aa r X i v : . [ m a t h . C O ] F e b CLUSTER DUALITY FOR LAGRANGIAN AND ORTHOGONALGRASSMANNIANS
CHARLES WANG
Abstract.
In [RW19] Rietsch and Williams relate cluster structures and mirror symmetryfor type A Grassmannians Gr( k, n ), and use this interaction to construct Newton-Okounkovbodies and associated toric degenerations. In this article we define a cluster seed for theLagrangian Grassmannian, and prove that the associated Newton-Okounkov body agreesup to unimodular equivalence with a polytope obtained from the superpotential defined byPech and Rietsch on the mirror Orthogonal Grassmannian in [PR13].
Contents
1. Introduction 12. The co-rectangles Seed 23. Polytopes 64. ∆ co-rect ∼ = Γ 125. Future Work 18References 181. Introduction
In [RW19] Rietsch and Williams view open subsets of the Langlands dual GrassmanniansGr( n − k, n ) \ D ac ∼ = Gr( k, n ) \ D ∨ ac in two ways: one as an X -cluster variety and the other asan A -cluster variety. Roughly speaking, cluster varieties are unions of algebraic tori ( C ∗ ) N ,indexed by combinatorial objects called seeds , identified along certain mutation maps . Theythen study combinatorial data on both sides: the Newton-Okounkov body associated to an X -torus and the superpotential polytope associated to a corresponding dual A -torus. Theirmain result is that these data agree, i.e. the polytopes are equal. Their strategy is to identifya particular seed for which they prove the equality “by hand,” and then to argue that thepolytopes transform in the same way under seed mutations.Let X = LGr( n, n ) be the variety of n -dimensional Lagrangian subspaces of C n withrespect to the symplectic form ω ij = ( − j δ i, n +1 − j . X is a homogeneous space of Dynkintype C , i.e. it can be written as Sp n /P for a parabolic subgroup P ⊂ Sp n . We considerits embedding as a subvariety of Gr( n, n ) in its Pl¨ucker embedding Gr( n, n ) ֒ → P ( ∧ n C n ).We will index Pl¨ucker coordinates on Gr( n, n ) by elements of (cid:0) [2 n ] n (cid:1) , the set of n -subsetsof [2 n ] := { , , . . . , n } , or alternatively by Young diagrams λ ⊂ n × n fitting inside the n × n square. We translate between these two notations by the following bijection. Considerlattice paths in the n × n rectangle which start in the upper right corner and end in the lowerleft corner with unit steps (either down or to the left) labelled sequentially by 1 , , . . . , n .Associate to I ∈ (cid:0) [2 n ] n (cid:1) the partition λ I lying above the lattice path whose vertical steps are labelled by the elements of I . For example, for n = 3, the partition associated to { , , } is. In particular, Pl¨ucker coordinates for X will be labelled interchangeably by n -subsets of[2 n ] and Young diagrams contained in the n × n square. X has dimension N := (cid:0) n +12 (cid:1) , and adistinguished anticanonical divisor D ac = D + · · · + D n made up of the n + 1 hyperplanes D i = { p n × i = 0 } = { p ( n − i +1) ... (2 n − i ) } , where n × i denotes the corresponding Young diagram.In this article, X takes the role of the X -cluster variety.Unlike the situation for the type A Grassmannians, the Langlands dual Grassmannian X ∨ is not isomorphic to X . Roughly speaking, we can associate to the Lie group G = Sp n thedata of the character lattice χ of a maximal torus T , and the root system Φ ⊂ χ . Then thereis a unique Lie group G ∨ , called the Langlands dual group , having as root system the corootsΦ ∨ and as character lattice the cocharacter lattice χ ∨ . The parabolic subgroup P ⊂ Sp n above then corresponds to some P ∨ ⊂ G ∨ , and we then set X ∨ = P ∨ \ G ∨ , and call this the Langlands dual Grassmannian .For X = LGr( n, n ), X ∨ is the orthogonal Grassmannian OG co ( n +1 , n +1) of co-isotropic( n + 1)-dimensional subspaces of C n +1 with respect to a quadratic form Q . (It is isomorphicto the orthogonal Grassmannian OG( n, n + 1) of isotropic n -dimensional subspaces of C n +1 with respect to Q .) Following [PR13], we consider X ∨ in its minimal embedding X ∨ ֒ → P ( V ∗ ),where V is the irreducible representation corresponding to the parabolic subgroup P ∨ ( P ∨ will be a maximal parabolic subgroup since P was). As noted in [PR13, § X is cominuscule , its cohomology is isomorphic (by the geometric Satake correspondence) to V .In this article, X ∨ takes the role of the A -cluster variety.In this article, we carry out the first step of the [RW19] strategy for X and X ∨ . We identifya particular seed, which we call the co-rectangles seed , and show that the Newton-Okounkovbody corresponding to this seed is unimodularly equivalent to the superpotential polytopedefined using the Landau-Ginzburg model studied in [PR13].The outline of this article is as follows. In section 2, we define our co-rectangles seed. Insection 3, we define the Newton-Okounkov body ∆ co-rect and the superpotential polytope Γ,and relate Γ to a chain polytope. In section 4, we prove that ∆ co-rect and Γ are unimodularlyequivalent, and in section 5, we describe upcoming work. Acknowledgements.
