Assembling Crystals of Type A

Abstract

Regular 𝐴_𝑛-crystals are certain edge-colored directed graphs, which are related to representations of the quantized universal enveloping algebra 𝑈_𝑞(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 enables us to give a recursive description of the combinatorial structure of 𝐾 via subcrystals and develop an efficient procedure of assembling 𝐾.

1. Introduction

Crystals are certain “exotic” edge-colored graphs. This graph-theoretic 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 𝐾 such that the edges are partitioned into 𝑛 subsets, or color classes, labeled 1,…,𝑛, each connected monochromatic subgraph of 𝐾 is a simple directed path, and there are certain interrelations between the lengths of such paths, which depend on the 𝑛×𝑛 Cartan matrix 𝑀=(𝑚_𝑖𝑗) 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 𝑈_𝑞(𝔤).

This paper continues our combinatorial study of crystals begun in [3, 4] and considers 𝑛-colored regular crystals of type A, where the number 𝑛 of colors is arbitrary. Recall that type A concerns 𝔤=𝔰𝔩_𝑛+1; in this case the Cartan matrix 𝑀 is viewed as 𝑚_𝑖𝑗=−1 if |𝑖−𝑗|=1, 𝑚_𝑖𝑗=0 if |𝑖−𝑗|>1, and 𝑚_𝑖𝑖=2. We will refer to a regular 𝑛-colored crystal of type A as an 𝐴_𝑛-crystal and omit the term 𝑛 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 𝐾 possesses the following properties. (i) 𝐾 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 𝐾. (ii) For any 𝐼⊆{1,…,𝑛}, each (inclusion-wise) maximal connected subgraph of 𝐾 whose edges have colors from 𝐼 is a crystal related to the corresponding 𝐼×𝐼 submatrix of the Cartan matrix for 𝐾. Throughout, speaking of a subcrystal of 𝐾, 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 𝐾^′ be a two-colored subcrystal with colors 𝑖,𝑗 in an A-crystal 𝐾. Then 𝐾^′ is the Cartesian product of a path with color 𝑖 and a path with color 𝑗 (forming an 𝐴₁×𝐴₁-crystal) when |𝑖−𝑗|>1, and an 𝐴₂-crystal when |𝑖−𝑗|=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 𝐴₁×𝐴₁ or 𝐴₂; for more details, see [6].)

Another important fact is that for any 𝑛-tuple 𝑐=(𝑐₁,…,𝑐_𝑛) of nonnegative integers, there exists exactly one 𝐴_𝑛-crystal 𝐾 such that each 𝑐_𝑖 is equal to the length of the maximal path with color 𝑖 beginning at the source (for short combinatorial explanations, see [4, Section 2]. We denote the crystal 𝐾 determined by 𝑐 in this way by 𝐾(𝑐) and refer to 𝑐 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 [7–10]. 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 𝐴₂-crystals is given in [3]. According to that construction, any 𝐴₂-crystal can be obtained from an 𝐴₁×𝐴₁-crystal by replacing each monochromatic path of the latter by a graph viewed as a triangle-shaped half of a directed square grid.

When 𝑛>2, the combinatorial structure of 𝐴_𝑛-crystals becomes much more complicated, even for 𝑛=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 𝐴_𝑛-crystal 𝐾=𝐾(𝑐). In particular, 𝐾 has the so-called principal lattice, a vertex subset Π with the following nice properties:(P1)Π contains the source and sink of 𝐾, and the vertices 𝑣∈Π are bijective to the elements of the integer box ℬ(𝑐)∶={𝑎∈ℤ^𝑛∶0≤𝑎≤𝑐}; we write 𝑣=𝑣[̆𝑎];(P2) for any 𝑎,𝑎^′∈ℬ(𝑐) with 𝑎≤𝑎^′, the interval of 𝐾 from 𝑣[̆𝑎] to 𝑣[̆𝑎^′] (i.e., the subgraph of 𝐾 formed by the vertices and edges contained in directed paths from 𝑣[̆𝑎] to 𝑣[̆𝑎^′]) is isomorphic to the 𝐴_𝑛-crystal 𝐾(𝑎^′−𝑎), and its principal lattice consists of the principal vertices 𝑣[̆𝑎^′′] of 𝐾 with 𝑎≤𝑎^′′≤𝑎^′;(P3) the set 𝒦^(−𝑛) of (𝑛−1)-colored subcrystals 𝐾^′ of 𝐾 having colors 1,…,𝑛−1 is bijective to Π; more precisely, 𝐾^′∩Π consists of exactly one vertex, called the heart of 𝐾^′ w.r.t. 𝐾, and similarly for the set 𝒦⁽⁻¹⁾ of subcrystals of 𝐾 with colors 2,…,𝑛.

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 𝐵₂-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 𝑎∈ℬ(𝑐), let 𝐾^↑[𝑎] (resp., 𝐾^↓[𝑎]) denote the subcrystal in 𝒦^(−𝑛) (resp., in 𝒦⁽⁻¹⁾) that contains the principal vertex 𝑣[̆𝑎]; we call it the upper (resp., lower) subcrystal at 𝑎. It is shown in [4] that the parameter of this subcrystal is expressed by a linear function of 𝑐 and 𝑎, and that the total amount of upper (lower) subcrystals with a fixed parameter 𝑐^′ is expressed by a piecewise linear function of 𝑐 and 𝑐^′.

In this paper, we further essentially use the crossing model, aiming to obtain a refined description of the structure of an 𝐴_𝑛-crystal 𝐾. We study the intersections of subcrystals 𝐾^↑[𝑎] and 𝐾^↓[𝑏] for any 𝑎,𝑏∈ℬ(𝑐). This intersection may be empty or consist of one or more subcrystals with colors 2,…,𝑛−1, called middle subcrystals of 𝐾. Each of these middle subcrystals 𝐾^∼ is therefore a lower subcrystal of 𝐾^↑[𝑎] and an upper subcrystal of 𝐾^↓[𝑏]; so 𝐾^∼ has a unique vertex 𝑧 in the principal lattice Π^↑ of the former, and a unique vertex 𝑧^′ in the principal lattice Π^↓ of the latter. Our main structural results—Theorems 7 and 8—give explicit expressions showing how the “loci” 𝑎 and 𝑏 in Π, the “deviation” of (the heart of) 𝐾^∼ from 𝑧 in Π^↑, and the “deviation” of 𝐾^∼ from 𝑧^′ in Π^↓ are interrelated.

This gives rise to a recursive procedure of assembling of the 𝐴_𝑛-crystal 𝐾(𝑐). More precisely, suppose that the (𝑛−1)-colored crystals 𝐾^↑[𝑎] and 𝐾^↓[𝑏] for all 𝑎,𝑏∈ℬ(𝑐) are already constructed. Then we can combine these subcrystals to obtain the desired crystal 𝐾(𝑐), by properly identifying the corresponding middle subcrystals (if any) for each pair 𝐾^↑[𝑎],𝐾^↓[𝑏]. This recursive method is implemented as an efficient algorithm which, given a parameter 𝑐∈ℤ^𝑛₊, outputs the crystal 𝐾(𝑐). The running time of the algorithm and the needed space are bounded by 𝐶𝑛²|𝐾(𝑐)|, where 𝐶 is a constant and |𝐾(𝑐)| is the size of 𝐾(𝑐). (It may be of practical use for small 𝑛 and 𝑐; in general, an 𝐴_𝑛-crystal has “dimension” 𝑛(𝑛+1)/2 and its size grows sharply by increasing 𝑐.)

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 𝐴_𝑛-crystal and relations between Π and the (𝑛−1)-colored subcrystals from [4]. Section 3 states Theorems 7 and 8 and gives a recursive description of the structure of an 𝐴_𝑛-crystal 𝐾 and the algorithm of assembling 𝐾. These theorems are proved in Section 4. Section 5 illustrates our assembling method for two special cases of A-crystals: for an arbitrary 𝐴₂-crystal (in which case the method can be compared with the explicit combinatorial construction in [3]) and for the particular 𝐴₃-crystal 𝐾(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 𝐵_𝑛-crystal (𝐶_𝑛-crystal) can be extracted, in a certain way, from a symmetric 𝐴_2𝑛−1-crystal (resp., 𝐴_2𝑛-crystal); this is discussed in detail in ([11], Sections 5–8). Here an 𝐴_𝑘-crystal with parameter (𝑐₁,…,𝑐_𝑘) is called symmetric if 𝑐_𝑖=𝑐_{𝑘+1−𝑖}.

2. 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.

2.1. 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].

Let 𝐾=(𝑉(𝐾),𝐸(𝐾)) be a directed graph whose edge set is partitioned into 𝑛 subsets 𝐸₁,…,𝐸_𝑛, denoted as 𝐾=(𝑉(𝐾),𝐸₁⊔⋯⊔𝐸_𝑛). We assume 𝐾 to be (weakly) connected. We say that an edge 𝑒∈𝐸_𝑖 has color 𝑖 or is an 𝑖-edge.

Unless explicitly stated otherwise, by a path we mean a simple finite directed path, that is, a sequence of the form (𝑣₀,𝑒₁,𝑣₁,…,𝑒_𝑘,𝑣_𝑘), where 𝑣₀,𝑣₁,…,𝑣_𝑘 are distinct vertices and each 𝑒_𝑖 is an edge from 𝑣_𝑖−1 to 𝑣_𝑖 (admitting 𝑘=0).

The first axiom concerns the structure of monochromatic subgraphs of 𝐾.  (A1) For 𝑖=1,…,𝑛, each connected subgraph of (𝑉(𝐾),𝐸_𝑖) is a path.

So each vertex of 𝐾 has at most one incoming 𝑖-edge and at most one outgoing 𝑖-edge, and therefore one can associate to the set 𝐸_𝑖 a partial invertible operator 𝐹_𝑖 acting on vertices: (𝑢,𝑣) is an 𝑖-edge if and only if 𝐹_𝑖 acts at 𝑢 and 𝐹_𝑖(𝑢)=𝑣 (or 𝑢=𝐹_𝑖⁻¹(𝑣), where 𝐹_𝑖⁻¹ is the partial operator inverse to 𝐹_𝑖). Since 𝐾 is connected, one can use the operator notation to express any vertex via another one. For example, the expression 𝐹₁⁻¹𝐹²₃𝐹₂(𝑣) determines the vertex 𝑤 obtained from a vertex 𝑣 by traversing 2-edge (𝑣,𝑣^′), followed by traversing 3 edges (𝑣^′,𝑢) and (𝑢,𝑢^′), followed by traversing 1-edge (𝑤,𝑢^′) 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 𝐾.

We refer to a monochromatic path with color 𝑖 on the edges as an 𝑖-path. So each maximal 𝑖-path is an 𝐴₁-subcrystal with color 𝑖 in 𝐾. The maximal 𝑖-path passing a given vertex 𝑣 (possibly consisting of the only vertex 𝑣) is denoted by 𝑃_𝑖(𝑣), its part from the first vertex to 𝑣 by 𝑃ⁱⁿ_𝑖(𝑣), and its part from 𝑣 to the last vertex by 𝑃^out_𝑖(𝑣) (the tail and head parts of 𝑃 w.r.t. 𝑣). The lengths (i.e., the numbers of edges) of 𝑃ⁱⁿ_𝑖(𝑣) and 𝑃^out_𝑖(𝑣) are denoted by 𝑡_𝑖(𝑣) and ℎ_𝑖(𝑣), respectively.

