Combinatorics of canonical bases revisited: String data in type $A$

We give a formula for the crystal structure on the integer points of the string polytopes and the $*$-crystal structure on the integer points of the string cones of type $A$ for arbitrary reduced words. As a byproduct we obtain defining inequalities for Nakashima-Zelevinsky string polytopes. Furthermore, we give an explicit description of the Kashiwara $*$-involution on string data for a special choice of reduced word.


Introduction
Let g be a simple complex Lie algebra of rank n − 1 and V a finite dimensional representation of g. Much information of V is encoded in a directed graph with arrows colored by {1, 2, . . . , n − 1}, called the crystal graph of V [K91]. For instance, this crystal graph is connected if and only if V is irreducible, the character of V is encoded in the vertices of the crystals graph and there exists a simple notion of the tensor product of two crystal graphs yielding the crystal graph of the tensor product of two representations.
For V irreducible, its crystal graph has a unique source corresponding to a highest weight vector of V . Making use of this fact, Littelmann [Lit98] and Berenstein-Zelevinsky [BZ93,BZ01] gave a bijection between the vertices of this graph as integer points of a rational convex polytope, called the Littelmann-Berenstein-Zelevinsky string polytope.
The rule for assigning an integer point in the Littelmann-Berenstein-Zelevinsky string polytope to a vertex v is as follows. Let x 1 be the largest integer such that there are x 1 consecutive arrows of color i 1 ending in v. Let v 1 be the source of this sequence of arrows. Let x 2 be the length of the longest sequence of arrows of a color i 2 ending in v 1 and so on. If we pick the colors i 1 , i 2 , . . . , i N according to the appearance in a reduced decomposition of the longest Weyl group element of g, this procedure ends at the source of the graph. Then the vertex v maps to the integer point (x 1 , x 2 , . . . , x N ) ∈ N N , called the string datum of v.
We consider the following problem for the string polytope of an irreducible representation V associated to the reduced word i = (i 1 , i 2 , . . . , i N ) of the longest Weyl group element of g.
Problem 0.1. Give a formula for the operator f a on the integer points of the string polytope P defined as follows. For two integer points x and x ′ in P we have f a x = x ′ , if the corresponding vertices v and v ′ in the crystal graph are connected by an arrow of color a.
There is, however, no obvious solution for arbitrary a. For sl 3 (C) and the reduced word s 1 s 2 s 1 , one can deduce from an explicit construction of the crystal graph ( [DKKA07]) that f 2 (x 1 , x 2 , x 3 ) is equal to (x 1 , x 2 + 1, x 3 ) if x 1 ≤ x 2 − x 3 and (x 1 − 1, x 2 + 1, x 3 + 1) otherwise. In this work we solve Problem 0.1 by establishing a formula for the operator f a for any a in the case that g = sl n (C).
For a ∈ {1, 2, . . . , n − 1} and a reduced word i = (i 1 , i 2 , . . . , i N ) of the longest element of the Weyl group of sl n (C) we define in Section 4 finitely many sequences γ = (γ j ) of positive roots of sl n (C) with certain properties which we call a-crossings. These sequences come with an order relation ⪯. We further introduce maps r, s associating to γ the vectors r(γ), s(γ) ∈ Z N .
Our main result reads as follows, where ⟨⋅, ⋅⟩ is the standard scalar product on Z N .
Theorem 5.1 is in analogy to the Crossing Formula established in [GKS16, Theorem 2.13, Proposition 2.20], which computes the operator f a on the polytopes arising from Lusztig's parametrizations of the crystal graph. Indeed, the two formulae may be viewed as dual since the roles of maximum and minimum and the vectors r(γ), s(γ) interchange. We elaborate on this duality in [GKS19].
Theorem 5.1 gives rise to two applications. The Verma module of g of weight 0 has a crystal graph B(∞) with a unique source. Kashiwara [K93] defined an involution * on the vertices of B(∞), leading to a second crystal graph B(∞) * with the same set of vertices. Namely, there is an arrow from v 1 to v 2 of color a in B(∞) * if and only if there is an arrow from v * 1 to v * 2 of color a in B(∞).
Associating integer vectors to the vertices of B(∞) * by taking their string data, we obtain a rational polyhedral cone called the string cone [Lit98, BZ93,BZ01] which contains the Littelmann-Berenstein-Zelevinsky string polytope.