The author is grateful to Konstanze Rietsch, Bernd Sturmfels, andLauren Williams for many helpful discussions, comments, and suggestions.2.
The co-rectangles Seed
Seeds in a cluster structure of rank l for a commutative algebra are specified by a pair( x , B ) of cluster variables x = ( x , . . . , x m ) and an m × l extended exchange matrix B , forsome m ≥ l . If the topmost l × l square submatrix of B is skew-symmetric, we can replace B with a quiver Q , which is a directed, oriented graph which may have parallel edges, butno 2-cycles or loops, and some vertices designated as ’frozen.’ For brevity, we do not give afull definition of a cluster algebra, and instead refer to [FWZ16, § k, n ) studied in [RW19], whichwere first proven to give a cluster structure in [Sco06], can be described by quivers, and fur-thermore certain seeds admit an additional description in terms of certain planar, bicoloredgraphs called plabic graphs . When we wish to distinguish the vertices of a plabic graph ac-cording to the bicoloring, we will refer to them as hollow ( ◦ ) or filled ( • ). Roughly speaking, x corresponds to the set of face labels of a plabic graph G , and Q corresponds to the dual LUSTER DUALITY FOR LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS 3 graph of G . A more thorough exposition of plabic graphs can be found in [Pos06], wherethey were first introduced, and the relationship between cluster seeds and plabic graphs forGrassmannians can be found in [RW19, § The co-rectangles Symmetric Plabic Graph.
For the Lagrangian Grassmannian,the seed we are interested in studying is associated to an extended exchange matrix B co-rect whose top square submatrix is not skew-symmetric. This seems to be related to the factthat the type C n Dynkin diagram is not simply-laced. However, certain plabic seeds forthe Grassmannian can be used to obtain seeds for the Lagrangian Grassmannian by quiverfolding . We first recall the analogous notion of symmetric plabic graphs , due to [Kar18].
Definition 2.1 ([Kar18, Def. 5.1]) . A symmetric plabic graph for X is a plabic graph G with 2 n boundary vertices, labelled clockwise by 1 , . . . , n , and a distinguished diameter d of the bounding disk satisfying the following conditions:(1) d has one endpoint between vertices 2 n and 1, and the other between n and n + 1.(2) No vertex of G lies on d .(3) Reflecting G through d gives a graph identical to G with the colors of vertices reversed.Our seed comes from the co-rectangles symmetric plabic graph G co-rect n (face are labelledby co mplements of rect angular Young diagrams in the n × n square). We define G co-rect n byexample for n = 4 (see 2.1), and give the associated dual quiver in 2.1 and folding in 2.5. Theextension to arbitrary n is straightforward. We note that our G co-rect n is mutation equivalentto G rec n, n of [RW19], so in particular satisfies a technical assumption called reducedness .1234 5678 ◦◦◦ ••• ◦◦◦ ••• ◦◦◦ ••• ◦ • ∅ Figure 2.
The co-rectangles symmetric plabic graph G co-rect4 . CHARLES WANG
Remark 2.3.