Axioms (A2)–(A5) concern interrelations of different colors 𝑖, 𝑗. They say that each component of the two-colored graph (𝑉(𝐾),𝐸_𝑖⊔𝐸_𝑗) forms an 𝐴₂-crystal when colors 𝑖, 𝑗 are neighboring, which means that |𝑖−𝑗|=1, and forms an 𝐴₁×𝐴₁-crystal otherwise.

When we traverse an edge of color 𝑖, the head and tail part lengths of maximal paths of another color 𝑗 behave as follows. (A2) For different colors 𝑖,𝑗 and for an edge (𝑢,𝑣) with color 𝑖, one holds 𝑡_𝑗(𝑣)≤𝑡_𝑗(𝑢) and ℎ_𝑗(𝑣)≥ℎ_𝑗(𝑢). The value (ℎ_𝑗(𝑢)−𝑡_𝑗(𝑢))−(ℎ_𝑗(𝑣)−𝑡_𝑗(𝑣)) is the constant 𝑚_𝑖𝑗 equal to −1 if |𝑖−𝑗|=1, and 0 otherwise. Furthermore, ℎ_𝑗 is convex on each 𝑖-path, in the sense that if (𝑢,𝑣),(𝑣,𝑤) are consecutive 𝑖-edges, then ℎ_𝑗(𝑢)+ℎ_𝑗(𝑤)≥2ℎ_𝑗(𝑣).

These constants 𝑚_𝑖𝑗 are just the off-diagonal entries of the Cartan 𝑛×𝑛 matrix 𝑀 related to the crystal type A and the number 𝑛 of colors.

It follows that for neighboring colors 𝑖, 𝑗, each maximal 𝑖-path 𝑃 contains a unique vertex 𝑟 such that when traversing any edge 𝑒 of 𝑃 before 𝑟 (i.e., 𝑒∈𝑃ⁱⁿ_𝑖(𝑟)), the tail length 𝑡_𝑗 decreases by 1 while the head length ℎ_𝑗 does not change, and when traversing any edge of 𝑃 after 𝑟, 𝑡_𝑗 does not change while ℎ_𝑗 increases by 1. This 𝑟 is called the critical vertex for 𝑃, 𝑖, 𝑗. To each 𝑖-edge 𝑒=(𝑢,𝑣), we associate label ℓ_𝑗(𝑒)∶=ℎ_𝑗(𝑣)−ℎ_𝑗(𝑢); then ℓ_𝑗(𝑒)∈{0,1} and 𝑡_𝑗(𝑣)=𝑡_𝑗(𝑢)−1+ℓ_𝑗(𝑒). We emphasize that the critical vertices on a maximal 𝑖-path 𝑃 w.r.t. its neighboring colors 𝑗=𝑖−1 and 𝑗=𝑖+1 may be different (and so are the edge labels on 𝑃).

Two operators 𝐹=𝐹^𝛼_𝑖 and 𝐹^′=𝐹^𝛽_𝑗, where 𝛼,𝛽∈{1,−1}, are said to commute at a vertex 𝑣 if each of 𝐹,𝐹^′ acts at 𝑣 (i.e., corresponding 𝑖-edge and 𝑗-edge incident with 𝑣 exist) and 𝐹𝐹^′(𝑣)=𝐹^′𝐹(𝑣). The third axiom indicates situations when such operators commute for neighboring 𝑖,𝑗. (A3) Let |𝑖−𝑗|=1. (a) If a vertex 𝑢 has outgoing 𝑖-edge (𝑢,𝑣) and outgoing 𝑗-edge (𝑢,𝑣^′) and if ℓ_𝑗(𝑢,𝑣)=0, then ℓ_𝑖(𝑢,𝑣^′)=1 and 𝐹_𝑖, 𝐹_𝑗 commute at 𝑣. Symmetrically: (b) if a vertex 𝑣 has incoming 𝑖-edge (𝑢,𝑣) and incoming 𝑗-edge (𝑢^′,𝑣) and if ℓ_𝑗(𝑢,𝑣)=1, then ℓ_𝑖(𝑢^′,𝑣)=0 and 𝐹_𝑖⁻¹, 𝐹_𝑗⁻¹ commute at 𝑣. (See the following picture.) (1)

One easily shows that if four vertices are connected by two 𝑖-edges 𝑒, 𝑒^′ and two 𝑗-edges 𝑒^∼,𝑒^∼′ (forming a “square”), then ℓ_𝑗(𝑒)=ℓ_𝑗(𝑒^′)≠ℓ_𝑖(𝑒^∼)=ℓ_𝑖(𝑒^∼′) (as illustrated in the picture). Another important consequence of (A3) is that for neighboring colors 𝑖,𝑗, if 𝑣 is the critical vertex on a maximal 𝑖-path w.r.t. color 𝑗, then 𝑣 is also the critical vertex on the maximal 𝑗-path passing 𝑣 w.r.t. color 𝑖; that is, we can speak of common critical vertices for the pair {𝑖,𝑗}.

The fourth axiom points out situations when, for neighboring 𝑖,𝑗, the operators 𝐹_𝑖,𝐹_𝑗 and their inverse ones “remotely commute” (forming the “Verma relation of degree 4”).(A4) Let |𝑖−j|=1. (i) If a vertex 𝑢 has outgoing edges with color 𝑖 and color 𝑗 and if each edge is labeled 1 w.r.t. the other color, then 𝐹_𝑖𝐹²_𝑗𝐹_𝑖(𝑢)=𝐹_𝑗𝐹²_𝑖𝐹_𝑗(𝑢). Symmetrically, (ii) if 𝑣 has incoming edges with color 𝑖 and color 𝑗 and if both are labeled 0, then 𝐹_𝑖⁻¹(𝐹_𝑗⁻¹)²𝐹_𝑖⁻¹(𝑣)=𝐹_𝑗⁻¹(𝐹_𝑖⁻¹)²𝐹_𝑗⁻¹(𝑣). (See the following picture.)(2)

Again, one shows that the label w.r.t. 𝑖, 𝑗 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 |𝑖−𝑗|≥2. Then for any 𝐹∈{𝐹_𝑖,𝐹_𝑖⁻¹} and 𝐹^′∈{𝐹_𝑗,𝐹_𝑗⁻¹}, the operators 𝐹,𝐹^′ commute at each vertex where both act.

This is equivalent to saying that each component of the two-colored subgraph (𝑉(𝐾),𝐸_𝑖⊔𝐸_𝑗) is the Cartesian product of an 𝑖-path 𝑃 and a 𝑗-path 𝑃^′, or that each subcrystal of 𝐾 with nonneighboring colors 𝑖, 𝑗 is an 𝐴₁×𝐴₁-crystal.

One shows that any 𝐴_𝑛-crystal 𝐾 is finite and has exactly one zero-indegree vertex 𝑠_𝐾 and one zero-outdegree vertex 𝑡_𝐾, called the source and sink of 𝐾, respectively. Furthermore, the 𝐴_𝑛-crystals 𝐾 admit a nice parameterization: the lengths ℎ₁(𝑠_𝐾),…,ℎ_𝑛(𝑠_𝐾) of monochromatic paths starting at the source determine 𝐾, and for each tuple 𝑐=(𝑐₁,…,𝑐_𝑛) of nonnegative integers, there exists a (unique) 𝐴_𝑛-crystal 𝐾 such that 𝑐_𝑖=ℎ_𝑖(𝑠_𝐾) for 𝑖=1,…,𝑛. (See [4, 6].) We call 𝑐 the parameter of 𝐾 and denote 𝐾 by 𝐾(𝑐).

2.2. The Crossing Model for 𝐴_𝑛-Crystals

Following [4], the crossing model ℳ_𝑛(𝑐) generating the 𝐴_𝑛-crystal 𝐾=𝐾(𝑐) with a parameter 𝑐=(𝑐₁,…,𝑐_𝑛)∈ℤ^𝑛₊ consists of three ingredients:(i)a directed graph 𝐺_𝑛=𝐺=(𝑉(𝐺),𝐸(𝐺)) depending on 𝑛, called the supporting graph of the model;(ii)a set ℱ=ℱ(𝑐) of feasible functions on 𝑉(𝐺);(iii)a set ℰ=ℰ(𝑐) of transformations 𝑓↦𝑓^′ of feasible functions, called moves.

To explain the construction of the supporting graph 𝐺, we first introduce another directed graph 𝒢=𝒢_𝑛 that we call the protograph of 𝐺. Its node set consists of elements 𝑉_𝑖(𝑗) for all 𝑖,𝑗∈{1,…,𝑛} such that 𝑗≤𝑖. (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 (𝑉_𝑖(𝑗),𝑉_𝑖−1(𝑗)) (ascending edges) or (𝑉_𝑖(𝑗),𝑉_𝑖+1(𝑗+1)) (descending edges). We say that the nodes 𝑉_𝑖(1),…,𝑉_𝑖(𝑖) form 𝑖th level of 𝒢 and order them as indicated (by increasing 𝑗). 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 𝑛=4.(3)

The supporting graph 𝐺 is produced by replicating elements of 𝒢 as follows. Each node 𝑉_𝑖(𝑗) generates 𝑛−𝑖+1 nodes of 𝐺, denoted as 𝑣^𝑘_𝑖(𝑗) for 𝑘=𝑖−𝑗+1,…𝑛−𝑗+1, which are ordered by increasing 𝑘 (and accordingly follow from left to right in the visualization). We identify 𝑉_𝑖(𝑗) with the set of these nodes and call it a multinode of 𝐺. Each edge of 𝒢 generates a set of edges of 𝐺 (a multiedge) connecting elements with equal upper indices. More precisely, (𝑉_𝑖(𝑗),𝑉_𝑖−1(𝑗)) produces 𝑛−𝑖+1 ascending edges (𝑣^𝑘_𝑖(𝑗),𝑣^𝑘_𝑖−1(𝑗)) for 𝑘=𝑖−𝑗+1,…,𝑛−𝑗+1, and (𝑉_𝑖(𝑗),𝑉_𝑖+1(𝑗+1)) produces 𝑛−𝑖 descending edges (𝑣^𝑘_𝑖(𝑗),𝑣^𝑘_𝑖+1(𝑗+1)) for 𝑘=𝑖−𝑗+1,…,𝑛−𝑗.

The resulting 𝐺 is the disjoint union of 𝑛 directed graphs 𝐺¹,…,𝐺^𝑛, where each 𝐺^𝑘 contains all vertices of the form 𝑣^𝑘_𝑖(𝑗). Also 𝐺^𝑘 is isomorphic to the Cartesian product of two paths, with the lengths 𝑘−1 and 𝑛−𝑘. For example, for 𝑛=4, the graph 𝐺 is viewed as (4)

(where the multinodes are surrounded by ovals) and its components 𝐺¹, 𝐺², 𝐺³, 𝐺⁴ are viewed as (5)

So each node 𝑣=𝑣^𝑘_𝑖(𝑗) of 𝐺 has at most four incident edges, namely, (𝑣^𝑘_𝑖−1(𝑗−1),𝑣), (𝑣^𝑘_𝑖+1(𝑗),𝑣), (𝑣,𝑣^𝑘_𝑖−1(𝑗)), and (𝑣,𝑣^𝑘_𝑖+1(𝑗+1)); we refer to them, when exist, as the NW-, SW-, NE-, and SE-edges and denote them by 𝑒^NW(𝑣),𝑒^SW(𝑣),𝑒^NE(𝑣), and 𝑒^SE(𝑣), respectively.