A variation of Problem 0.1 now arises, replacing the Littelmann-Berenstein-Zelevinsky string polytope by the string cone and the crystal graph of an irreducible representation by B(∞) * . In Theorem 5.2 we provide a solution to this problem for g = sl n . Indeed the crystal graph of each irreducible representation V is a full subgraph of B(∞) * . Making use of this fact we deduce Theorem 5.2 from Theorem 5.1.
A second crystal graph for the irreducible representation V is obtained as a full subgraph of B(∞). The set of corresponding string parameters is, due to a result of Fujita-Naito [FN17], again the set of integer points in a rational polytope, called the Nakashima-Zelevinsky string polytope. These polytopes have been found to coincide with Newton-Okounkov bodies for flag varieties [FN17,FO17]. They also appear in [CFL] among Newton-Okounkov bodies inducing semitoric degenerations of Schubert varieties associated to maximal chains in the corresponding Bruhat graphs.
For Nakashima-Zelevinsky polytopes problem 0.1 has been solved in the work of Kashiwara [K93] and Nakashima-Zelevinsky [NZ97,N99]. It is, however, a difficult problem to compute the inequalities which cut the Nakashima-Zelevinsky polytopes out of the string cone. This is so far only known in a few special cases [N99, H05]. Using Theorem 5.2 we obtain these inequalities for all reduced words of the longest Weyl group element of sl n in Theorem 6.1.
The paper is organized as follows. In Section 1 we recall the background on crystals. In Section 2 we recall facts about reduced words for elements of the symmetric group. In Section 3 string cones and Littelmann-Berenstein-Zelevinsky string polytopes, as well as their crystal structures, are discussed.
In Section 4 we introduce the main combinatorial tools of this paper, namely the notion of wiring diagrams and Reineke crossings. The main result (Theorem 5.1), providing a formula for the crystal structure on Littelmann-Berenstein-Zelevinsky string polytopes, is stated in Section 5. We further prove the Dual Crossing Formula for the * -crystal structure on the string cone in this section.
In Section 6 Nakashima-Zelevinsky string polytopes are introduced and their defining inequalities are computed.
Section 7 deals with Lusztig's parametrization of the canonical basis and recalls facts from [GKS16] which are used in the proof of Theorem 5.1 which is presented in Section 8.
In Section 9 we give a description of the piecewise linear Kashiwara *involution on string data. In particular, we obtain a linear isomorphism between the Littelmann-Berenstein-Zelevinsky polytope and the Nakashima-Zelevinsky polytope for a specific reduced word.
Schumann was supported by the SFB/TRR 191. B. Schumann would furthermore like to thank Xin Fang, Peter Littelmann, Valentin Rappel, Christian Steinert and Shmuel Zelikson for helpful discussions.
Let U q (sl n ) be the Q(q)-algebra with generators E a , F a , K ±1 a , a ∈ [n − 1] and the following relations for b ∈ [n − 1] ∖ {a} . (1) For λ ∈ P + we denote by V (λ) the irreducible U q (sl n )-module of highest weight λ.
We finally denote by U − q ⊂ U q (sl n ) be the subalgebra generated by {F a } a∈[n−1] .
Here 0 is an element not included in B. The above maps satisfy the following axioms for a ∈ [n − 1] Here we put −∞ + k = −∞ for k ∈ Z.
Let B 1 and B 2 be crystals.
for all a ∈ [n − 1]. An injective strict morphism is called a strict embedding of crystals and a bijective strict morphism is called an isomorphism of crystals.
Definition 1.2. Let B 1 and B 2 be crystals. The set equipped with the following crystal structure is called the tensor product of B 1 and B 2 . For a ∈ [n − 1] 1.3. Crystals of representations. We recall the crystal bases B(∞) and B(λ) of U − q and V (λ), respectively, from [K91, Sections 2 and 3].
We define e ′ a (P ) = R. As vector spaces, we have We define the Kashiwara operators e a , f a on U − q for u ∈ ker(e ′ a ) by Let A be the subring of Q(q) consisting of rational functions g(q) without a pole at q = 0. Let L(∞) be the A-lattice generated by all elements of the form This endows B(∞) with the structure of an crystal (see Definition 1.1). We a (x) = (e a x * ) * and ε * a (x) = ε a (x * ) the * -twisted maps. This endows B(∞) with a second structure of a crystal. We denote the crystal given by the set B(∞) and the twisted maps by B(∞) * . By construction * induces a crystal isomorphism between B(∞) and B(∞) * . For where v λ is a highest weight vector of V (λ). The operators e a and f a defined in (2) descend to V (λ) and we denote by L(λ) the A-lattice generated by all elements of the form and by B(λ) ⊂ L(λ) qL(λ) the subsets of all residues of elements of the form (4).