The face labels in 2.1 are auxiliary data associated to the graph, and theprocedure to obtain them is described in [RW19, Definition 3.5].Next we give the dual quiver, which corresponds to an X -seed for Gr(4 , x x x x x ∅ xxx xxx xxx xxx Figure 4.
The dual quiver Q to G co-rect4 . Vertices of Q are labelled by the X -cluster variables. Edges of Q are directed such that when crossing an edgeof the plabic graph, the hollow vertex is to the left. Boxes have been drawnaround frozen vertices, which correspond to faces of the plabic graph adjacentto the boundary disk.The associated exchange matrix is the 17 × B whose rows are indexed by anyof the Young diagrams in 2.1 and whose columns are indexed by the non-boxed Youngdiagrams. The ( µ, ν ) entry of B is given by: B µ,ν = µ → ν is an edge in Q − µ ← ν is an edge in Q X -seed for X , we fold the quiver 2.1 by the involution induced bysending a vertex to its reflection about the dashed diagonal. Because we will not make useof it in this article, we do not define quiver folding, and we refer to [FWZ17, § × B co-rect4 of the folding: Example 2.5.
The columns of the folded matrix are indexed (in order) by the mutable orbitsof the involution. These are: { } , { } , { } , { , } , { , } , { , } . The LUSTER DUALITY FOR LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS 5 first six rows will be indexed in the same order. The remaining 5 rows will be indexed (inorder) by the frozen orbits of the involution { } , { , } , { , } , { , } , { ∅ } . B co-rect4 = − − − − − − −
10 2 − − − − −
10 0 − The Network Parametrization ( X -cluster seed) for X . We will now describe howto use a symmetric plabic graph to construct a network torus in X . This will allow us tocompute valuations associated to a seed using plabic graphs in the following section. Wesummarize the presentation in [RW19, § Definition 2.6. A perfect orientation O of a plabic graph G is an orientation of each edgeof G such that each filled internal vertex is incident to exactly one edge directed away fromit, and each hollow vertex is incident to exactly one edge directed towards it. The source set I O of O is the set of boundary vertex labels which are sources of G as a directed graph withedge directions O .Let G denote a plabic graph with a perfect orientation O . If we need further assumptionson G , they will be stated explicitly. Definition 2.7.
Let J be a subset of the boundary vertices of G with | J | = | I O | . A flow from I O to J is a collection of pairwise vertex-disjoint paths with sources I O \ ( I O ∩ J ) andsinks J \ ( I O ∩ J ).Because each path p in a flow F begins and ends at a boundary vertex of G , p partitionsthe faces of G into two sets, those to the left of p and those to the right of p in the directionof the path. Let p L denote the set of face labels to the left of p . Definition 2.8.
For a path p in a flow F , we define the weight of p to be wt( p ) = Q λ ∈ p L x λ .For a flow F , we define the weight to be wt( F ) = Q p ∈ F wt( p ). Finally, for a subset J of theboundary vertices of G with | J | = | I O | , let F denote the set of all flows from I O to J , anddefine the flow polynomial P GJ = P F ∈F wt( F ).Now let G = G co-rect n be the co-rectangles plabic graph. In what follows, we will need aperfect orientation O co-rect on G , defined as follows. Definition 2.9.
