Research Article | Open Access
Assembling Crystals of Type A
Regular -crystals are certain edge-colored directed graphs, which are related to representations of the quantized universal enveloping algebra . For such a crystal with colors , we consider its maximal connected subcrystals with colors and with colors and characterize the interlacing structure for all pairs of these subcrystals. This enables us to give a recursive description of the combinatorial structure of via subcrystals and develop an efficient procedure of assembling .
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 such that the edges are partitioned into subsets, or color classes, labeled , 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 . 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 .
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 ; in this case the Cartan matrix is viewed as if , if , and . 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 , 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  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 -crystal) when , and an -crystal when . (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 or ; for more details, see .)
Another important fact is that for any -tuple of nonnegative integers, there exists exactly one -crystal 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 [7–10]. In the last decade there appeared additional, more enlightening, descriptions. A short list of “local” defining axioms for A-crystals is pointed out in , and an explicit construction for -crystals is given in . According to that construction, any -crystal can be obtained from an -crystal by replacing each monochromatic path of the latter by a graph viewed as a triangle-shaped half of a directed square grid.
When , the combinatorial structure of -crystals becomes much more complicated, even for . Attempting to learn more about this structure, we elaborated in  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 with the following nice properties:(P1) contains the source and sink of , and the vertices are bijective to the elements of the integer box ; we write ;(P2) for any with , the interval of from to (i.e., the subgraph of formed by the vertices and edges contained in directed paths from to ) is isomorphic to the -crystal , and its principal lattice consists of the principal vertices of with ;(P3) the set of -colored subcrystals of having colors is bijective to ; more precisely, consists of exactly one vertex, called the heart of w.r.t. , and similarly for the set of subcrystals of with colors .
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 -crystals, see also ). 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 , let (resp., ) denote the subcrystal in (resp., in ) that contains the principal vertex ; we call it the upper (resp., lower) subcrystal at . It is shown in  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 . This intersection may be empty or consist of one or more subcrystals with colors , called middle subcrystals of . Each of these middle subcrystals is therefore a lower subcrystal of and an upper subcrystal of ; so has a unique vertex in the principal lattice of the former, and a unique vertex in the principal lattice of the latter. Our main structural results—Theorems 7 and 8—give explicit expressions showing how the “loci” and in , the “deviation” of (the heart of) from in , and the “deviation” of from in are interrelated.
This gives rise to a recursive procedure of assembling of the -crystal . More precisely, suppose that the -colored crystals and for all 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 , outputs the crystal . The running time of the algorithm and the needed space are bounded by , where is a constant and is the size of . (It may be of practical use for small and ; in general, an -crystal has “dimension” 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 of an -crystal and relations between and the -colored subcrystals from . 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 -crystal (in which case the method can be compared with the explicit combinatorial construction in ) and for the particular -crystal .
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 -crystal (resp., -crystal); this is discussed in detail in (, Sections 5–8). Here an -crystal with parameter is called symmetric if .
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  that will be needed later.
Stembridge  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 .
Let be a directed graph whose edge set is partitioned into subsets , denoted as . We assume to be (weakly) connected. We say that an edge has color or is an -edge.
Unless explicitly stated otherwise, by a path we mean a simple finite directed path, that is, a sequence of the form , where are distinct vertices and each is an edge from to (admitting ).
The first axiom concerns the structure of monochromatic subgraphs of . (A1) For , each connected subgraph of is a path.
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 , where 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 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 -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 , and its part from to the last vertex by (the tail and head parts of w.r.t. ). The lengths (i.e., the numbers of edges) of and are denoted by and , respectively.
Axioms (A2)–(A5) concern interrelations of different colors , . They say that each component of the two-colored graph forms an -crystal when colors , are neighboring, which means that , and forms an -crystal otherwise.
When we traverse an edge of color , the head and tail part lengths of maximal paths of another color behave as follows. (A2) For different colors and for an edge with color , one holds and . The value is the constant equal to if , and 0 otherwise. Furthermore, is convex on each -path, in the sense that if are consecutive -edges, then .