By a feasible function in the model (with a given 𝑐), we mean a function 𝑓∶𝑉(𝐺)→ℤ₊ satisfying the following three conditions, where for an edge 𝑒=(𝑢,𝑣), 𝜕𝑓(𝑒) denotes the increment 𝑓(𝑢)−𝑓(𝑣) of 𝑓 on 𝑒, and 𝑒 is called tight for 𝑓, or 𝑓- tight, if 𝜕𝑓(𝑒)=0: (i)(𝑒)(𝐺);()(𝑣)𝑓is𝑚𝑜𝑛𝑜𝑡𝑜𝑛𝑒ontheedges,inthesensethat𝜕𝑓≥0forall𝑒∈𝐸ii0≤𝑓≤𝑐_𝑘foreach𝑣∈𝑉𝐺^𝑘(),𝑘=1,…,𝑛;iiieachmultinode𝑉_𝑖𝑗containsanode𝑣withthefollowingproperty∶theedge𝑒^SE(𝑢)istightforeachnode𝑢∈𝑉_𝑖𝑗preceding𝑣,and𝑒^SW𝑢^′istightforeachnode𝑢^′∈𝑉_𝑖𝑗succeeding𝑣. (6)

The first node 𝑣=𝑣^𝑘_𝑖(𝑗) (i.e., with 𝑘 minimum) satisfying the property in (iii) is called the switch-node of the multinode 𝑉_𝑖(𝑗). These nodes play an important role in our transformations of feasible functions in the model.

To describe the rule of transforming 𝑓∈ℱ(𝑐), we first extend each 𝐺^𝑘 by adding extra nodes and extra edges (following [4] and aiming to slightly simplify the description). In the extended directed graph 𝐺^𝑘, the node set consists of elements 𝑣^𝑘_𝑖(𝑗) for all 𝑖=0,…,𝑛+1 and 𝑗=0,…,𝑛 such that 𝑗≤𝑖. The edge set of 𝐺^𝑘 consists of all possible pairs of the form (𝑣^𝑘_𝑖(𝑗),𝑣^𝑘_𝑖−1(𝑗)) or (𝑣^𝑘_𝑖(𝑗),𝑣^𝑘_𝑖+1(𝑗+1)). Then all 𝐺^𝑘 are isomorphic. The disjoint union of these 𝐺^𝑘 gives the extended supporting graph 𝐺. The creation of 𝐺² from 𝐺² for 𝑛=4 is illustrated in the picture: (7)

Each feasible function on 𝑉(𝐺) is extended to the extra nodes 𝑣=𝑣^𝑘_𝑖(𝑗) as follows: 𝑓(𝑣)∶=𝑐_𝑘 if there is a path from 𝑣 to a node of 𝐺^𝑘, and 𝑓(𝑣)∶=0 otherwise (one may say that 𝑣 lies on the left of 𝐺^𝑘 in the former case and on the right of 𝐺^𝑘 in the latter case; in the above picture, such nodes 𝑣 are marked by white and black circles, resp.). 𝜕𝑓 is extended to the extra edges accordingly. In particular, each edge 𝑒 of 𝐺 not incident with a node of 𝐺 is tight; that is, 𝜕𝑓(𝑒)=0. For a node 𝑣=𝑣^𝑘_𝑖(𝑗) with 1≤𝑗≤𝑖≤𝑛, define the value 𝜀(𝑣)=𝜀_𝑓(𝑣) by 𝜀(𝑣)∶=𝜕𝑓𝑒^NW(𝑣)−𝜕𝑓𝑒^SE(𝑢)=𝜕𝑓𝑒^SW(𝑣)−𝜕𝑓𝑒^NE(𝑢), (8) where 𝑢∶=𝑣^𝑘_𝑖(𝑗−1). For a multinode 𝑉_𝑖(𝑗), define the numbers 𝜀_𝑖𝑗∶=𝜀(𝑣)∶𝑣∈𝑉_𝑖𝑗𝜀_𝑖,𝑝,𝑗∶=𝜀_𝑖𝑝+𝜀_𝑖𝑝+1+⋯+𝜀_𝑖𝑗for1≤𝑝≤𝑗,_∼𝜀_𝑖𝑗∶=max0,min𝜀_𝑖𝑝,𝑗∶𝑝=1,…,𝑗. (9)(10) We call 𝜀(𝑣), 𝜀_𝑖(𝑗), and ^∼𝜀_𝑖(𝑗) the slack at a node 𝑣, the total slack at a multinode 𝑉_𝑖(𝑗), and the reduced slack at 𝑉_𝑖(𝑗), respectively. (Note that 𝜀, ^∼𝜀 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 𝑓 (or the moves from 𝑓). At most 𝑛 transformations 𝜙₁,…,𝜙_𝑛 are possible. Each 𝜙_𝑖 changes 𝑓 within level 𝑖 and is applicable when this level contains a multinode 𝑉_𝑖(𝑗^′) with ^∼𝜀_𝑖(𝑗^′)>0. In this case we take the multinode 𝑉_𝑖(𝑗) such that ^∼𝜀_𝑖𝑗>0,^∼𝜀_𝑖𝑞=0for𝑞>𝑗, (11) referring to it as the active multinode for the given 𝑓 and 𝑖. We increase 𝑓 by 1 at the switch-node in 𝑉_𝑖(𝑗), preserving 𝑓 on the other nodes of 𝐺. It is shown [4] that the resulting function 𝜙_𝑖(𝑓) is again feasible.

As a result, the model generates 𝑛-colored directed graph 𝒦(𝑐)=(ℱ,ℰ₁⊔⋯⊔ℰ_𝑛), where each color class ℰ_𝑖 is formed by the edges (𝑓,𝜙_𝑖(𝑓)) for all feasible functions 𝑓 to which the operator 𝜙_𝑖 is applicable. This graph is just an 𝐴_𝑛-crystal.

Theorem 1 (see [4, Th. 5.1]). For each 𝑐∈ℤ^𝑛₊, the 𝑛-colored graph 𝒦(𝑐) is exactly the 𝐴_𝑛-crystal 𝐾(𝑐).

2.3. Principal Lattice and (𝑛−1)-Colored Subcrystals of an 𝐴_𝑛-Crystal

Based on the crossing model, [4] reveals some important ingredients and relations for an 𝐴_𝑛-crystal 𝐾=𝐾(𝑐). One of them is the so-called principal lattice, which is defined as follows.

Let 𝑎∈ℤ^𝑛₊ and 𝑎≤𝑐. One easily checks that the function on the vertices of the supporting graph 𝐺 that takes the constant value 𝑎_𝑘 within each subgraph 𝐺^𝑘 of 𝐺, 𝑘=1,…,𝑛, is feasible. We denote this function and the vertex of 𝐾 corresponding to it by 𝑓[𝑎] and 𝑣[̆𝑎], respectively, and call them principal ones. So the set of principal vertices is bijective to the integer box ℬ(𝑐)∶={𝑎∈ℤ^𝑛∶0≤𝑎≤𝑐}; this set is called the principal lattice of 𝐾 and denoted by Π=Π(𝑐). When it is not confusing, the term “principal lattice” may also be applied to ℬ(𝑐).

The following properties of the principal lattice will be essentially used later.

Proposition 2 (see [4, Statement (6.4)]). Let 𝑎∈ℬ(𝑐), 𝑘∈{1,…,𝑛}, and 𝑎^′∶=𝑎+1_𝑘 (where 1_𝑘 is 𝑖th unit base vector in ℝ^𝑛). The principal vertex 𝑣[̆𝑎^′] is obtained from 𝑣[̆𝑎] by applying the operator string 𝑆_𝑛,𝑘∶=𝑤_{𝑛,𝑘,𝑛−𝑘+1}⋯𝑤_𝑛,𝑘,2𝑤_𝑛,𝑘,1, (12) where for 𝑗=1,…,𝑛−𝑘+1, the substring 𝑤_{𝑛,𝑘,𝑗} is defined as 𝑤_{𝑛,𝑘,𝑗}∶=𝐹_𝑗𝐹_𝑗+1⋯𝐹_{𝑗+𝑘−1}. (13) When acting on Π, any two (applicable) strings 𝑆_𝑛,𝑘,𝑆_𝑛,𝑘′ commute. In particular, any principal vertex 𝑣[̆𝑎] is expressed via the source 𝑠_𝐾=𝑣[̆0] as 𝑣̆[𝑎]=𝑆^𝑎𝑛_𝑛,𝑛𝑆^𝑎𝑛−1_{𝑛,𝑛−1}⋯𝑆^𝑎1_𝑛,1𝑠_𝐾. (14)

Proposition 3 (see [4, Prop. 6.1]). For 𝑐^′,𝑐^′′∈𝑍^𝑛₊ with 𝑐^′≤𝑐^′′≤𝑐, let 𝐾(𝑐^′∶𝑐^′′) be the subgraph of 𝐾(𝑐) formed by the vertices and edges contained in (directed) paths from 𝑣[̆𝑐^′] to 𝑣[̆𝑐^′′] (the interval of 𝐾(𝑐) from 𝑣[̆𝑐^′] to 𝑣[̆𝑐^′′]). Then 𝐾(𝑐^′∶𝑐^′′) is isomorphic to the 𝐴_𝑛-crystal 𝐾(𝑐^′′−𝑐^′), and the principal lattice of 𝐾^′ consists of the principal vertices 𝑣[̆𝑎] of 𝐾(𝑐) with 𝑐^′≤𝑎≤𝑐^′′.

Let 𝒦^(−𝑛)(𝑐) denote the set of subcrystals with colors 1,…,𝑛−1, and 𝒦⁽⁻¹⁾ the set of subcrystals with colors 2,…,𝑛 in 𝐾 (recall that a subcrystal is assumed to be connected and maximal for the corresponding subset of colors).

Proposition 4 (see [4, Prop. 7.1]). Each subcrystal in 𝒦^(−𝑛) (in 𝒦⁽⁻¹⁾) contains precisely one principal vertex. This gives a bijection between 𝒦^(−𝑛) and Π (resp., between 𝒦⁽⁻¹⁾ and Π).

We refer to the members of 𝒦^(−𝑛) and 𝒦⁽⁻¹⁾ as upper and lower ((𝑛−1)-colored) subcrystals of 𝐾, respectively. For 𝑎∈ℬ(𝑐), the upper subcrystal containing the vertex 𝑣[̆𝑎] is denoted by 𝐾^↑[𝑎]. This subcrystal has its own principal lattice of dimension 𝑛−1, which is denoted by Π^↑[𝑎]. We say that the coordinate tuple 𝑎 is the locus of 𝐾^↑[𝑎] (and of Π^↑[𝑎]) in Π. Analogously, for 𝑏∈ℬ(𝑐), the lower subcrystal containing 𝑣[̆𝑏] is denoted by 𝐾^↓[𝑏] and its principal lattice by Π^↓[𝑏]; we say that 𝑏 is the locus of 𝐾^↓[𝑏] (and of Π^↓[𝑏]) in Π. It turns out that the parameters of upper and lower subcrystals can be expressed explicitly as follows.