This endows B(λ) with the structure of a crystal (see Definition 1.1).
into a product of simple transposition with a minimal possible number of factors. We call k the length ℓ(w) of w. For a reduced expression of w ∈ S n we write i ∶= (i 1 , i 2 , . . . , i N ) and call i a reduced word (for w). The set of reduced words for w is denoted by W(w). The group S n has a unique longest element w 0 of length N ∶= n(n−1)

2
. We have two operations on the set of reduced words W(w 0 ).
A reduced word j = (j 1 , . . . , j N ) is said to be obtained from . Let I(w) be the set of inversions for w ∈ S n . We have the I(w) is equal to the length ℓ(w).
It is well known that the sets W(w) and I(w) are in natural bijection (see e.g. [D93, Proposition 2.13]). Under this bijection the reflection order where α p,q is defined in Section 1.1. The reflection order corresponding to i induces a total ordering on Φ + in this case.
where b ∞ is the element in B(∞) of highest weight. By (7) the map str i is injective. We denote by S i = str i (B(∞)) the image of str i . Let S R i ⊂ R N be the cone spanned by S i . By [Lit98, Proposition 1.5], [BZ01, Proposition 3.5] S R i is a rational polyhedral cone, called the string cone, and S i are the integral points of S R i . Recall the definition of ε * a and e * a from Section 1.3. Now let we obtain (8). Now (9) follows by applying (7) to str i (b * ).
3.2. Crystal structures on string data. In this section we equip S i with two crystal structures isomorphic to B(∞).
For a ∈ [n − 1] and k ∈ Z let b a (k) be a formal symbol. We denote by By [K93, Theorem 2.2.1] there exists for any a ∈ [n − 1] a unique strict embedding of crystals given by In [K93, Theorem 2.2.1 and its proof] (see also [NZ97, Section 2.4]) the following statement is proved.
Lemma 3.2 naturally provides two crystal structures on S i as follows. Let i = (i 1 , . . . , i N ) ∈ W(w 0 ). We iterate the map (11) along i by setting Combining Lemma 3.1 with Lemma 3.2 we obtain the strict embedding we obtain the following explicit description of the crystal structure on S i resulting from B(∞). Let (c i,j ) be the Cartan matrix of sl n (C). For k ∈ [N ] and x ∈ S i we set Lemma 3.3. The crystal structure on S i obtained from B(∞) via the bijection b ↦ str * i (b) is given as follows. For x ∈ S i and a ∈ [n − 1] where is maximal with i ℓx = a and η ℓx (x) = ε a (x).

The crystal structure on
as follows.
. For arbitrary i, j ∈ W(w 0 ) we define Ψ i j ∶ S i → S j as the composition of the transition maps corresponding to a sequence of 2− and 3−moves transforming i into j.
Proof. The statement follows from Lemma 3.2 and (14).
In Theorem 5.2 we give a formula for the crystal structure of Lemma 3.4.
3.3. String polytopes and their crystals structures. Let λ ∈ P + and i ∈ W(w 0 ). Recall from (5) that the crystal B(λ) is isomorphic to the subcrystal B(λ) of B(∞) ⊗ R λ . Hence, using (7) we get a bijection between B(λ) and In [Lit98, Proposition 1.5] it is shown that S * i (λ) is the set of integer points of the rational polytope We call S * i (λ) R the Littelmann-Berenstein-Zelevinsky string polytope. By (16) we obtain the following crystal structure isomorphic to B(λ) on S * i (λ) ⊂ S i . Denoting by ι λ ∶ S * i (λ) ↪ S i the natural embedding we obtain In Theorem 5.1 we give a formula for the crystal structure of Lemma 3.5.

Wiring diagrams and Reineke crossings
Following [BFZ96], we introduce the notion of a wiring diagram which is a graphical presentation of the reduced word i ∈ W(w 0 ).   = (2, 1, 2, 3, 4, 3, 2, 1, 3, 2). The corresponding wiring diagram D i is depicted below. The condition i ∈ W(w 0 ) implies that two lines p, q with p ≠ q in D i intersect exactly once.
