Affine type A geometric crystal structure on the Grassmannian

We construct a type A n−1 affine geometric crystal structure on the Grassmannian Gr(k, n). The tropicalization of this structure recovers the combinatorics of crystal operators on semistandard Young tableaux of rectangular shape (with n − k rows), including the affine crystal operator ẽ0. In particular, the promotion operation on these tableaux essentially corresponds to cyclically shifting the Plücker coordinates of the Grassmannian. Résumé. Nous munissons la grassmannienne Gr(k, n) d’une structure de cristal géométrique affine en type A n−1. La tropicalisation de cette structure recouvre la combinatoire des opérateurs de cristal sur les tableaux de Young semistandards de forme rectangulaire (avec n− k lignes), y compris l’opérateur affine ẽ0. En particulier, l’opération de promotion sur ces tableaux correspond essentiellement au décalage cyclique des coordonées plückeriennes de la grassmannienne.


Introduction
Kashiwara's theory of crystal bases provides a combinatorial model for the representation theory of semisimple Lie algebras, and more generally of Kac-Moody algebras ( [Kas91]).In type A n−1 , this theory brings to light an intimate connection between the representation theory of sl n and the combinatorics of semistandard Young tableaux (SSYTs).The operations on tableaux that arise in this theory, such as promotion, evacuation, the "crystal operators," and the Robinson-Schensted-Knuth correspondence, are classically defined in terms of combinatorial algorithms on the individual entries in the tableau, such as bumping, sliding, or bracketing rules.Interestingly, when these operations are transferred from tableaux to Gelfand-Tsetlin patterns, they are given by piecewise-linear formulas ( [Kir01], [KB96]).This suggests that there should be a way to lift all of these operations to subtraction-free rational functions on some algebraic variety, in such a way that the behavior of these rational functions parallels that of the combinatorial operations.One advantage of such a rational lift is that certain properties of the combinatorial maps may become more transparent in the rational setting.For example, this technique was used in [LP13] to study the intrinsic energy function of tensor products of one-row tableaux; in [LPS14] to compute the lengths of solitons in a discrete integrable system called the box-ball system; and in [GR14] to prove a result on the order of rowmotion.
Berenstein and Kazhdan's work on geometric crystals shows how to lift a large part of classical crystal combinatorics to the rational setting ( [BK07]).For every reductive Lie group, they define rational maps on the corresponding Schubert cells and partial flag varieties.These maps behave like rational versions of Kashiwara's crystal operators, and under a carefully chosen parametrization and tropicalization, they recover the combinatorics of Kashiwara's crystal bases.
Nakashima extended Berenstein and Kazhdan's geometric crystal theory to the Kac-Moody setting, and in particular to affine Lie algebras ([Nak05]).Affine Lie algebras have an important class of finitedimensional representations, known as Kirillov-Reshetikhin modules.In type A (1) n−1 , Kirillov-Reshetikhin modules correspond to rectangular partitions, and their crystal bases are modeled by semistandard Young tableaux of rectangular shape.In addition to the classical crystal operators defined on tableaux of all shapes, there is an extra "affine crystal operator" e 0 defined on rectangular tableaux, corresponding to the action of the additional "affine root" of the Lie algebra sl n [Shi02].In [KNO08], the authors constructed geometric crystals corresponding to the Kirillov-Reshetikhin modules for symmetric powers of the standard representation in various affine types; in type A (1) n−1 , their construction provides a rational lift of the combinatorics of one-row tableaux.In [MN13], the authors constructed a geometric crystal corresponding to the second exterior power of the standard representation in type A (1) n−1 , thereby giving a rational lift of the combinatorics of two-row rectangular tableaux.By duality, lifting the combinatorics of k-row tableaux is essentially equivalent to lifting the combinatorics of (n − k)-row tableaux.However, we do not know of any previous construction of an affine geometric crystal corresponding to k-row rectangular tableaux, for min(k, n − k) > 2.
Our main result is the following: For each k ≤ n − 1, there exists a structure of A (1) n−1 geometric crystal on the Grassmannian Gr(n − k, n), such that the tropicalization of this geometric crystal, with respect to a "Gelfand-Tsetlin parametrization" defined below, recovers the crystal bases of Kirillov-Reshetikhin modules corresponding to rectangular partitions with k rows, and any number of columns.
We also show that a "twisted" version of the cyclic shifting of Plücker coordinates (Definition 3.4) tropicalizes to the promotion map on rectangular SSYTs.Cyclic shifting on the Grassmannian has previously been applied to the combinatorics of rectangular tableaux in the study of birational rowmotion on rectangular posets ( [GR14]), and in the study of cyclic sieving phenomena ( [Rho10]).In fact, the map induced by cyclic shifting of Plücker coordinates maps the Kazhdan-Lusztig basis element corresponding to a rectangular tableaux T to the basis element corresponding to the promotion of T (up to a sign) ( [Rho10]).This result was our inspiration for using the Grassmannian to lift promotion.
The outline of this paper is as follows.In Section 2, we review the Gelfand-Tsetlin parametrization of rectangular SSYTs, and we explain how to lift this parametrization to the Grassmannian (this lift was previously used in [NY04]).In Section 3, we review the definition of a geometric crystal, and define an affine geometric crystal structure on the Grassmannian.We state the main results of this paper (Theorems 3.6, 3.12, and 3.14).Proofs of these results will appear in a forthcoming work ( [Fri]).Section 4 illustrates these results with an extended example.