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., ), the tail length decreases by 1 while the head length does not change, and when traversing any edge of after , does not change while increases by 1. This is called the critical vertex for , , . To each -edge , we associate label ; then and . We emphasize that the critical vertices on a maximal -path w.r.t. its neighboring colors and may be different (and so are the edge labels on ).
Two operators and , where , are said to commute at a vertex if each of acts at (i.e., corresponding -edge and -edge incident with exist) and . The third axiom indicates situations when such operators commute for neighboring . (A3) Let . (a) If a vertex has outgoing -edge and outgoing -edge and if , then and , commute at . Symmetrically: (b) if a vertex has incoming -edge and incoming -edge and if , then and , commute at . (See the following picture.) (1)
One easily shows that if four vertices are connected by two -edges , and two -edges (forming a “square”), then (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 “remotely commute” (forming the “Verma relation of degree 4”).(A4) Let . (i) If a vertex has outgoing edges with color and color and if each edge is labeled 1 w.r.t. the other color, then . Symmetrically, (ii) if has incoming edges with color and color and if both are labeled 0, then . (See the following picture.)(2)
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 . Then for any and , 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 -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 of monochromatic paths starting at the source determine , and for each tuple of nonnegative integers, there exists a (unique) -crystal such that for . (See [4, 6].) We call the parameter of and denote by .
2.2. The Crossing Model for -Crystals
Following , the crossing model generating the -crystal with a parameter consists of three ingredients:(i)a directed graph depending on , called the supporting graph of the model;(ii)a set of feasible functions on ;(iii)a set of transformations of feasible functions, called moves.
To explain the construction of the supporting graph , we first introduce another directed graph that we call the protograph of . Its node set consists of elements for all 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 (ascending edges) or (descending edges). We say that the nodes form th level of and order them as indicated (by increasing ). We visualize 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 .(3)
The supporting graph is produced by replicating elements of as follows. Each node generates nodes of , denoted as for , which are ordered by increasing (and accordingly follow from left to right in the visualization). We identify with the set of these nodes and call it a multinode of . Each edge of generates a set of edges of (a multiedge) connecting elements with equal upper indices. More precisely, produces ascending edges for , and produces descending edges for .
The resulting is the disjoint union of directed graphs , where each contains all vertices of the form . Also is isomorphic to the Cartesian product of two paths, with the lengths and . For example, for , the graph is viewed as (4)
(where the multinodes are surrounded by ovals) and its components , , , are viewed as (5)
So each node of has at most four incident edges, namely, , , , and ; we refer to them, when exist, as the NW-, SW-, NE-, and SE-edges and denote them by , and , respectively.
By a feasible function in the model (with a given ), we mean a function satisfying the following three conditions, where for an edge , denotes the increment of on , and is called tight for , or - tight, if :
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 , we first extend each by adding extra nodes and extra edges (following  and aiming to slightly simplify the description). In the extended directed graph , the node set consists of elements for all and such that . The edge set of consists of all possible pairs of the form or . Then all are isomorphic. The disjoint union of these gives the extended supporting graph . The creation of from for is illustrated in the picture: (7)
Each feasible function on is extended to the extra nodes as follows: if there is a path from to a node of , and 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, . For a node with , define the value by where . For a multinode , define the numbers We call , , and 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 , 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 are possible. Each changes within level and is applicable when this level contains a multinode with . 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  that the resulting function is again feasible.
As a result, the model generates -colored directed graph , where each color class is formed by the edges for all feasible functions to which the operator is applicable. This graph is just an -crystal.
Theorem 1 (see [4, Th. 5.1]). For each , the -colored graph is exactly the -crystal .
2.3. Principal Lattice and -Colored Subcrystals of an -Crystal
Based on the crossing model,  reveals some important ingredients and relations for an -crystal . One of them is the so-called principal lattice, which is defined as follows.
Let and . One easily checks that the function on the vertices of the supporting graph that takes the constant value within each subgraph of , , 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 ; this set is called the principal lattice of and denoted by . When it is not confusing, the term “principal lattice” may also be applied to .
The following properties of the principal lattice will be essentially used later.