Proposition 5 (see [4, Props. 7.2, 7.3]). For 𝑎∈ℬ(𝑐), the upper subcrystal 𝐾^↑[𝑎] is isomorphic to the 𝐴_𝑛−1-crystal 𝐾(𝑐^↑), where 𝑐^↑ is the tuple (𝑐^↑₁,…,𝑐^↑_𝑛−1) defined by 𝑐^↑_𝑖∶=𝑐_𝑖−𝑎_𝑖+𝑎_𝑖+1,𝑖=1,…,𝑛−1. (15) The principal vertex 𝑣[̆𝑎] is contained in the upper lattice Π^↑[𝑎] and its coordinate ℏ^↑=(ℏ^↑₁,…,ℏ^↑_𝑛−1) in Π^↑[𝑎] satisfies ℏ^↑_𝑖=𝑎_𝑖+1,𝑖=1,…,𝑛−1. (16)
Symmetrically, for 𝑏∈ℬ(𝑐), the lower subcrystal 𝐾^↓[𝑏] is isomorphic to the 𝐴_𝑛−1-crystal 𝐾(𝑐^↓) with colors 2,…,𝑛, where 𝑐^↓ is defined by 𝑐^↓_𝑖∶=𝑐_𝑖−𝑏_𝑖+𝑏_𝑖−1,𝑖=2,…,𝑛. (17) The principal vertex 𝑣[̆𝑏] is contained in the lower lattice Π^↓[𝑏] and its coordinate ℏ^↓=(ℏ^↓₂,…,ℏ^↓_𝑛) in Π^↓[𝑏] satisfies ℏ^↓_𝑖=𝑏_𝑖−1,𝑖=2,…,𝑛. (18)

We call 𝑣[̆𝑎] the heart of 𝐾^↑[𝑎] w.r.t. 𝐾, and similarly for lower subcrystals.

(One more result given in [4, Remark 5] is a piecewise linear formula to compute, for an (𝑛−1)-tuple 𝑞, the number of upper subcrystals of 𝐾(𝑐) with the parameter equal to 𝑞, 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 𝑐∈ℤ^𝑛₊, form the partition 𝜆=(𝜆₁≥𝜆₂≥⋯≥𝜆_𝑛≥𝜆_𝑛+1=0) by setting 𝜆_𝑖∶=𝑐[1∶𝑛−𝑖+1], where 𝑐[𝑝∶𝑞] denotes 𝑐_𝑝+𝑐_𝑝+1+⋯+𝑐_𝑞. A GT-pattern for 𝜆 is a triangular array 𝑋=(𝑥_𝑖𝑗)_{1≤𝑗≤𝑖≤𝑛} of integers satisfying (a) 𝑥_𝑖𝑗≥𝑥_{𝑖−1,𝑗},𝑥_{𝑖+1,𝑗+1} and (b) 𝜆_𝑖≥𝑥_𝑛,𝑖≥𝜆_𝑖+1, for all possible 𝑖,𝑗. It is shown in [4] that the set of feasible functions 𝑓 in the crossing model ℳ_𝑛(𝑐) is bijective to the set of GT-patterns 𝑋 for 𝜆; such a correspondence is given by 𝑥_𝑖,𝑗∶=𝑓_𝑖(𝑗)+𝑐[1∶𝑖−𝑗], where 𝑓_𝑖(𝑗) denotes the sum of values of 𝑓 over the multinode 𝑉_𝑖(𝑗). However, it is not clear how to visualize, and work with, principal vertices directly in terms of GT-patterns, whereas such vertices are well visualized and fit to handle in the crossing model.

3. Assembling an 𝐴_𝑛-Crystal

As mentioned in the Introduction, the structure of an 𝐴_𝑛-crystal 𝐾=𝐾(𝑐) can be described in a recursive manner. The idea is as follows. We know that 𝐾 contains |Π|=(𝑐₁+1)×⋯×(𝑐_𝑛+1) upper subcrystals (with colors 1,…,𝑛−1) and |Π| lower subcrystals (with colors 2,…,𝑛). Moreover, the parameters of these subcrystals are expressed explicitly by (15) and (17). Assume that the set 𝒦^(−𝑛) of upper subcrystals and the set 𝒦⁽⁻¹⁾ of lower subcrystals are already available. Then in order to assemble 𝐾, it suffices to point out, in appropriate terms, the intersection 𝐾^↑[𝑎]∩𝐾^↓[𝑏] for all pairs 𝑎,𝑏∈ℬ(𝑐) (the intersection may either be empty or consist of one or more (𝑛−2)-colored subcrystals with colors 2,…,𝑛−1 in 𝐾). We give an appropriate characterization in Theorems 7 and 8.

To state them, we need additional terminology and notation. Consider a subcrystal 𝐾^↑[𝑎], and let 𝑐^↑,ℏ^↑ be defined as in (15), (16). For 𝑝=(𝑝₁,…,𝑝_𝑛−1)∈ℬ(𝑐^↑), the vertex in the upper lattice Π^↑[𝑎] having the coordinate 𝑝 is denoted by 𝑣^↑[𝑎,𝑝]. We call the vector Δ∶=𝑝−ℏ^↑ the deviation of 𝑣^↑[𝑎,𝑝] from the heart 𝑣[̆𝑎] in Π^↑[𝑎] and we will use the alternative notation 𝑣^↑[𝑎∣Δ] for this vertex. In particular, 𝑣[̆𝑎]=𝑣^↑[𝑎,ℏ^↑]=𝑣^↑[𝑎∣0].

Similarly, for a lower subcrystal 𝐾^↓[𝑏], let 𝑐^↓,ℏ^↓ be as in (17), (18). For 𝑞=(𝑞₂,…,𝑞_𝑛)∈ℬ(𝑐^↓), the vertex with the coordinate 𝑞 in Π^↓[𝑏] is denoted by 𝑣^↓[𝑎,𝑞]. Its deviation is ∇∶=𝑞−ℏ^↓, and we may alternatively denote this vertex by 𝑣^↓[𝑏∣∇].

We call an (𝑛−2)-colored subcrystal with colors 2,…,𝑛−1 in 𝐾 a middle subcrystal and denote the set of these by 𝒦^{(−1,−𝑛)}. Each middle subcrystal 𝐾^↑↓ is a lower subcrystal of some upper subcrystal 𝐾^′=𝐾^↑[𝑎] of 𝐾. By Proposition 4 applied to 𝐾^′, 𝐾^↑↓ has a unique vertex 𝑣^↑[𝑎∣Δ] in the lattice Π^↑[𝑎]. So each 𝐾^↑↓ can be encoded by a pair (𝑎,Δ) formed by a point 𝑎∈ℬ(𝑐) and a deviation Δ in Π^↑[𝑎]. At the same time, 𝐾^↑↓ is an upper subcrystal of some lower subcrystal 𝐾^↓[𝑏] of 𝐾 and has a unique vertex 𝑣^↓[𝑏∣∇] in Π^↓[𝑏]. Therefore, the members of 𝒦^{(−1,−𝑛)} determine a bijection 𝜁∶(𝑎,Δ)↦(𝑏,∇) (19) between all pairs (𝑎,Δ) concerning upper subcrystals and all pairs (𝑏,∇) concerning lower subcrystals.

The map 𝜁 is expressed explicitly in the following two theorems. Here for a tuple 𝜌=(𝜌_𝑖∶𝑖∈𝐼) of reals, we denote by 𝜌⁺ (𝜌⁻) the tuple formed by 𝜌⁺_𝑖∶=max{0,𝜌_𝑖} (resp., 𝜌⁻_𝑖∶=min{0,𝜌_𝑖}), 𝑖∈𝐼.

Theorem 7 (on two deviations). Let 𝑎∈ℬ(𝑐) and let Δ=(Δ₁,…,Δ_𝑛−1) be a deviation in Π^↑[𝑎] (from the heart of 𝐾^↑[𝑎]). Let (𝑏,∇)=𝜁(𝑎,Δ). Then ∇_𝑖=−Δ_𝑖−1,𝑖=2,…,𝑛. (20)

Theorem 8 (on two loci). Let 𝑎,𝑏,Δ,∇ be as in the previous theorem. Then 𝑏_𝑖=𝑎_𝑖+Δ⁺_𝑖+Δ⁻_𝑖−1,𝑖=1,…,𝑛, (21) letting Δ₀=Δ_𝑛∶=0.

Proofs of these theorems will be given in the next section.

Based on Theorems 7 and 8, the crystal 𝐾(𝑐) 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 𝐾^↑[𝑎]≃𝐾(𝑐^↑), its principal lattice is distinguished by the use of the corresponding injective map 𝜎∶ℬ(𝑐^↑)→𝑉(𝐾(𝑐^↑)), and similarly for the lower subcrystals. We delete the edges with color 1 in each 𝐾^↑[𝑎] and extract the components of the resulting graphs, forming the set 𝒦^{(−1,−𝑛)} (arranged as a list) of all middle subcrystals of 𝐾(𝑐). Each 𝐾^↑↓∈𝒦^{(−1,−𝑛)} is encoded by a corresponding pair (𝑎,Δ), where 𝑎∈ℬ(𝑐) and the deviation Δ in Π^↑[𝑎] is determined by the use of 𝜎 as above. Acting similarly for the lower subcrystals 𝐾^↓[𝑏] (by deleting the edges with color 𝑛), we obtain the same set of middle subcrystals (arranged as another list), each of which being encoded by a corresponding pair (𝑏,∇), where 𝑏∈ℬ(𝑐) and ∇ is a deviation in Π^↓(𝑏). 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 𝑛, we obtain the desired crystal 𝐾(𝑐). The corresponding map ℬ(𝑐)→𝑉(𝐾(𝑐)) is constructed easily (e.g., by the use of operator strings as in Proposition 2).

We conclude this section with several remarks.