Gelfand-Tsetlin parametrization of the Grassmannian
In this section, we recall the the Gelfand-Tsetlin parametrization of semistandard Young tableaux, and explain how to lift it to a parametrization of the Grassmannian.

Gelfand-Tsetlin patterns
Fix an integer n ≥ 2. Definition 2.1 A semistandard Young tableaux (SSYT) is a filling of a Young diagram with numbers in {1, . . ., n}, so that the rows are weakly increasing, and the columns are strictly increasing.The partition corresponding to the Young diagram is the shape of the tableau.
A Gelfand-Tsetlin pattern (GT pattern) is a triangular array of nonnegative integers (A ij ) 1≤i≤j≤n satisfying the inequalities for Gelfand-Tsetlin patterns can be represented pictorially as triangular arrays, where the jth row in the triangle lists the numbers A ij for i ≤ j.For example, if n = 4, then a Gelfand-Tsetlin pattern looks like: There is a natural bijection between Gelfand-Tsetlin patterns and SSYTs.Given a Gelfand-Tsetlin pattern (A ij ), the associated tableau T is described as follows: the number of j's in the ith row of T is A ij − A i,j−1 .Equivalently, the jth row of the pattern is the shape of T ≤j , the part of T obtained by removing numbers larger than j.In particular, the nth row of the pattern is the shape of T .
Example 2.2 Here is an example of a Gelfand-Tsetlin pattern, and the corresponding SSYT.
In this paper, we will restrict our attention to rectangular SSYTs, that is, tableaux consisting of k rows of some common length L. We will typically fix the number of rows k ≤ n − 1, and allow the row length L to vary.

Definition 2.3 A k-row rectangular GT pattern is an array of nonnegative integers
Clearly k-row rectangular GT patterns are in natural bijection with k-row rectangular SSYTs.
Example 2.4 Suppose n = 5 and k = 2.Here is an example of a 2-row rectangular GT pattern, and the corresponding SSYT.The 6 in the bottom left corner is L, the length of each row.(If this number is removed from the diagram, the resulting array is a rectangle; hence the adjective "rectangular".)