Proposition 2 (see [4, Statement (6.4)]). Let , , and (where is th unit base vector in ). The principal vertex is obtained from by applying the operator string where for , the substring is defined as When acting on , any two (applicable) strings commute. In particular, any principal vertex is expressed via the source as
Proposition 3 (see [4, Prop. 6.1]). For with , let be the subgraph of formed by the vertices and edges contained in (directed) paths from to (the interval of from to ). Then is isomorphic to the -crystal , and the principal lattice of consists of the principal vertices of with .
Let denote the set of subcrystals with colors , and the set of subcrystals with colors in (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 (in ) contains precisely one principal vertex. This gives a bijection between and (resp., between and ).
We refer to the members of and as upper and lower (-colored) subcrystals of , respectively. For , the upper subcrystal containing the vertex is denoted by . This subcrystal has its own principal lattice of dimension , which is denoted by . We say that the coordinate tuple is the locus of (and of ) in . Analogously, for , the lower subcrystal containing is denoted by and its principal lattice by ; we say that is the locus of (and of ) in . 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 , the upper subcrystal is isomorphic to the -crystal , where is the tuple defined by
The principal vertex is contained in the upper lattice and its coordinate in satisfies
Symmetrically, for , the lower subcrystal is isomorphic to the -crystal with colors , where is defined by The principal vertex is contained in the lower lattice and its coordinate in satisfies
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 -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 . More precisely, for , form the partition by setting , where denotes . A GT-pattern for is a triangular array of integers satisfying (a) and (b) , for all possible . It is shown in  that the set of feasible functions in the crossing model is bijective to the set of GT-patterns for ; such a correspondence is given by , where denotes the sum of values of over the multinode . 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 -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 upper subcrystals (with colors ) and lower subcrystals (with colors ). Moreover, the parameters of these subcrystals are expressed explicitly by (15) and (17). Assume that the set of upper subcrystals and the set of lower subcrystals are already available. Then in order to assemble , it suffices to point out, in appropriate terms, the intersection for all pairs (the intersection may either be empty or consist of one or more -colored subcrystals with colors in ). We give an appropriate characterization in Theorems 7 and 8.
To state them, we need additional terminology and notation. Consider a subcrystal , and let be defined as in (15), (16). For , the vertex in the upper lattice having the coordinate is denoted by . We call the vector the deviation of from the heart in and we will use the alternative notation for this vertex. In particular, .
We call an -colored subcrystal with colors in a middle subcrystal and denote the set of these by . Each middle subcrystal is a lower subcrystal of some upper subcrystal of . By Proposition 4 applied to , has a unique vertex in the lattice . So each can be encoded by a pair formed by a point and a deviation in . At the same time, is an upper subcrystal of some lower subcrystal of and has a unique vertex in . Therefore, the members of determine a bijection between all pairs concerning upper subcrystals and all pairs concerning lower subcrystals.
The map is expressed explicitly in the following two theorems. Here for a tuple of reals, we denote by () the tuple formed by (resp., ), .
Theorem 7 (on two deviations). Let and let be a deviation in (from the heart of ). Let . Then
Theorem 8 (on two loci). Let be as in the previous theorem. Then letting .
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 , 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 (arranged as a list) of all middle subcrystals of . Each is encoded by a corresponding pair , where and the deviation in 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 and is a deviation in . 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 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 and each vertex in the upper lattice , one can express the parameter of the middle subcrystal containing , as well as the coordinate of its heart w.r.t. in the principal lattice of . Indeed, since is a lower subcrystal of , one can apply relations as in (17) and (18). Denoting the parameter of by and the coordinate of its heart in by , letting , and using (15), (16), we have for ,
Symmetrically, if is contained in and has deviation in , then for , where is the coordinate of the heart of w.r.t. in the principal lattice of (note that may differ from ). We will use (22) and (24) in Section 4.
Remark 10. A straightforward implementation of the above recursive method of constructing takes time and space, where is a polynomial in and is the number of vertices of . Here the factor appears because the total number of vertices in the upper and lower subcrystals is (implying that there appear 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 , 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 . We call a subcrystal of this sort a color-interval subcrystal, or a CI-subcrystal, of . In fact, every CI-subcrystal of appears in the process. Since the number of intervals is and the CI-subcrystals concerning equal intervals are pairwise disjoint, the total number of vertices of all CI-subcrystals of is . 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.