Remark 9. For each 𝑎∈ℬ(𝑐) and each vertex 𝑣=𝑣^↑[𝑎,𝑝] in the upper lattice Π^↑[𝑎], one can express the parameter 𝑐^↑↓=(𝑐₂^↑↓,…,𝑐^↑↓_𝑛−1) of the middle subcrystal 𝐾^↑↓ containing 𝑣, as well as the coordinate ℏ^↑↓=(ℏ₂^↑↓,…,ℏ^↑↓_𝑛−1) of its heart w.r.t. 𝐾^↑[𝑎] in the principal lattice of 𝐾^↑↓. Indeed, since 𝐾^↑↓ is a lower subcrystal of 𝐾^↑[𝑎], one can apply relations as in (17) and (18). Denoting the parameter of 𝐾^↑[𝑎] by 𝑐^↑ and the coordinate of its heart in Π^↑[𝑎] by ℏ^↑, letting Δ∶=𝑝−ℏ^↑, and using (15), (16), we have for 𝑖=2,…,𝑛−1, 𝑐_𝑖^↑↓=𝑐^↑_𝑖−𝑝_𝑖+𝑝_𝑖−1=𝑐_𝑖−𝑎_𝑖+𝑎_𝑖+1−𝑎_𝑖+1+Δ_𝑖+𝑎_𝑖+Δ_𝑖−1=𝑐_𝑖−Δ_𝑖+Δ_i−1,ℏ_𝑖^↑↓=𝑝_𝑖−1=ℏ^↑_𝑖−1+Δ_𝑖−1=𝑎_𝑖+Δ_𝑖−1. (22)(23)
Symmetrically, if 𝐾^↑↓ is contained in 𝐾^↓[𝑏] and has deviation ∇ in Π^↓[𝑏], then for 𝑖=2,…,𝑛−1, 𝑐_𝑖^↑↓=𝑐_𝑖−∇_𝑖+∇_𝑖+1,ℏ_𝑖^↓↑=𝑏_𝑖+∇_𝑖+1, (24) where ℏ^↓↑ is the coordinate of the heart of 𝐾^↑↓ w.r.t. 𝐾^↓[𝑏] in the principal lattice of 𝐾^↑↓ (note that ℏ^↓↑ may differ from ℏ^↑↓). We will use (22) and (24) in Section 4.

Remark 10. A straightforward implementation of the above recursive method of constructing 𝐾=𝐾(𝑐) takes 𝑂(2^𝑞(𝑛)𝑁) time and space, where 𝑞(𝑛) is a polynomial in 𝑛 and 𝑁 is the number of vertices of 𝐾. Here the factor 2^𝑞(𝑛) appears because the total number of vertices in the upper and lower subcrystals is 2𝑁 (implying that there appear 4𝑁 vertices in total on the previous step of the recursion, and so on). Therefore, such an implementation has polynomial complexity of the size of the output for each fixed 𝑛, 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,…,𝑛). We call a subcrystal of this sort a color-interval subcrystal, or a CI-subcrystal, of 𝐾. In fact, every CI-subcrystal of 𝐾 appears in the process. Since the number of intervals is 𝑛(𝑛+1)/2 and the CI-subcrystals concerning equal intervals are pairwise disjoint, the total number of vertices of all CI-subcrystals of 𝐾 is 𝑂(𝑛²𝑁). It is not difficult to implement the recursive process so that each CI-subcrystal 𝐾^′ be explicitly constructed only once. As a result, we obtain the following.

Proposition 11. Let 𝑐∈ℤ^𝑛₊. The 𝐴_𝑛-crystal 𝐾(𝑐) and all its CI-subcrystals can be constructed in 𝑂(𝑞^′(𝑛)|𝑉(𝐾(𝑐))|) time and space, where 𝑞^′(𝑛) is a polynomial in 𝑛.

Remark 12. Relation (21) shows that the intersection of 𝐾^↑[𝑎] and 𝐾^↓[𝑏] may consist of many middle subcrystals. Indeed, if Δ_𝑖>0 and Δ_𝑖−1<0 for some 𝑖, then 𝑏 does not change by simultaneously decreasing Δ_𝑖 by 1 and increasing Δ_𝑖−1 by 1. The number of common middle subcrystals of 𝐾^↑[𝑎] and 𝐾^↓[𝑏] for arbitrary 𝑎,𝑏∈ℬ(𝑐) can be expressed by an explicit piecewise linear formula, using (21) and the box constraints −𝑎_𝑖+1≤Δ_𝑖≤𝑐_𝑖−𝑎_𝑖, 𝑖=1,…,𝑛−1, on the deviations Δ in Π^↑[𝑎] (which follow from (15) and (16)).

4. Proofs of Theorems 7 and 8

Let 𝑎, Δ, 𝑏, ∇ be as in the hypotheses of Theorem 7. First we show that Theorem 7 follows from Theorem 8.

Proof of (20) (in the assumption that (21) is valid). The middle subcrystal 𝐾^↑↓ determined by (𝑎,Δ) is the same as the one determined by (𝑏,∇). The parameter 𝑐^↑↓ of 𝐾^↑↓ is expressed simultaneously by (22) and by (24). Then 𝑐_𝑖−Δ_𝑖+Δ_𝑖−1=𝑐_𝑖−∇_𝑖+∇_𝑖+1 for 𝑖=2,…,𝑛−1. Therefore, Δ₁+∇₂=Δ₂+∇₃=⋯=Δ_𝑛−1+∇_𝑛=∶𝛼. (25)
In order to obtain (20), one has to show that 𝛼=0. We argue as follows. Renumber the colors 1,…,𝑛 as 𝑛,…,1, respectively; this yields the crystal 𝐾=𝐾(𝑐)̂ symmetric to 𝐾(𝑐). Then 𝐾^↓[𝑏] turns into the upper subcrystal 𝐾^↑[𝑏̂] of 𝐾, where (𝑏̂₁,…,𝑏̂_𝑛)=(𝑏_𝑛,…,𝑏₁). Also the deviation ∇ in Π^↓[𝑏] turns into the deviation ∇=(∇₁,…,∇_𝑛−1)=(∇_𝑛,…,∇₂) in the principal lattice of 𝐾^↑[𝑏̂]. Applying relations as in (21) to (𝑏̂,∇), we have 𝑎_𝑖=𝑏̂_𝑖+∇⁺_𝑖+∇⁻_𝑖−1=𝑏_{𝑛−𝑖+1}+∇⁺_{𝑛−𝑖+1}+∇⁻_{𝑛−𝑖+2},𝑖=1,…,𝑛, (26) where 𝑎_𝑖∶=𝑎_{𝑛−𝑖+1} and ∇⁺_𝑛∶=∇⁻₀∶=0. On the other hand, (21) for (𝑎,Δ) gives 𝑏̂_𝑖=𝑏_{𝑛−𝑖+1}=𝑎_{𝑛−𝑖+1}+Δ⁺_{𝑛−𝑖+1}+Δ⁻_𝑛−𝑖,𝑖=1,…,𝑛. (27)
Relations (26) and (27) imply 𝑎_{𝑛−𝑖+1}=𝑏_{𝑛−𝑖+1}+∇⁺_{𝑛−𝑖+1}+∇⁻_{𝑛−𝑖+2}=𝑎_{𝑛−𝑖+1}+Δ⁺_{𝑛−𝑖+1}+Δ⁻_𝑛−𝑖+∇⁺_{𝑛−𝑖+1}+∇⁻_{𝑛−𝑖+2}, (28) whence Δ⁺_{𝑛−𝑖+1}+Δ⁻_𝑛−𝑖+∇⁺_{𝑛−𝑖+1}+∇⁻_{𝑛−𝑖+2}=0,𝑖=1,…,𝑛. (29) Adding up the latter equalities, we obtain Δ₁+⋯+Δ_𝑛−1+∇₂+⋯+∇_𝑛=0. (30) This and (25) imply (𝑛−1)𝛼=0. Hence 𝛼=0, yielding (20) and Theorem 7.

