comparing with in the node := ( ) = + −1 ( ) and its adjacent nodes, we observe that has no -tight entering edge and that ( ) > 0. Also for any other node in level + − 1, one can see that if has a tight entering edge for , then this property holds for as well, and that ( ) ≥ ( ) ≥ 0. Using this, properties (42), (43) (iii), and condition (11), one can conclude that the total and reduced slacks for at the multinode := + −1 ( ) are positive, that is the active multinode for in level + −1, and that ( , ) can be applied only at , yielding (41) again.
Thus, (40) is valid in Case 1. Case 2 [Δ < 0]. We assert that in this case the string −1, acts within the right rectangle +1 of the subgraph +1 (note that +1 is of size × ( − )). More precisely, each operator ( , ) modifies the current function by increasing its value at the node +1 ( ) by 1.
Thus, we have the desired property (40) in both Cases 1 and 2, and statement (ii) in Lemma 13 follows.
This completes the proof of relation (21), yielding Theorem 8.

Illustrations
In this concluding section, we give two illustrations to the above assembling construction for A-crystals. The first one refines the interrelation between upper and lower subcrystals in an arbitrary 2 -crystal; this can be compared with the explicit construction (the so-called "sail model") for 2crystals in [3]. The second one visualizes the subcrystals structure for one instance of 3 -crystals, namely, (1, 1, 1).

2 -Crystals.
The subcrystals structure becomes simpler when we deal with an 2 -crystal = ( 1 , 2 ). In this case the roles of upper, lower, and middle subcrystals are played by 1-paths, 2-paths, and vertices of , respectively, where by an -path we mean a maximal path of color .
Using (46) and (47), one can enumerate the sets of 1paths and 2-paths and properly intersect corresponding pairs, obtaining the 2 -crystal ( ). It is rather routine to check that the resulting graph coincides with the one generated by the sail model from [3]. Next we outline that construction.
Given ∈ Z 2 + , the 2 -crystal ( ) is produced from two particular two-colored graphs and , called the right sail of size 1 and the left sail of size 2 , respectively. The vertices of correspond to the vectors ( , ) ∈ Z 2 such that 0 ≤ ≤ ≤ 1 , and the vertices of to the vectors ( , ) ∈ Z 2 such that 0 ≤ ≤ ≤ 2 . In both and , the edges of color 1 are all possible pairs of the form (( , ), ( + 1, )), and the edges of color 2 are all possible pairs of the form (( , ), ( , + 1)).
The case ( 1 , 2 ) = (1, 2) is drawn in the picture; here the critical (principal) vertices are indicated by big circles, 1-edges by horizontal arrows, and 2-edges by vertical arrows: In particular, the sail model shows that the numbers of edges of each color in an 2 -crystal are the same. This implies a similar property for any -crystal. Crystal (1, 1,1). Next we illustrate the 3 -crystal = (1, 1, 1). It has 64 vertices and 102 edges, is rather puzzling, and drawing it in full would be cumbersome and take too much space; for this reason, we expose it by fragments;