Each vertex of the wiring diagram D i , i ∈ W(w 0 ), corresponds to an inversion (p, q) ∈ I(w 0 ), where p and q are the labels of the wires intersecting in that vertex. Thus the vertices of D i are in bijection with the positive roots by (6). The reflection order on I(w 0 ) and the induced total order on Φ + can be read off of D i by reading the vertices from left to right. We identify Definition 4.6 (Reineke crossings). For a ∈ [n − 1] an a-crossing is an oriented path γ = (v 1 , . . . , v k ) in D i (a) which starts with the leftmost vertex of the wire a and ends with the leftmost vertex of the wire a+1. Additionally γ satisfies the following condition: Whenever v j , v j+1 , v j+2 lie on the same wire p in D i and the vertex v j+1 lies on the intersection the wires p and q, we have In other words, the path γ avoids the following two fragments. Remark 4.7. Reineke crossings appear as rigorous paths in [GP00].
Example 4.8. Let n = 5. The vertices lying on the red path below form the 3−Reineke crossing γ = (v 3,2 , v 3,1 , v 1,2 , v 2,5 , v 2,4 , v 4,5 , v 4,1 ). In the remainder of this section we adopt the following convention: We label each vertex v = v p,q ∈ γ by the wires p and q that intersect in this edge where p is the wire of the oriented edge whose source in γ is v p,q .

Using the identification (18) we introduce
Definition 4.11. The maps r ∶ Γ a → Z N and s ∶ Γ a → Z N are given by else. Definition 4.13. Let γ 1 , γ 2 ∈ Γ a . We say γ 1 ⪯ γ 2 if all vertices of γ 1 lie in the region of D i cut out by γ 2 .
Example 4.14. Let γ be as in Example 4.8 and γ ′ = (v 3,2 , v 2,1 , v 1,4 ). In the picture below the region cut out by γ is shaded grey while γ ′ consists of all vertices lying on the red path. Thus γ ′ ⪯ γ.

Dual Crossing Formula for string parametrizations
Let λ ∈ P + and i ∈ W(w 0 ). In this section we state our main result which is a formula for the crystal structure on the integer points of the Littelmann-Berenstein-Zelevinsky string polytope S * i (λ) R defined in (17). Recall the notion of the set of a-Reineke crossings Γ a from Definition 4.6 and their associated vectors from Definition 4.11. We denote by ⟨⋅, ⋅⟩ the standard scalar product on Z N . The crystal structure on S * i (λ) from Lemma 3.5 is explicitly computed by where γ x ∈ Γ a is minimal with ⟨x, r(γ x )⟩ = ε a (x) and γ x ∈ Γ a is maximal with ⟨x, r(γ x )⟩ = ε a (x).
Theorem 5.1 is proved in Section 8. A formula for the * -crystal structure on S i given in Lemma 3.4 can directly deduced from Theorem 5.1: where γ x ∈ Γ a is minimal with ⟨x, r(γ x )⟩ = ε * a (x) and γ x ∈ Γ a is maximal with ⟨x, r(γ x )⟩ = ε * a (x). Proof. Since S i = ∪ λ∈P + S * i (λ) we can find for each x ∈ S i a λ ∈ P + such that f * a x ∈ S * i (λ) = {x ∈ S i ε a (x) ≤ λ a ∀a ∈ [n − 1]}. Thus the claim follows from Lemma 3.5 and Theorem 5.1.
Remark 5.3. The * -crystal structure on the string cone S i is dual to the crystal structure on Lusztig data, which is governed by the Crossing Formula 7.3 recalled below. By duality we understand the following: Maximum and minimum swap place as do the maps r ∶ Γ a → Z N and s ∶ Γ a → Z N .
The * -crystal structure on Lusztig data x ∈ N N is described by the * -Crossing Formula [GKS16, Theorem 2.20], which is completely analogous to the Crossing Formula for Lusztig data. In [GKS16,Theorem 4.4] we show that S i is polar to the set i.e. the vectors f * a x − x of the * -crystal structure on Lusztig data provide defining inequalities for S i . For the special case of reduced words adapted to quivers (23) was obtained in [Z13].
Similarly, the set of Lusztig data N N is polar to i.e. the vectors f a x − x of the crystal structure (13) on S i provide defining inequalities for the cone of Lusztig data N N .