Proof of Theorem 8. It is more intricate and essentially uses the crossing model.
For a feasible function 𝑓∈ℱ(𝑐) and its corresponding vertex 𝑣 in 𝐾=𝐾(𝑐), we may denote 𝑣 as 𝑣_𝑓 and 𝑓 as 𝑓_𝑣. From the crossing model it is seen that ifavertex𝑣∈𝑉(𝐾)belongsto𝐾^↑[𝑎]andto𝐾^↓[𝑏],thenthetuples𝑎and𝑏areexpressedviathevaluesof𝑓=𝑓_𝑣𝑎inlevels𝑛and1asfollows∶_𝑘=𝑓𝑣^𝑘_𝑛(𝑛−𝑘+1),and𝑏_𝑘=𝑓𝑣^𝑘₁(1)for𝑘=1,…,𝑛. (31) Indeed, the principal vertex 𝑣[̆𝑎] is reachable from 𝑣 by applying operators 𝐹_𝑖 or 𝐹_𝑖⁻¹ with 𝑖≠𝑛. The corresponding moves in the crossing model do not change 𝑓 within level 𝑛. Similarly, 𝑣[̆𝑏] is reachable from 𝑣 by applying operators 𝐹_𝑖 or 𝐹_𝑖⁻¹ with 𝑖≠1, and the corresponding moves do not change 𝑓 within level 1. Also the first (second) equality in (31) is valid for the principal function 𝑓_̆𝑣[𝑎] (resp., 𝑓_̆𝑣[𝑏]).
Next we introduce special functions on the node set 𝑉(𝐺) of the supporting graph 𝐺=𝐺_𝑛. Consider a component 𝐺^𝑘=(𝑉^𝑘,𝐸^𝑘) of 𝐺. It is a rectangular grid of size 𝑘×(𝑛−𝑘+1) (rotated by 45° in the visualization of 𝐺), and its vertex set is 𝑉^𝑘=𝑣^𝑘_𝑖𝑗∶𝑗=1,…,𝑛−𝑘+1,𝑖=𝑗,…,𝑗+𝑘−1. (32) To represent it in a more convenient form, let us rename 𝑣^𝑘_𝑖(𝑗) as 𝑥^𝑘_{𝑖−𝑗+1}(𝑗) or 𝑥_{𝑖−𝑗+1}(𝑗) (as though rotating 𝐺^𝑘 by 45°). Then 𝑉^𝑘=𝑥_𝑚𝑗∶𝑚=1,…,𝑘,𝑗=1,…,𝑛−𝑘+1, (33) the SE-edges of 𝐺^𝑘 become of the form (𝑥_𝑚(𝑗),𝑥_𝑚(𝑗+1)), and the NE-edges become of the form (𝑥_𝑚(𝑗),𝑥_𝑚−1(𝑗)). We distinguish the following subsets of 𝑉^𝑘: (i)the SW-side 𝑃^𝑘∶={𝑥_𝑘(1),…,𝑥_𝑘(𝑛−𝑘+1)};(ii)the right rectangle 𝑅^𝑘∶={𝑥_𝑚(𝑗)∶1≤𝑚≤𝑘−1,1≤𝑗≤𝑛−𝑘+1};(iii)the left rectangle 𝐿^𝑘∶={𝑥_𝑚(𝑗)∶1≤𝑚≤𝑘,1≤𝑗≤𝑛−𝑘}. Denote the characteristic functions (in ℝ^𝑉𝑘) of 𝑃^𝑘,𝑅^𝑘, and 𝐿^𝑘 as 𝜋^𝑘, 𝜌^𝑘,and𝜆^𝑘, respectively.
Return to 𝑎 and Δ as above. We associate to (𝑎,Δ) the functions 𝑓^𝑘_𝑎,Δ∶=𝑎_𝑘𝜋^𝑘+𝑎_𝑘+Δ⁻_𝑘−1𝜌^𝑘+Δ⁺_𝑘𝜆^𝑘 (34) on 𝑉^𝑘 for 𝑘=1,…,𝑛 (see Figure 1) and define 𝑓_𝑎,Δ to be the function on 𝑉(𝐺) whose restriction to each 𝑉^𝑘 is 𝑓^𝑘_𝑎,Δ.
In view of (31), 𝑓=𝑓_𝑎,Δ takes the values in levels 𝑛 and 1 as required in (21) (with 𝑘 in place of 𝑖); namely, 𝑓(𝑣^𝑘_𝑛(𝑛−𝑘+1))=𝑎_𝑘 and 𝑓(𝑣^𝑘₁(1))=𝑎_𝑘+Δ⁺_𝑘+Δ⁻_𝑘−1 for 𝑘=1,…,𝑛. Therefore, to obtain (21) it suffices to show the following.
Lemma 13. (i) The function 𝑓=𝑓_𝑎,Δ is feasible. (ii) The vertex 𝑣_𝑓 belongs to Π^↑[𝑎] and has the deviation Δ in it; in other words, 𝑓=𝑓_{̆𝑣[𝑎∣Δ]}.
Proof. First we prove assertion (i). Let 𝑘∈{1,…,𝑛}. We partition 𝑉^𝑘 into four subsets (rectangular pieces): 𝑍^𝑘₁∶=𝑃^𝑘−𝑥^𝑘_𝑘(,𝑍𝑛−𝑘+1)_𝑘2∶=𝐿^𝑘⧵𝑃^𝑘,𝑍^𝑘₃∶=𝑥^𝑘_𝑘(𝑛−𝑘+1),𝑍^𝑘₄∶=𝑅^𝑘⧵𝐿^𝑘, (35) where 𝑍^𝑘₂=𝑍^𝑘₄=∅ when 𝑘=1, and 𝑍^𝑘₁=𝑍^𝑘₂=∅ when 𝑘=𝑛. By (34), 𝑓takesaconstantvaluewithineachpiece𝑍^𝑘_𝑞𝑎;namely,_𝑘+Δ⁺_𝑘on𝑍^𝑘₁,𝑎_𝑘+Δ⁺_𝑘+Δ⁻_𝑘−1on𝑍^𝑘₂;𝑎_𝑘on𝑍^𝑘₃,𝑎_𝑘+Δ⁻_𝑘−1on𝑍^𝑘₄ (36) (as illustrated in Figure 1). Also each edge of 𝐺^𝑘 connecting different pieces goes either from 𝑍^𝑘₁ to 𝑍^𝑘₂∪𝑍^𝑘₃ or from 𝑍^𝑘₂∪𝑍^𝑘₃ to 𝑍^𝑘₄. This and (36) imply that 𝜕𝑓(𝑒)≥0 for each edge 𝑒∈𝐸^𝑘, whence 𝑓 satisfies (6)(i).
The deviation Δ is restricted as −ℏ^↑≤Δ≤𝑐^↑−ℏ^↑, where 𝑐^↑ is the parameter of the subcrystal 𝐾^↑[𝑎] and ℏ^↑ is the coordinate of its heart 𝑣[̆𝑎] in Π^↑[𝑎]. Formulas (15) and (16) for 𝑐^↑ and ℏ^↑ give −𝑎_𝑘+1≤Δ_𝑘≤𝑐_𝑘−𝑎_𝑘,−𝑎_𝑘≤Δ_𝑘−1≤𝑐_𝑘−1−𝑎_𝑘−1. (37) The inequalities Δ_𝑘≤𝑐_𝑘−𝑎_𝑘 and 𝑎_𝑘≤𝑐_𝑘 imply 𝑎_𝑘+Δ⁺_𝑘≤𝑐_𝑘. The inequalities −𝑎_𝑘≤Δ_𝑘−1 and 𝑎_𝑘≥0 imply 𝑎_𝑘+Δ⁻_𝑘−1≥0. Then, in view of (36), we obtain 0≤𝑓(𝑣)≤𝑐_𝑘 for each node 𝑣 of 𝐺^𝑘, yielding (6)(ii).
To verify the switch condition (6)(iii), consider a multinode 𝑉_𝑖(𝑗) with 𝑖<𝑛. It consists of 𝑛−𝑖+1 nodes 𝑣^𝑘_𝑖(𝑗), where 𝑖−𝑗+1≤𝑘≤𝑛−𝑗+1.
Let 𝑖≤𝑛−2. Suppose that 𝑣=𝑣^𝑘_𝑖(𝑗) is a node whose SW-edge 𝑒=(𝑢,𝑣) exists and is not 𝑓-tight. This is possible only if 𝑢∈𝑍^𝑘₁ and 𝑣∈𝑍^𝑘₂. In this case, 𝑘 is determined as 𝑘=𝑖−𝑗+2; that is, 𝑣 is the second node in 𝑉_𝑖(𝑗). We observe that (a) for the first node 𝑣_𝑖^𝑘−1(𝑗) of 𝑉_𝑖(𝑗), both ends of its SE-edge 𝑒^′ belong to the piece 𝑍₁^𝑘−1; and (b) for any node 𝑣^𝑘′_𝑖(𝑗) with 𝑘^′>𝑘 in 𝑉_𝑖(𝑗), both ends of its SW-edge 𝑒^′′ belong either to 𝑍^𝑘′₂ or to 𝑍^𝑘′₄. So, such 𝑒^′ and 𝑒^′′ are 𝑓-tight. Therefore, the node 𝑣 satisfies the condition in (6)(iii) for 𝑉_𝑖(𝑗).
Now let 𝑖=𝑛−1. Then 𝑉_𝑖(𝑗) consists of two nodes 𝑣=𝑣^𝑛−𝑗_𝑛−1(𝑗) and 𝑣^′=𝑣^{𝑛−𝑗+1}_𝑛−1(𝑗). Put 𝑘∶=𝑛−𝑗. Then the edge 𝑒=𝑒^SE(𝑣) goes from 𝑍^𝑘₁ to 𝑍^𝑘₃={𝑥^𝑘_𝑘(𝑛−𝑘+1)}, and the edge 𝑒^′=𝑒^SW(𝑣^′) goes from 𝑍₃^𝑘+1={𝑥^𝑘+1_𝑘+1(𝑛−𝑘)} to 𝑍₄^𝑘+1. By (36), we have 𝜕𝑓(𝑒)=(𝑎_𝑘+Δ⁺_𝑘)−𝑎_𝑘=Δ⁺_𝑘 and 𝜕𝑓(𝑒^′)=𝑎_𝑘+1−(𝑎_𝑘+1+Δ⁻_𝑘)=−Δ⁻_𝑘. Since at least one of Δ⁺_𝑘,Δ⁻_𝑘 is zero, we conclude that at least one of 𝑒,𝑒^′ is tight. So (6)(iii) is valid again.
Next we prove assertion (ii) in the lemma. (The idea is roughly as follows. For each 𝑘, compare the function 𝑓^𝑘_𝑎,Δ=∶𝑔 with the function ℎ on 𝑉^𝑘 taking the constant value 𝑎_𝑘. By (34), 𝑔=ℎ+Δ⁺_𝑘𝜆^𝑘+Δ⁻_𝑘−1𝜌^𝑘. In other words, 𝑔 is obtained from 𝑓_{̆𝑣[𝑎]∣𝑉𝑘} by adding Δ⁺_𝑘 times the “left rectangle function” 𝜆^𝑘, followed by subtracting |Δ⁻_𝑘−1| times the “right rectangle function” 𝜌^𝑘. A crucial observation is that adding 𝜆^𝑘 corresponds to applying the operator string 𝑆_{𝑛−1,𝑘} (or shifting by 𝑘th unit base vector in the upper principal lattice Π^↑[𝑎]), while subtracting 𝜌^𝑘 corresponds to applying 𝑆⁻¹_{𝑛−1,𝑘−1} (or shifting by minus (𝑘−1)th unit base vector in Π^↑[𝑎]). This is because the substrings 𝑤 in 𝑆_{𝑛−1,𝑘} correspond to the SW-NE paths in 𝐿^𝑘, and the substrings in 𝑆_{𝑛−1,𝑘−1} to similar paths in 𝑅^𝑘≃𝐿^𝑘−1).
Now we give a more careful and formal description. We use induction on 𝜂(Δ)∶=Δ₁+⋯+Δ_𝑛−1. (38)
In view of (37), 𝜂(Δ)≥−𝑎₂−⋯−𝑎_𝑛. Suppose that this turns into equality. Then Δ_𝑘=−𝑎_𝑘+1≤0 for 𝑘=1,…,𝑛−1, and 𝑓=𝑓_𝑎,Δ takes the following values within each 𝑉^𝑘 (cf. (36)): 𝑓(𝑣)=𝑎_𝑘 if 𝑣∈𝑃^𝑘, and 𝑓(𝑣)=0 if 𝑣∈𝑉^𝑘−𝑃^𝑘. This 𝑓 is the minimal feasible function whose values in level 𝑛 correspond to 𝑎; that is, 𝑣_𝑓 is the source of 𝐾^↑[𝑎]. Then 𝑣_𝑓 is the minimal vertex 𝑣[̆𝑎,0] in Π^↑[𝑎], and its deviation in Π^↑[𝑎] is just Δ, as required. This gives the base of our induction.
Now consider an arbitrary Δ satisfying (37). Let 𝑘 be such that Δ_𝑘<𝑐_𝑘−𝑎_𝑘 (if any) and define Δ^′_𝑘∶=Δ_𝑘+1 and Δ^′_𝑖∶=Δ_𝑖 for 𝑖≠𝑘. Then 𝜂(Δ)<𝜂(Δ^′). We assume by induction that assertion (ii) is valid for 𝑓_𝑎,Δ, and our aim is to show validity of (ii) for 𝑓_𝑎,Δ′.
In what follows 𝑓 stands for the former function 𝑓_𝑎,Δ.