Set { , , . . . , n } to be sources, and { n + 1 , n + 2 , . . . , n } to be sinks. (Theedges adjacent to vertices 1 ≤ i ≤ n will be directed away from i , and the edges adjacent tovertices n + 1 ≤ i ≤ n will be directed towards i .) Because symmetric plabic graphs arealso usual plabic graphs, then there is a unique such perfect orientation by [PSW09, Lemma4.5], see [RW19, Remark 6.4]. We call this O co-rect . CHARLES WANG
This is the choice of perfect orientation we will use for the rest of the article. For anexample of the above definitions, see 3.4, where we give our perfect orientation for n = 3,and compute a flow polynomial.Next, let S = { x µ | µ is a face label of G co-rect n } be the set of face labels of the co-rectanglesplabic graph. We think of these as coordinates on the network torus T G ∼ = ( C ∗ ) | S | , and weuse the flow polynomials to define an embedding of T G into Gr( n, n ). Theorem 2.10 ([Pos06, Theorem 12.7],[RW19, Theorem 6.8]) . Let G be the co-rectanglesplabic graph, and J ∈ (cid:0) [2 n ] n (cid:1) . Consider the map Φ : T G → Gr( n, n ) defined by sending ( x µ | µ ∈ S ) ∈ T G ( P GJ ( x µ ) | J ∈ (cid:0) [2 n ] n (cid:1) ) ∈ Gr( n, n ) . Then Φ is well-defined, and givesan embedding T G ֒ → Gr( n, n ) . Finally, let G = G co-rect n be the co-rectangles symmetric plabic graph. We define theequivalence relation ∼ on S given by x µ ∼ x µ T , where µ T denotes the transpose partition to µ . In Karpman’s language, this corresponds to taking a symmetric weighting , and Karpmanshows that restricting to these weightings gives an embedding whose image lands inside of X ⊂ Gr( n, n ). We think of S/ ∼ as coordinates on the network torus T co-rect ∼ = ( C ∗ ) | S/ ∼| ,and we use the flow polynomials to define an embedding of T co-rect into X . Theorem 2.11 ([Kar18, Theorem 5.15]) . Let G = G co-rect n be the co-rectangles symmetricplabic graph, and J ∈ (cid:0) [2 n ] n (cid:1) . Consider the map Φ : T co-rect ֒ → X which is defined by sending ( x µ | µ ∈ S/ ∼ ) ∈ T co-rect ( P GJ ( x µ = x µ T ) | J ∈ (cid:0) [2 n ] n (cid:1) ) ∈ X . Then Φ is well-defined, andgives an embedding T co-rect ֒ → X . Remark 2.12.
Although Karpman’s paper is written in terms of edge weightings, the trans-lation to face weightings can be found in [Pos06, Lemma 11.2].Thus we associate to G co-rect n and O co-rect a dense torus T co-rect ֒ → X . On the level ofcoordinate rings, this induces an injection C [ X ] ֒ → C [ T co-rect ], so we may express polynomialsin the Pl¨ucker coordinates on X as Laurent polynomials in the coordinates on T co-rect .3. Polytopes
The Newton-Okounkov body ∆ co-rect . We associate to the co-rectangles symmetricplabic graph G co-rect n and ample divisor D = D n a Newton Okounkov body ∆ co-rect ( D ) by thefollowing procedure, following [RW19, Definition 8.1]. First, we define a valuation using theinclusion C [ X ] ֒ → C [ T co-rect ] obtained at the end of the previous section. Definition 3.1.
Fix a total order on the torus coordinates S defined at the end of theprevious section. Then we define the valuation val co-rect : C [ X ] \ { } → Z | S | by sending f ∈ C [ X ] to the exponent vector of the lexicographically minimal term when f is viewed asan element of C [ T co-rect ], i.e. as a Laurent polynomial in the torus coordinates.Now, using this valuation, we define the Newton-Okounkov body: Definition 3.2.
Let val co-rect be as above. Then we define∆ co-rect = conv ∞ [ r =1 r val co-rect ( H ( X , O ( rD ))) ! LUSTER DUALITY FOR LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS 7
Concretely, the nonzero sections in H ( X , O ( rD )) can be identified with Laurent polyno-mials whose numerators are degree r homogeneous polynomials in the Pl¨ucker coordinatesof X , and whose denominators are the Pl¨ucker coordinate p rn × n . For G co-rect n and O co-rect , theonly flow from [ n ] to [ n ] is the empty flow, so the expression of p n × n on the torus T co-rect is 1.Therefore, computing valuations of sections H ( X , O ( rD )) reduces to computing valuationsof elements of C [ X ], so we can use the valuation 3.1. Remark 3.3.
Although the valuation val co-rect depended upon a choice of total order of thetorus coordinates S , the Newton-Okounkov body ∆ co-rect does not, and we will not make useof any choice of total order in our proofs. Example 3.4.
For LGr(3 , • ◦• ◦•◦ • ◦• ◦ ∅ Figure 5.
The co-rectangles symmetric plabic graph for n = 3, with acyclicperfect orientation, and (minimal) flow from { , , } to { , , } in purple.123 456 • ◦• ◦•◦ • ◦• ◦ ∅ Figure 6.