Gelfand-Tsetlin coordinates and Pl ücker coordinates
In this section, we encode the Gelfand-Tsetlin coordinates of a k-row rectangular SSYT using Plücker coordinates on the Grassmannian Gr(n − k, n).The main result of this section is Lemma 2.10, which explains how to recover the Gelfand-Tsetlin coordinates from the Plücker coordinates.
Definition 2.5 Let E ij denote the matrix with a 1 in position (i, j), and zeros elsewhere.
For 1 ≤ i ≤ n − 1 and c ∈ C × , define For For example, if n = 4, then we have Definition 2.6 For i ≤ j, define Example 2.7 Suppose n = 5 and k = 3.Then we have where Definition 2.9 Let M be a matrix representative for a point in the Grassmannian Gr(n − k, n).We use the convention that M is an n × (n − k) matrix.For each subset J ⊂ {1, . . ., n} of size n − k, let P J denote the maximal minor of M using the rows in J.The P J are the Plücker coordinates of M .
Lemma 2.10 The map inverse is given by (M, L) → (A ij , L), where Here [i, j] denotes the interval {i, i + 1, . . ., j} if i ≤ j, and the empty set if i > j.
We call Θ k the Gelfand-Tsetlin paramaterization of Gr(n − k, n) × C × .See Section 4 for an explicit example of the maps Θ k and Θ −1 k .Remark 2.11 The matrix in Example 2.7 is the "weight matrix" of the planar network shown in Figure 1.That is, the matrix entry in position (i, j) is equal to where the sum is over directed paths γ starting at i and ending at j , and the weight of a path is the product of the weights of the edges in the path.In general, the matrix Φ k (A ij , L) has such a network representation, and one can apply the Lindström-Gessel-Viennot Lemma (i) to the network to obtain formula (3) for Θ −1 k , and to show that all the Plücker coordinates of Θ k (A ij , L) are given by subtraction-free rational expressions in the variables A ij , L.
(i) See [FZ00] for a nice exposition of this lemma and some of its applications.
3 Geometric crystal structure and tropicalization

Geometric crystals
In this section we define a type A • for each i ∈ Z/nZ, a rational map φ i : X → C × • for each i ∈ Z/nZ, a rational unital action e .i of C × on X, where we denote the action of c ∈ C × on x ∈ X by e c i (x).
These data must satisfy the following three properties: where with c in the ith component and c −1 in the (i + 1)st component (mod n).
2. For i ∈ Z/nZ, x ∈ X and all c ∈ C × , we have φ i (e c i (x)) = c −1 φ i (x).A geometric crystal is decorated if there is a rational function f : , with indices taken mod n.
Fix k ≤ n − 1.We will now define a type A n−1 geometric crystal structure on the variety Gr(k, n) × C × .Let M be a matrix representative for a point in Gr(k, n).As in the previous section, we use the convention that M is an n × k matrix, and we denote the Plücker coordinates of M by P J .Definition 3.2 For i ∈ Z/nZ, let i denote the cyclic interval {i, i − 1, . . ., i − k + 1} ⊂ Z/nZ.Definition 3.3 The data for our (decorated) affine geometric crystal structure on Gr(k, n) × C × are defined as follows: 1.The map γ : Gr(k, n) × C × → (C × ) n is defined by γ(M, L) = (γ 1 , . . ., γ n ), where

The map
3. The map e c i : Here x i (a) is the n × n matrix with 1's on the diagonal, a in position (i, i + 1), and 0's elsewhere.Note that x 0 (a) has an a in position (n, 1).Although this map is not part of the data for a geometric crystal, it is an essential tool for studying the affine geometric crystal structure, due to the following lemma.

The decoration
Lemma 3.5 For each i ∈ Z/nZ, we have , where γ i = γ i (M, L) are defined in Definition 3.3.
Theorem 3.6 The data (Gr(k, n) × C × , γ, φ i , e .i , f ) define a decorated geometric crystal of type A (1) n−1 .For an explicit computation of some of the maps defined above, see Section 4.
Remark 3.7 In [BK07], the authors construct a type A n−1 geometric crystal on the variety X − P = U Z(L P )w P U ∩ B − ⊂ GL n , for each parabolic subgroup P .Let P k be the maximal parabolic subgroup in type A n−1 corresponding to the subset of Dynkin nodes {1, . . ., k − 1, k + 1, . . ., n − 1}.There is a rational isomorphism from X − P k to Gr(n − k, n) × C × , essentially given by projecting from GL n to the first n − k columns, as in Section 2.2.This isomorphism commutes with the classical part of the crystal structure defined above (i.e., everything other than φ 0 and e c 0 ).One advantage of projecting to the Grassmannian is that it brings out the cyclic symmetry which is "invisible" in X − P k , making it clear how to define promotion and the affine operator e c 0 .The Gelfand-Tsetlin parametrization used in this paper is closely related to the family of parametrizations given in [BK07], although it is not a member of that family.