Let 𝑣^′ be the vertex with the deviation Δ^′ in Π^↑[𝑎]. Both 𝑣_𝑓 and 𝑣^′ are principal vertices of the subcrystal 𝐾^↑[𝑎], and the coordinate of 𝑣^′ in Π^↑[𝑎] is obtained from the one of 𝑣_𝑓 by increasing its 𝑘th entry by 1. According to Proposition 2 (with 𝑛 is replaced by 𝑛−1), 𝑣^′ is obtained from 𝑣_𝑓 by applying the operator string 𝑆_{𝑛−1,𝑘}=𝑤_{𝑛−1,𝑘,𝑛−𝑘}⋯𝑤_{𝑛−1,𝑘,1}, (39) where 𝑤_{𝑛−1,𝑘,𝑗}=𝐹_𝑗⋯𝐹_{𝑗+𝑘−1} (cf. (12)). In light of this, we have to show that whenthesequenceofmovescorrespondingto𝑆_{𝑛−1,𝑘}𝑓isappliedto𝑓,theresultingfeasiblefunctionisexactly_𝑎,Δ′. (40)
For convenience, 𝑚th term 𝐹_{𝑗+𝑚−1} in the substring 𝑤_{𝑛−1,𝑘,𝑗} will be denoted by 𝜙(𝑗,𝑚), 𝑚=1,…,𝑘. So 𝑤_{𝑛−1,𝑘,𝑗}=𝜙(𝑗,1)𝜙(𝑗,2)⋯𝜙(𝑗,𝑘).
We distinguish between two cases: Δ_𝑘≥0 and Δ_𝑘<0.
Case 1 [Δ_𝑘≥0]. An essential fact is that the number 𝑘(𝑛−𝑘) of operators in 𝑆_{𝑛−1,𝑘} is equal to the number of nodes in the left rectangle 𝐿^𝑘 of 𝐺^𝑘. Moreover, the level of each node 𝑥_𝑚(𝑗) of 𝐿^𝑘 is equal to the “color” of the operator 𝜙(𝑗,𝑚) (indeed, 𝑥_𝑚(𝑗)=𝑣^𝑘_{𝑗+𝑚−1}(𝑗) and 𝜙(𝑗,𝑚)=𝐹_{𝑗+𝑚−1}).
Let 𝑓^𝑗,𝑚 denote the current function on 𝑉(𝐺) just before the application of 𝜙(𝑗,𝑚) (when the process starts with 𝑓=𝑓_𝑎,Δ). We assert that foreach𝑚,theapplicationof𝜙𝑗,𝑚to𝑓^𝑗,𝑚increasesthevalueatthenode𝑥^𝑘_𝑚𝑗by1, (41) whence (40) will immediately follow.
In order to show (41), we first examine tight edges and the slacks 𝜀(𝑣) of the nodes 𝑣 in levels <𝑛 for the initial function 𝑓. One can see from (36) that for𝑘^′=1,…,𝑛,eachnode𝑣ofthesubgraph𝐺^𝑘′hasatleastoneenteringedgei.e.,𝑒^SW(𝑣)or𝑒^NW(𝑣)whichis𝑓-tight,except,possibly,forthenodes𝑣^𝑘′_𝑘′(1),𝑣^𝑘′_𝑘′−1(1),𝑣^𝑘′_𝑛𝑛−𝑘^′+1,𝑣^𝑘′_𝑛−1𝑛−𝑘^′+1 (42)
indicated by stars in Figure 1.
Claim. For 𝑘^′=1,…,𝑛 and a node 𝑣 of 𝐺^𝑘′ at a level <𝑛,(a)if 𝑣≠𝑣^𝑘′_𝑘′(1),𝑣^𝑘′_𝑘′−1(1), then 𝜀(𝑣)=0;(b)if 𝑣=𝑣^𝑘′_𝑘′(1), then 𝜀(𝑣)=𝑐_𝑘′−𝑎_𝑘′−Δ⁺_𝑘′≥0;(c)if 𝑣=𝑣^𝑘′_𝑘′−1(1), then 𝜀(𝑣)=−Δ⁻_𝑘′≥0.
Proof of Claim. Let 𝑣=𝑣^𝑘′_𝑖(𝑗) and 𝑖<𝑛. By (8), the slack 𝜀(𝑣) is equal to 𝑓(𝑤)+𝑓(𝑧)−𝑓(𝑢)−𝑓(𝑣), where 𝑤∶=𝑣^𝑘′_𝑖−1(𝑗−1), 𝑧∶=𝑣^𝑘′_𝑖+1(𝑗), 𝑢∶=𝑣^𝑘′_𝑖(𝑗−1) (these vertices belong to the extended graph 𝐺^𝑘′). We consider possible cases and use (36).(i)If 𝑤,𝑧,𝑢 are in 𝐺^𝑘′, then 𝜕𝑓(𝑤,𝑣)=𝜕𝑓(𝑢,𝑧).(ii)If both 𝑣,𝑤 are in the piece 𝑍^𝑘′₁ of 𝐺^𝑘′, then 𝑓(𝑤)=𝑓(𝑣) and 𝑓(𝑢)=𝑓(𝑧)=𝑐_𝑘′.(iii)If 𝑗=1 and 𝑖≤𝑘^′−2, then 𝑓(𝑣)=𝑓(𝑧) and 𝑓(𝑢)=𝑓(𝑤)=𝑐_𝑘′. So in these cases we have 𝜀(𝑣)=0, yielding (a).(iv)Let 𝑣=𝑣^𝑘′_𝑘′(1). Then 𝑓(𝑣)=𝑎_𝑘′+Δ⁺_𝑘′ and 𝑓(𝑢)=𝑓(𝑤)=𝑓(𝑧)=𝑐_𝑘′. This gives 𝜀(𝑣)=𝑐_𝑘′−𝑎_𝑘′−Δ⁺_𝑘′, yielding (b).(v)Let 𝑣=𝑣^𝑘′_𝑘′−1(1). Then 𝑓(𝑣)=𝑎_𝑘′+Δ⁺_𝑘′+Δ⁻_𝑘′−1, 𝑓(𝑧)=𝑎_𝑘′+Δ⁺_𝑘′ and 𝑓(𝑢)=𝑓(𝑤)=𝑐_𝑘′. This gives 𝜀(𝑣)=−Δ⁻_𝑘′−1, yielding (c).
This claim and the relations Δ⁻_𝑘=0 and Δ_𝑘<𝑐_𝑘−𝑎_𝑘 enable us to estimate the total slacks 𝜀_𝑖(𝑗) for 𝑓 at the multinodes 𝑉_𝑖(𝑗) with 𝑖<𝑛: (i)theedge𝑒^SW𝑣_𝑘^𝑘+1(1)is𝑓-tight,𝜀𝑣^𝑘_𝑘(1)𝜀(𝑣)>0,and=0fortheothernodes𝑣in𝑉_𝑘(1);so𝜀_𝑘(1)()𝑣>0;iiif𝑖≠𝑘,𝑛,then𝜀^𝑖_𝑖(1),𝜀𝑣_𝑖^𝑖+1(1)≥0and𝜀(𝑣)=0fortheothernodes𝑣in𝑉_𝑖(1);so𝜀_𝑖(1)()(𝑣)𝑉≥0;iiiif𝑖≠𝑛and𝑗>1,then𝜀=0forallnodes𝑣in_𝑖𝑗;so𝜀_𝑖𝑗=0. (43)
Now we are ready to prove (41). When dealing with a current function 𝑓^𝑗,𝑚 and seeking for the node at level 𝑗+𝑚−1 where the operator 𝜙(𝑗,𝑚) should act to increase 𝑓^𝑗,𝑚, we can immediately exclude from consideration any node 𝑣 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 𝑓=𝑓^1,𝑘, there is only one node in level 𝑘 that has no tight entering edge, namely, 𝑣^𝑘_𝑘(1). So, at the first step of the process, the first operator 𝜙(1,𝑘) of 𝑆_{𝑛−1,𝑘} acts just at 𝑣^𝑘_𝑘(1), as required in (41).
Next consider a step with 𝑓^′∶=𝑓^𝑗,𝑚 for (𝑗,𝑚)≠(1,𝑘), assuming validity of (41) on the previous steps.
(A) Let 𝑗=1 (and 𝑚<𝑘). For 𝑣∶=𝑣^𝑘_𝑚(1) and 𝑧∶=𝑣^𝑘_𝑚+1(1), we have 𝑓^′(𝑣)=𝑓(𝑣)≤𝑓(𝑧)=𝑓^′(𝑧)−1. So the unique edge 𝑒=(𝑧,𝑣) entering 𝑣 is not 𝑓^′-tight. By (42), there are at most two other nodes in level 𝑚 that may have no tight entering edges for 𝑓 (and therefore, for 𝑓^′), namely, 𝑣^𝑚_𝑚(1) and 𝑣_𝑚^𝑚+1(1). Then 𝜙(1,𝑚) must act at 𝑣, as required in (41) (since the nontightness of the SW-edge 𝑒 of 𝑣 implies that none of the nodes 𝑣^𝑚′_𝑚(1) in 𝑉_𝑚(1) preceding 𝑣 (i.e., with 𝑚^′<𝑘) can be the switch-node).
(B) Let 𝑗>1. Comparing 𝑓^′ with 𝑓 in the node 𝑣∶=𝑥^𝑘_𝑚(𝑗)=𝑣^𝑘_{𝑗+𝑚−1}(𝑗) and its adjacent nodes, we observe that 𝑣 has no 𝑓^′-tight entering edge and that 𝜀_𝑓′(𝑣)>0. Also for any other node 𝑣^′ in level 𝑗+𝑚−1, one can see that if 𝑣^′ has a tight entering edge for 𝑓, then this property holds for 𝑓^′ as well, and that 𝜀_𝑓(𝑣^′)≥𝜀_𝑓′(𝑣^′)≥0. Using this, properties (42), (43) (iii), and condition (11), one can conclude that the total and reduced slacks for 𝑓^′ at the multinode 𝑉^′∶=𝑉_{𝑗+𝑚−1}(𝑗) are positive, that 𝑉^′ is the active multinode for 𝑓^′ in level 𝑗+𝑚−1, and that 𝜙(𝑗,𝑚) can be applied only at 𝑣, yielding (41) again.
Thus, (40) is valid in Case 1.
Case 2 [Δ_𝑘<0]. We assert that in this case the string 𝑆_{𝑛−1,𝑘} acts within the right rectangle 𝑅^𝑘+1 of the subgraph 𝐺^𝑘+1 (note that 𝑅^𝑘+1 is of size 𝑘×(𝑛−𝑘)). More precisely, eachoperator𝜙𝑗,𝑚modifiesthecurrentfunctionbyincreasingitsvalueatthenode𝑥_𝑚^𝑘+1𝑗by1. (44)
Then the resulting function in the process is just 𝑓_𝑎,Δ′ (in view of (Δ^′)⁻_𝑘=Δ⁻_𝑘+1), yielding (40).
To show (44), we argue as in the previous case and use (42) and the previous claim. Since Δ_𝑘<0, part (i) in (43) for the initial function 𝑓 is modified as follows: 𝑥for𝑗=1,…,𝑛−𝑘,theSW-edgeofeachnode_𝑘𝑘+1𝑗=𝑣^𝑘+1_{𝑗+𝑘−1}𝑗isnot𝑓-tight,𝜀𝑣_𝑘^𝑘+1(1)𝜀𝑣>0,_𝑘𝑘(1)≥0,and𝜀(𝑣)=0fortheothernodes𝑣in𝑉_𝑘(1);so𝜀_𝑘(1)>0, (45) while properties (ii) and (iii) preserve.
By (42) and (45), there are only two nodes in level 𝑘 that have no 𝑓-tight entering edges, namely, 𝑣^𝑘_𝑘(1) and 𝑣_𝑘^𝑘+1(1). Also 𝑒=𝑒^SW(𝑣_𝑘^𝑘+1(1)) is not tight. So, at the first step, 𝜙(1,𝑘) must act at 𝑣_𝑘^𝑘+1(1), as required in (44) (since the nontightness of 𝑒 implies that the node 𝑣^𝑘_𝑘(1) preceding 𝑣_𝑘^𝑘+1(1) cannot be the switch-node in 𝑉_𝑘(1)).
The fact that 𝜙(1,𝑚) with 𝑚<𝑘 acts at 𝑣_𝑚^𝑘+1(1) is shown by arguing as in (A) above. And for 𝑗>1, to show that 𝜙(𝑗,𝑚)=𝐹_{𝑗+𝑚−1} acts at 𝑥_𝑚^𝑘+1(𝑗)=𝑣^𝑘+1_{𝑗+𝑚−1}(𝑗), we argue as in (B) above. Here, when 𝑚=𝑘, we also use the fact that the edge 𝑒^SW(𝑥_𝑘^𝑘+1(𝑗)) is not 𝑓-tight (by (45) ), whence both edges entering 𝑥_𝑘^𝑘+1(𝑗) are not tight for the current function. So (44) is always valid.
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.