A second flow from { } to { } for n = 3. CHARLES WANG
For the top flow above 3.4, there are no face labels to the left of the path 1 →
1. The facelabels to the left of 3 → → x , , x , x (3 , , x (3 , , x , , .For the bottom flow above 3.4, there are no face labels to the left of the path 1 → → → , , , , , , , contributing a monomial x , , x , x (3 , , x (3 , , x , , x (3 , , These are the only flows from { , , } to { , , } for G = G co-rect n and O co-rect , so the flowpolynomial is the sum of these P G { , , } = ( x , , x , x (3 , , x (3 , , x , , )(1 + x (3 , , )The minimal term is ( x , , x , x (3 , , x (3 , , x , , ), so the valuation is (0 , , , , , Definition 3.7.
For any skew partition ν ⊂ n × n , we define maxdiag( ν ) to be the maximumnumber of boxes along any diagonal of slope − Proposition 3.8.
For µ ⊂ n × n a face label of G and λ ⊂ n × n arbitrary , we have val co-rect ( p λ ) µ = ( maxdiag( µ \ λ ) + maxdiag( µ T \ λ ) µ = µ T maxdiag( µ \ λ ) µ = µ T Proof.
Because symmetric plabic graphs are also plabic graphs in the usual sense, then by[RW19, Lemma 6.3] we may choose a perfect orientation O with source set { , , . . . , n } .Recall that we think of λ also as an n -subset of [2 n ]. Then, because flows of the symmetricplabic graph are the same as flows in the underlying plabic graph, there is a minimal flow F from [ n ] to λ by [RW19, Corollary 12.4], and val co-rect ( p λ ) is the number of paths in F which have the face labelled by µ to the left, plus the number of paths in F which have theface labelled by µ T to the left if µ = µ T . Finally, by [RW19, Corollary 16.19], this is equalto maxdiag( µ \ λ ) if µ = µ T , and maxdiag( µ \ λ ) + maxdiag( µ T \ λ ) if µ = µ T . (cid:3) Example 3.9.
For LGr(3 , , , , , , , , , , , )): I ∈ (cid:0) [6]3 (cid:1) val co-rect ( p I )123 (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , I ∈ (cid:0) [6]3 (cid:1) val co-rect ( p I )146 = 245 (1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , LUSTER DUALITY FOR LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS 9 and the inequalities defining the convex hull of these coordinates: − − − − − − − − − − − − − − − p p p p p p ≥ f -vector (14 , , , , ,
10) and volume 16 =deg LGr(3 , { , , } with this face to the left, so every val co-rect ( p λ ) = 0 for any λ . Thus we exclude this coordinate in order to work with a full-dimensional polytope.The fact that the volume of the convex hull of the valuations of the Pl¨ucker coordinatesis equal to the degree of LGr(3 ,
6) in the example above is not an accident. In fact, as wewill see in the proof of 4.14:
Theorem 3.10. conv( { val co-rect ( p λ ) | λ ⊂ n × n } ) = ∆ co-rect is a Newton-Okounkov bodyfor X with respect to the valuation val co-rect . (Equivalently, the Pl¨ucker coordinates form aKhovanskii basis for C [ X ] with respect to the valuation val co-rect .) The superpotential polytope Γ . We use the Laurent polynomial expression for therestriction of the superpotential W q to a torus ( C ∗ )( n +12 ) ֒ → X ∨ for the Landau-Ginzburgmodel for X found by Pech and Rietsch: Definition 3.11 ([PR13, Prop. A.1]) . Let coordinates on the torus above be given by a ij for 1 ≤ i ≤ j ≤ n , and let Λ denote the set of strict partitions with at least one part of size n that are contained in the maximal, right-justified staircase in the n × n square. For any λ ∈ Λ, label each box by ( i, j ) where i indexes the row and j the column. Then set λ j tobe the largest index such that ( λ j , j ) ∈ λ . Then the restriction of the superpotential to thistorus is given by W q = X i ≤ j ∈ [ n ] a ij + X λ ∈ Λ q Q j ∈ [ n ] a λ j j Example 3.12.