6. Defining inequalities of Nakashima-Zelevinsky string polytopes Theorem 5.1 provides a formula for the crystal structure on the Littelmann-Berenstein-Zelevinsky string polytope S * i (λ). Switching the roles of B(∞) and B(∞) * in the definition of S * i (λ) one arrives at Building up on [NZ97], S i (λ) and its crystal structure is defined in [N99].
By Lemma 3.4 the set S i (λ) consists of the integer points of the Nakashima-Zelevinsky string polytope (15). By [FN17] the convex polytope S i (λ) R is rational. In this section we solve the problem of deriving defining inequalities for S i (λ) R ⊂ R N .
The Dual Crossing Formula (Theorem 5.2) immediately implies and for all γ ∈ Γ a . Using the explicit description of defining inequalities of S R i obtained in [GP00] we obtain defining inequalities of S i (λ) R ⊂ R N . We recall the result of [GP00] for the convenience of the reader.
Using the notation of Section 4 let D i be the wiring diagram associated to i ∈ W(w 0 ). For a ∈ [n − 1] let D i (a) ∨ be the graph obtained from D i (a) by reversing all arrows. For a ∈ [n − 1] an a-rigorous path is an oriented path γ = (v 1 , . . . , v k ) in D i (a) ∨ which starts with the rightmost vertex of the wire a and ends with the rightmost vertex of the wire a+1. Additionally γ satisfies the following condition: Whenever v j , v j+1 , v j+2 lie on the same wire p in D i and the vertex v j+1 lies on the intersection the wires p and q, we have We denote the set of all a-rigorous paths by Γ * a . For γ ∈ Γ * a we define the set of turning points and the vector r(γ) as in Definitions 4.9 and 4.11, respectively.
As a direct consequence of [GP00, Corollary 5.8] and Theorem 6.1 we obtain Corollary 6.2. The Nakashima-Zelevinsky string polytope S i (λ) R is explicitly described by For the sake of completeness we recall the crystal structure on S i (λ). For k ∈ [N ] we consider the function η k on S i (λ) defined in (12). Analogously to Lemma 3.5 we have Lemma 6.3 ( [N99]). The following defines a crystal structure on S i (λ) isomorphic to B(λ). For x ∈ S i (λ) and a ∈ [n − 1] is maximal with i ℓx = a and η ℓx (x) = ε a (x).

The Crossing Formula on Lusztig data
The main ingredient in the proof of Theorem 5.1 is the Crossing Formula proved in [GKS16], which we recall in this section. 7.1. Lusztig's parametrization of the canonical basis. Lusztig [L90] associated to a reduced word i = (i 1 , i 2 , . . . , i N ) ∈ W(w 0 ) a PBW-type basis B i of U − q as follows. Let β 1 < β 2 < . . . < β N be the total ordering of Φ + corresponding to i via Remark 2.2. We set where T i acts via the braid group action defined in [Lu90, Section 1.3]. The divided powers x (m) for x ∈ U − q are defined in (1). Then the PBW-type basis is in natural bijection with the canonical basis B of U − q (see [L90, Proposition 2.3, Theorem 3.2]).
Crystal structures on Lusztig's parametrizations. Let i and j be two reduced words for w 0 . A piecewise linear bijection Φ i j ∶ N N → N N from the set of i-Lusztig data to the set of j-Lusztig data is defined in [L90, Section 2.1] using the fact that any reduced word j can be obtained from any other reduced word i by applying a sequence of 2-and 3-moves given in Definition 2.1.
Proposition 7.2. Let a ∈ [n−1] and j ∈ W(w 0 ) with j 1 = a. For an i-Lusztig The main result of [GKS16] is the Crossing Formula for the crystal structure from Proposition 7.2. Using (5) this leads for λ ∈ P + to a formula for the crystal structure on Theorem 7.3 ([GKS16, Theorem 2.13, Proposition 2.20]). For λ ∈ P + , x ∈ L i (λ) and a ∈ [n − 1] we have where γ x ∈ Γ a is minimal with ⟨x, s(γ x )⟩ = ε a (x) and γ x ∈ Γ a is maximal with ⟨x, s(γ x )⟩ = ε a (x).