Figure 1 

The partition of 𝑉𝑘 (see Section 4).

5. 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 𝐴₂-crystal; this can be compared with the explicit construction (the so-called “sail model”) for 𝐴₂-crystals in [3]. The second one visualizes the subcrystals structure for one instance of 𝐴₃-crystals, namely, 𝐾(1,1,1).

5.1. 𝐴₂-Crystals

The subcrystals structure becomes simpler when we deal with an 𝐴₂-crystal 𝐾=𝐾(𝑐₁,𝑐₂). In this case the roles of upper, lower, and middle subcrystals are played by 1-paths, 2-paths, and vertices of 𝐾, respectively, where by an 𝑖-path we mean a maximal path of color 𝑖.

Consider an upper subcrystal in 𝐾. This is a 1-path 𝑃=(𝑣₀,𝑣₁,…,𝑣_𝑝) containing exactly one principal vertex 𝑣[̆𝑎] of 𝐾 (the heart of 𝑃); here 𝑣_𝑖 stands for 𝑖th vertex in 𝑃, 𝑎=(𝑎₁,𝑎₂)∈ℤ²₊, and 𝑎≤𝑐. Let 𝑣[̆𝑎]=𝑣_ℎ. Formulas (15) and (16) give |𝑃|=𝑝=𝑐₁−𝑎₁+𝑎₂,ℎ=𝑎₂. (46) Fix a vertex 𝑣=𝑣_𝑖 of 𝑃. It belongs to some 2-path (lower subcrystal) 𝑄=(𝑢₁,𝑢₂,…,𝑢_𝑞). Let 𝑣=𝑢_𝑗 and let 𝑣[̆𝑏]=𝑢_ℎ be the principal vertex of 𝐾 occurring in 𝑄 (the heart of 𝑄). The vertex 𝑣 forms a middle subcrystal of 𝐾; its deviations from the heart of 𝑃 and from the heart of 𝑄 are equal to 𝑖−ℎ=∶𝛿 and 𝑗−ℎ=∶𝛿, respectively. By (20) in Theorem 7, we have 𝛿=−𝛿. Then we can compute the coordinates 𝑏 by the use of (21) and, further, apply (17) and (18) to compute the length of 𝑄 and the locus of its heart. This gives the following: (i)if𝛿≥0i.e.,𝑎₂≤𝑖≤𝑐₁−𝑎₁+𝑎₂𝑏,then₁=𝑎₁+𝛿=𝑎₁+𝑖−𝑎₂,𝑏₂=𝑎₂,|𝑄|=𝑐₂−𝑏₂+𝑏₁=𝑐₂−2𝑎₂+𝑎₁|𝑄|−+𝑖,andℎ=|𝑄|−𝑏₁=𝑐₂−𝑎₂;(ii)if𝛿≤0i.e.,0≤𝑖≤𝑎₂,then𝑏₁=𝑎₁,𝑏₂=𝑎₂+𝛿=𝑎₂+𝑖−𝑎₂|𝑄|=𝑖,=𝑐₂−𝑏₂+𝑏₁=𝑐₂−𝑖+𝑎₁,andℎ=𝑏₁=𝑎₁. (47)

Using (46) and (47), one can enumerate the sets of 1-paths and 2-paths and properly intersect corresponding pairs, obtaining the 𝐴₂-crystal 𝐾(𝑐). 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 𝑐∈ℤ²₊, the 𝐴₂-crystal 𝐾(𝑐) is produced from two particular two-colored graphs 𝑅 and 𝐿, called the right sail of size 𝑐₁ and the left sail of size 𝑐₂, respectively. The vertices of 𝑅 correspond to the vectors (𝑖,𝑗)∈ℤ² such that 0≤𝑗≤𝑖≤𝑐₁, and the vertices of 𝐿 to the vectors (𝑖,𝑗)∈ℤ² such that 0≤𝑖≤𝑗≤𝑐₂. In both 𝑅 and 𝐿, the edges of color 1 are all possible pairs of the form ((𝑖,𝑗),(𝑖+1,𝑗)), and the edges of color 2 are all possible pairs of the form ((𝑖,𝑗),(𝑖,𝑗+1)). (Observe that both 𝑅 and 𝐿 satisfy axioms (A1)–(A4), 𝑅 is isomorphic to 𝐾(𝑐₁,0), 𝐿 is isomorphic to 𝐾(0,𝑐₂), and their critical vertices are the “diagonal vertices” (𝑖,𝑖).)

In order to produce 𝐾(𝑐), take 𝑐₂ disjoint copies 𝑅₁,…,𝑅_𝑐2 of 𝑅 and 𝑐₁ disjoint copies 𝐿₁,…,𝐿_𝑐1 of 𝐿, referring to 𝑅_𝑗 as 𝑗th right sail and to 𝐿_𝑖 as 𝑖th left sail. Let 𝐷(𝑅_𝑗) and 𝐷(𝐿_𝑖) denote the sets of diagonal vertices in 𝑅_𝑗 and 𝐿_𝑖, respectively. For all 𝑖=1,…,𝑐₁ and 𝑗=1,…,𝑐₂, we identify the diagonal vertices (𝑖,𝑖)∈𝐷(𝑅_𝑗) and (𝑗,𝑗)∈𝐷(𝐿_𝑖). The resulting graph is just the desired 𝐾(𝑐). The edge colors of 𝐾(𝑐) are inherited from 𝐿 and 𝑅. One checks that 𝐾(𝑐) has (𝑐₁+1)×(𝑐₂+1) critical vertices; they coincide with the diagonal vertices of the sails. The principal lattice of 𝐾(𝑐) is just constituted by the critical vertices.

The case (𝑐₁,𝑐₂)=(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: (48)

In particular, the sail model shows that the numbers of edges of each color in an 𝐴₂-crystal are the same. This implies a similar property for any 𝐴_𝑛-crystal.

5.2. 𝐴₃-Crystal 𝐾(1,1,1)

Next we illustrate the 𝐴₃-crystal 𝐾=𝐾(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; namely, we demonstrate all of its upper and lower subcrystals. We abbreviate notation 𝑣[̆(𝑖,𝑗,𝑘)] for principal vertices to (𝑖,𝑗,𝑘) for short. So the principal lattice consists of eight vertices (0,0,0),…,(1,1,1), as drawn in the picture (where the arrows indicate moves by principal operator strings 𝑆_3,𝑘 as in (12)): (49)

Thus, 𝐾 has eight upper subcrystals 𝐾^↑[𝑖,𝑗,𝑘] and eight lower subcrystals 𝐾^↓[𝑖,𝑗,𝑘] (writing 𝐾^•[𝑖,𝑗,𝑘] for 𝐾^•[(𝑖,𝑗,𝑘)]); they are drawn in Figures 2 and 3. Here the directions of edges of colors 1,2,3 are as indicated in the upper left corner. In each subcrystal, we indicate its critical vertices by black circles and the unique principal vertex of 𝐾 occurring in it (the heart) by a big white circle. 𝐾 has 30 middle subcrystals (paths of color 2), which are labeled as 𝐴,…,𝑍,Γ,Δ,Φ,Ψ (note that 𝐵,𝐹,𝐺,𝑁,𝑃,𝑇,𝑉,andΦ consist of single vertices).

Figure 2 

The upper subcrystals in 𝐾(1,1,1) (see Section 5).

Figure 3 

The lower subcrystals in 𝐾(1,1,1) (see Section 5).

For each upper subcrystal 𝐾^↑[𝑖,𝑗,𝑘], its parameter 𝑐^↑ and heart locus ℏ^↑, computed by use of (15) and (16), are as follows (where 𝐾^∼,^∼𝑠,and𝑧 denote the current subcrystal, its source, and its heart, resp.):(i)for 𝐾^↑[0,0,0]: 𝑐^↑₁=1−0+0=1, 𝑐^↑₂=1−0+0=1, and ℏ^↑₁=ℏ^↑₂=0 (so 𝐾^∼ is isomorphic to 𝐾(1,1) and 𝑧 coincides with ^∼𝑠);(ii)for 𝐾^↑[1,0,0]: 𝑐^↑₁=1−1+0=0, 𝑐^↑₂=1−0+0=1, and ℏ^↑₁=ℏ^↑₂=0;(iii)for 𝐾^↑[0,1,0]: 𝑐^↑₁=1−0+1=2, 𝑐^↑₂=1−1+0=0, ℏ^↑₁=1, and ℏ^↑₂=0 (so 𝐾^∼≃𝐾(2,0) and 𝑧 is located at 𝑆_2,1(^∼𝑠)=𝐹₂𝐹₁(^∼𝑠));(iv)for 𝐾^↑[0,0,1]: 𝑐^↑₁=1−0+0=1, 𝑐^↑₂=1−0+1=2, ℏ^↑₁=0, and ℏ^↑₂=1 (so 𝐾^∼≃𝐾(1,2) and 𝑧 is located at 𝑆_2,2(^∼𝑠)=𝐹₁𝐹₂(^∼𝑠));(v)for 𝐾^↑[1,1,0]: 𝑐^↑₁=1−1+1=1, 𝑐^↑₂=1−1+0=0, ℏ^↑₁=1, and ℏ^↑₂=0;(vi)for 𝐾^↑[1,0,1]: 𝑐^↑₁=1−1+0=0, 𝑐^↑₂=1−0+1=2, ℏ^↑₁=0, and ℏ^↑₂=1;(vii)for 𝐾^↑[0,1,1]: 𝑐^↑₁=1−0+1=2, 𝑐^↑₂=1−1+1=1, and ℏ^↑₁=ℏ^↑₂=1 (so 𝐾^∼≃𝐾(2,1) and 𝑧 is located at 𝑆_2,2𝑆_2,1(^∼𝑠)=𝐹₁𝐹₂𝐹₂𝐹₁(^∼𝑠));(viii)for 𝐾^↑[1,1,1]: 𝑐^↑₁=1−1+1=1, 𝑐^↑₂=1−1+1=1, and ℏ^↑₁=ℏ^↑₂=1.

Since 𝐾(1,1,1) is “symmetric,” so are its upper and lower subcrystals; that is, each 𝐾^↓[𝑖,𝑗,𝑘] is obtained from 𝐾^↑[𝑘,𝑗,𝑖] by replacing color 1 by 3. In Figure 3, when writing 𝐾^↓[𝑖,𝑗,𝑘]≃𝐾(𝛼,𝛽), the parameters 𝛼,𝛽 concern colors 3 and 2, respectively.

Now the desired 𝐾(1,1,1) is assembled by gluing the fragments in Figures 2 and 3 along the 2-paths 𝐴,…,Ψ.

Acknowledgments

This paper is supported by RFBR Grant 10-01-9311-CNRSL_a. A part of this research was done while Alexander V. Karzanov was visiting Equipe Combinatoire et Optimisation, Université de Paris, France and the Institut für Diskrete Mathematik, Universität Bonn, Germany.