For n = 3, the superpotential has (cid:0) (cid:1) + 2 − = 10 terms: a + a + a + a + a + a + qa a a + qa a a + qa a a + qa a a where the last four terms correspond to the diagrams
11 12 13 ,
11 12 1323 ,
11 12 1322 23 , and
11 12 1322 2333
In order to define the superpotential polytope Γ, we first define tropicalization for Laurentpolynomial whose coefficients are all positive, real numbers.
Definition 3.13 ([RW19, Def. 10.7]) . For any Laurent polynomial h in variables z , . . . , z k with coefficients in R > , we define Trop( h ) : R k → R inductively as follows. First, we set Trop( z i )( y , . . . , y k ) = y i , and we denote this tropicalization by a capital letter Trop( z i ) = Z i .Next, if h and h are any Laurent polynomials with positive coefficients, and c , c are anypositive real numbers, thenTrop( c h + c h ) = min(Trop( h ) , Trop( h )) and Trop( h h ) = Trop( h ) + Trop( h )This inductively defines Trop( h ).Following [RW19, Def. 10.14], we make the following definition for the Γ. Definition 3.14.
Consider W q : R ( n ) × R → R as a Laurent polynomial with positivecoefficients in the variables a ij (corresponding to the first factor of R ( n ) and q correspondingto the second factor of R . Then the superpotential polytope Γ is defined byΓ = { y ∈ R ( n ) | Trop( W q )( y, ≥ } Implicitly, we are “tropicalizing” q i to i by the evaluation Trop( W q )( y, Example 3.15.
The superpotential polytope for n = 3 corresponding to the potential 3.12is a polytope in R ( ), with coordinates indexed by the A ij ordered lexicographically, definedby the inequalities: A ij ≥ − A − A − A ≥ − A − A − A ≥ − A − A − A ≥ − A − A − A ≥ Remark 3.16.
The choice of 1 in the formula Trop( W q )( x, ≥ D = 1 ∗ D n . In fact, [RW19] defines ∆ G ( D ) and Γ G ( D )for more general divisors D than just D n and more general seeds G . However, we havesuppressed the dependence of Γ on G and D because we have not discussed the clusterstructure for X ∨ . This will be part of upcoming work [SW21].3.3. Poset polytope combinatorics.
In [Sta86], Stanley associated two polytopes to aposet P : the order polytope and the chain polytope . The chain polytope lives in R | P | , and isdefined by the inequalities e b ≥ b ∈ P and for any chain b < b < · · · < b k of P ,we have e b + e b + · · · + e b k ≤
1. In particular, because of the positivity inequalities, it isenough to consider the chain inequalities e b + e b + · · · + e b k ≤ b < b < · · · < b k isany maximal chain of P .Let P n be the poset on the elements { b ij | ≤ i ≤ j ≤ n } , with the cover relations b ij > b i +1 j +1 , b ij +1 . The superpotential polytope Γ produced above is the chain polytope of P n : the terms a ij correspond to the positivity inequalities, and the terms q Q j ∈ [ n ] a ijj correspondto maximal chain inequalities. Example 3.17.
The four maximal chains of P are b ≤ b ≤ b , b ≤ b ≤ b , b ≤ b ≤ b , and b ≤ b ≤ b . These correspond exactly to the ‘ q ’ terms of thesuperpotential above, so the chain polytope of P coincides with the superpotential polytope.By [Sta86, Thm. 2.2 and Cor. 4.2], the superpotential polytope has as many vertices asantichains in P n , and volume equal to the number of linear extensions of P n . LUSTER DUALITY FOR LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS 11 b b b b b b Figure 18.
Hasse diagram of P Lemma 3.19.
The number of antichains of P n is C n +1 = n +2 (cid:0) n +2 n +1 (cid:1) .Proof. Antichains are in bijection with order ideals, and for this poset the set of order idealsis in bijection with the set of Dyck paths of length 2 n +2 (Draw the Hasse diagram of P n , andrepresent each poset element by a box, so that the resulting picture is of a tilted staircase.Then the desired bijection is obtained by associating to an antichain the Dyck path whichpasses over only those boxes in the corresponding order ideal.) The number of such Dyckpaths is C n +1 . (cid:3) Hence the superpotential polytope has C n +1 many vertices. This is also the number ofYoung diagrams contained in the n × n square up to transpose, and hence the number ofdistinct Pl¨ucker coordinates for the Lagrangian Grassmannian X . This bijection is describedin more detail later as part of the proof that the superpotential polytope coincides with theNewton-Okounkov body. Example 3.20.