Proposition 11. Let . The -crystal and all its CI-subcrystals can be constructed in time and space, where is a polynomial in .
Remark 12. Relation (21) shows that the intersection of and may consist of many middle subcrystals. Indeed, if and for some , then does not change by simultaneously decreasing by 1 and increasing by 1. The number of common middle subcrystals of and for arbitrary can be expressed by an explicit piecewise linear formula, using (21) and the box constraints , , on the deviations in (which follow from (15) and (16)).
Proof of (20) (in the assumption that (21) is valid). The middle subcrystal determined by is the same as the one determined by . The parameter of is expressed simultaneously by (22) and by (24). Then for . Therefore,
In order to obtain (20), one has to show that . We argue as follows. Renumber the colors as , respectively; this yields the crystal symmetric to . Then turns into the upper subcrystal of , where . Also the deviation in turns into the deviation in the principal lattice of . Applying relations as in (21) to , we have where and . On the other hand, (21) for gives
Relations (26) and (27) imply whence Adding up the latter equalities, we obtain This and (25) imply . Hence , yielding (20) and Theorem 7.
Proof of Theorem 8. It is more intricate and essentially uses the crossing model.
For a feasible function and its corresponding vertex in , we may denote as and as . From the crossing model it is seen that Indeed, the principal vertex is reachable from by applying operators or with . The corresponding moves in the crossing model do not change within level . Similarly, is reachable from by applying operators or with , and the corresponding moves do not change within level 1. Also the first (second) equality in (31) is valid for the principal function (resp., ).
Next we introduce special functions on the node set of the supporting graph . Consider a component of . It is a rectangular grid of size (rotated by 45° in the visualization of ), and its vertex set is To represent it in a more convenient form, let us rename as or (as though rotating by 45°). Then the SE-edges of become of the form , and the NE-edges become of the form . We distinguish the following subsets of : (i)the SW-side ;(ii)the right rectangle ;(iii)the left rectangle . Denote the characteristic functions (in ) of , and as , , respectively.
Return to and as above. We associate to the functions on for (see Figure 1) and define to be the function on whose restriction to each is .
In view of (31), takes the values in levels and 1 as required in (21) (with in place of ); namely, and for . Therefore, to obtain (21) it suffices to show the following.
Lemma 13. (i) The function is feasible. (ii) The vertex belongs to and has the deviation in it; in other words, .
Proof. First we prove assertion (i). Let . We partition into four subsets (rectangular pieces): where when , and when . By (34), (as illustrated in Figure 1). Also each edge of connecting different pieces goes either from to or from to . This and (36) imply that for each edge , whence satisfies (6)(i).
The deviation is restricted as , where is the parameter of the subcrystal and is the coordinate of its heart in . Formulas (15) and (16) for and give The inequalities and imply . The inequalities and imply . Then, in view of (36), we obtain for each node of , yielding (6)(ii).
To verify the switch condition (6)(iii), consider a multinode with . It consists of nodes , where .
Let . Suppose that is a node whose SW-edge exists and is not -tight. This is possible only if and . In this case, is determined as ; that is, is the second node in . We observe that (a) for the first node of , both ends of its SE-edge belong to the piece ; and (b) for any node with in , both ends of its SW-edge belong either to or to . So, such and are -tight. Therefore, the node satisfies the condition in (6)(iii) for .
Now let . Then consists of two nodes and . Put . Then the edge goes from to , and the edge goes from to . By (36), we have and . Since at least one of is zero, we conclude that at least one of is tight. So (6)(iii) is valid again.
Next we prove assertion (ii) in the lemma. (The idea is roughly as follows. For each , compare the function with the function on taking the constant value . By (34), . In other words, is obtained from by adding times the “left rectangle function” , followed by subtracting times the “right rectangle function” . A crucial observation is that adding corresponds to applying the operator string (or shifting by th unit base vector in the upper principal lattice ), while subtracting corresponds to applying (or shifting by minus th unit base vector in ). This is because the substrings in correspond to the SW-NE paths in , and the substrings in to similar paths in ).
Now we give a more careful and formal description. We use induction on
In view of (37), . Suppose that this turns into equality. Then for , and takes the following values within each