Proof of Theorem 5.1
We fix i = (i 1 , . . . , i N ) ∈ W(w 0 ) as well as λ = ∑ b∈[n] λ b ω b ∈ P + and set 8.1. A bijection between string and Lusztig data. Let (c i,j ) be the Cartan matrix of sl n . For x ∈ Z N we define The bijection G λ i between S * i (λ) and L i (λ * ) intertwines the crystal structures given in Lemma 3.5 and Proposition 7.2 as follows.
Proof. Clearly, (25) and (26) hold for i 1 = a and thus by Proposition 8.1 for arbitrary i ∈ W(w 0 ). By (26) and the crystal axiom (C3) in Definition 1.1 it is enough to show (27) for the highest weight element x λ of S * i (λ). By (25) we have for a ′ ∈ [n − 1] 8.2. Reineke crossings and the bijection G λ i . For a ∈ [n − 1] we attach in Definition 4.11 to γ ∈ Γ a the vectors s(γ), r(γ) ∈ Z N . In [G18,Theorem 3.11] it is shown that the map F i relates s(γ) and r(γ) ∈ Z N as follows: In this section we use Proposition 8.3 to show For this we define for a ∈ [n − 1] the function To prove Proposition 8.4 we use Lemma 8.5. For a, b ∈ [n − 1] and γ ∈ Γ a we have ℓ b (s(γ)) = δ a,b .
Proof of Proposition 8.4. From Proposition 8.3 we obtain Furthermore, since Thus, by Lemma 8.5 Combining (28), (29) and (30) yields It remains to prove Lemma 8.5. Recall the notion of the level of a vertex v of D i from Definition 4.1. For each vertex v of γ, we define the oriented edge of D i (a) with target v that γ follows is headed upwards, the oriented edge of D i (a) with source v that γ follows is headed downwards, level(v) + 1 the oriented edge of D i (a) with source v that γ follows is headed upwards.
Here we understand "headed upwards" and "headed downwards" with respect to a small neighborhood around the vertex v.
We give an example for this notion.
Example 8.6. Let n = 5. And γ = (v 3,2 , v 3,1 , v 1,2 , v 2,5 , v 2,4 , v 4,5 , v 4,1 ) the 3−Reineke crossing from Example 4.8 colored red below. We have Note that, by definition, for γ = (v 1 , v 2 , . . . , v m ) ∈ Γ a , we have level − γ (v 1 ) = a, level + γ (v ℓ ) = level − γ (v ℓ+1 ) and level + γ (v m ) = a + 1. Thus, Lemma 8.5 is now a direct consequence of Proof. Assume that the vertex v ℓ = v p,q of γ lies at the intersection of wires p and q, where p is the oriented wire with source v ℓ . We assume first p ≤ a, hence the wire p is oriented from left to right in D i (a). We proceed by a case by case analysis.
The argument for the assumption a + 1 ≤ p is symmetrical.
The proof of (22) works analogously to the proof of (21).

Kashiwara * -involution on String data
In this section we denote by S i and S * i the set of i-string data equipped with the crystal structure inherited from B(∞) and B(∞) * , respectively, via the bijection str * i (see (13) and (15)). We denote by L i = N N and L * i = N N the set of i-Lusztig data with the crystal structure inherited from B(∞) and B(∞) * , respectively, via the bijection b i defined in (24). We write L R i ∶= R N ≥0 . Using ε a from the crystal L i and ε * a from L * i we define the polytopes L i (λ) R ∶= {x ∈ L R i ε * a (x) ≤ λ a ∀a ∈ [n − 1]}, L i (λ) R ∶= {x ∈ L R i ε a (x) ≤ λ a ∀a ∈ [n − 1]}. The integral points of L i (λ) R and L * i (λ) R are L i (λ) and L * i (λ) respectively. For a reduced word i = (i 1 , . . . , i N ) ∈ W(w 0 ) we define i * ∶= (n − i 1 , . . . , n − i N ) ∈ W(w 0 ), i op ∶ = (i N , . . . , i 1 ) ∈ W(w 0 ).
Since i 0 and i op 0 are related by a sequence of 2-moves the isomorphism of crystals Φ i 0 i 0 ∶ S * i (λ) R ∼ → S j (λ) R .
For λ ∈ P + we thus have the following commutative diagrams of isomorphisms of crystals which are linear for i = i 0 .
Furthermore, the following are commutative diagrams of volume preserving piecewise linear bijections which are linear for i = i 0 .