Combinatorial crystals
Here we list several important properties of the combinatorial maps on tableaux that show up in Kashiwara's crystal theory in type A (1) n−1 .Let B k,L be the set of rectangular semistandard Young tableaux with k rows and L columns (and entries in {1, . . ., n}).We will identify B k,L with the set of k-row rectangular GT patterns with length parameter L.

Fact 3.8
1.There is a weight map γ : B k,L → Z n , which maps a tableau to the vector (a 1 , . . ., a n ), where a i is the number of i's in the tableau.
2. For each i ∈ Z/nZ, there is a map φ i : B k,L → Z, and a crystal operator e i : B k,L → B k,L ∪{0}.
3. There is a promotion operation pr : B k,L → B k,L , which satisfies the following properties.
(b) For i ∈ Z/nZ, we have φ i • pr = φ i−1 and pr −1 • e i • pr = e i−1 , where subscripts are taken mod n.
(c) pr n (T ) = T for all T ∈ B k,L .
Proofs of these facts, along with combinatorial descriptions of the maps involved, can be found in [Shi02].

Tropicalization
Tropicalization is the procedure which transforms a subtraction-free rational function into a piecewiselinear function by replacing multiplication by addition, division by subtraction, and addition by the operation min. (ii) If g : R n → R m is a subtraction-free rational map, then we denote by Trop(g) its tropicalization, i.e., the map from R n to R m obtained by tropicalizing each component function of g.
Recall the parametrization is a subtraction-free rational map in the coordinates A ij , L. Thus, their tropicalizations are defined.Definition 3.11 Set g := Trop(g • Θ k ) if g is any of the functions γ, φ i , f .Also, set PR := Trop(Θ −1 k • PR •Θ k ), and e i := (Trop(Θ −1 k • e c i • Θ k ))| c=1 .Theorem 3.12 Suppose (A ij , L) is a k-row rectangular GT pattern.Then we have where pr is the (combinatorial) promotion operation.
The proof of this result (which will appear in [Fri]) is based on Kirillov and Berenstein's piecewise-linear description of Bender-Knuth moves in [KB96].
Theorem 3.14 The following statements hold for k-row rectangular GT patterns (A ij , L).
Proof (Sketch): The first statement is easy.For the second and third statements, we compute directly from the definitions that they hold when i = 1.Then we apply PR and invoke Fact 3.8(3b) and Theorem 3.12 to prove the statements for all i. 2 Remark 3.15 A result similar to Theorem 3.14 is shown to hold for a large class of the geometric crystals considered in [BK07].These results can be viewed as a precise formulation of the statement that geometric crystals tropicalize to combinatorial crystals, or that geometric crystals are a rational lift of combinatorial crystals.

An example
In this section, we explicitly write down the maps e c 0 and PR, as well as their tropicalizations, in the case , and Observe that e c 0 changes the A ij as follows: A We verify that these piecewise-linear formulas agree with e 0 for a particular tableau.Consider the following 2-row tableau T , and its corresponding rectangular GT pattern: The image of T under e 0 (and its corresponding GT pattern) are: One can easily verify that the second GT pattern is obtained from the first according to the piecewise-linear formulas (6).For example, substituting the numbers in the GT pattern for T into the piecewise-linear formula for A 22 in (6), we compute that the value of A 22 after applying e 0 is 1 + min(4, 6) − min(5, 6) = 1 + 4 − 5 = 0, which agrees with the value computed combinatorially by e 0 .Now we write down the map PR in terms of the coordinates A ij and L. By definition, PR acts on Plücker coordinates as follows: Tropicalizing these formulas, we get the following piecewise-linear formulas for the action of PR on the GT coordinates A ij : We leave it to the reader to verify that the GT pattern for pr(T ) agrees with the output of the piecewiselinear formulas for the action of PR on the A ij , in accordance with Theorem 3.12.

Fig. 1 :
Fig. 1: A network corresponding to the matrix in Example 2.7.Edges are directed from left to right, and the weight of an edge is shown directly above it.Edges with no weight label (such as the diagonal edges) are assumed to have weight 1.