We illustrate the bijection between Dyck paths and antichains. ·· ·· · ·· · · ·· · · · · b b b b b b Figure 21.
Dyck path (dashed, purple) corresponding to the antichain { a } ,with corresponding order filter marked in purple. Lemma 3.22.
The number of linear extensions of P n is the degree of the Lagrangian Grass-mannian X .Proof. For any linear extension L of a poset P , we can consider L itself as a poset, and takeits dual L ′ . Then L ′ is a linear extension of the dual poset P ′ , for if x ≤ y in P ′ , then x ≥ y in P , and hence x ≥ y in L , so that x ≤ y in L ′ . Hence it suffices to count the numberof linear extensions of the dual of P n . Such a linear extension is equivalent to assigning adistinct integer c x in [ (cid:0) n +12 (cid:1) ] to each element x of P n , such that if x ≥ y in P n , then c x ≤ c y ,with equality iff x = y . This is exactly a standard Young tableaux on the staircase Youngdiagram ( n, n − , . . . , P n is the number of standard Young tableaux on the staircase diagram ( n, n − , . . . , X . (See A005118.) (cid:3) Hence the superpotential polytope has volume equal to deg( X ). By our choice of D = D n (sections of O ( rD ) correspond to degree r homogeneous polynomials) and G (the valuationis full rank, for example from 4.5), the volume of the Newton-Okounkov bodies constructedabove should also be equal to deg( X ) (see e.g. [KK12, Cor. 3.2] or [KM19], noting that weare not using the normalized volume, so should disregard the normalizing factor of dim( X )!).In particular, it is reasonable to expect that the superpotential polytopes and the Newton-Okounkov bodies constructed in this section should be unimodularly equivalent, and we willprove this in the following section. 4. ∆ co-rect ∼ = ΓNow we show that for the seed G = G co-rect n and corresponding valuation val co-rect , theNewton-Okounkov body ∆ co-rect and superpotential polytope Γ defined above are unimodu-larly equivalent, i.e. that there is a lattice isomorphism sending one polytope to the other.Our strategy is as follows. First, we define a linear map M n : R ( n +12 ) → R ( n +12 ) with integerentries. We show next that M n is unimodular, and finally that M R (Γ) = ∆ co-rect .Recall first the superpotential polytope Γ ⊂ R ( n +12 ) with coordinates ( A ij ) ordered lexico-graphically by the indices and the Newton-Okounkov body ∆ co-rect ⊂ R ( n +12 ) with coordinates( p λ ), where λ is the complement of a rectangle, ordered reverse lexicographically by the in-dices.Consider the Pl¨ucker coordinates p λ such that ( n − × ( n − ⊆ λ ( n × n and there are atleast as many boxes to the right of the main diagonal (upper left to lower right) as there arebelow. Note that there are (cid:0) n +12 (cid:1) such Pl¨ucker coordinates: for each pair (0 ≤ i ≤ j ≤ n − n − × ( n −
1) rectangle with j additionalboxes to the right of the diagonal, and i additional boxes below the diagonal.Form the (cid:0) n +12 (cid:1) × (cid:0) n +12 (cid:1) matrix M n whose columns are the valuations of the above Pl¨uckercoordinates, ordered as follows: order first by decreasing order (i.e. 2 n ≤ n − ≤ · · · ≤ increasing order in the first to last, second to last, etc.entries. On the level of Young diagrams, this corresponds to ordering first by increasingnumber of additional boxes below the main diagonal, and then by dexreasing number ofadditional boxes right of the main diagonal. When these diagrams are indexed by pairs( i, j ), the ordering is ( i, j ) ≤ ( i ′ , j ′ ) if i < i ′ or i = i ′ and j ≥ j ′ . Example 4.1.
For n = 3, the Pl¨ucker coordinates (in order) indexing the columns of M are: ( p , p , p , p , p , p ) or Young diagrams, ( p , p , p , p , p , p ), orpairs ((0 , , (0 , , (0 , , (1 , , (1 , , (2 , p , p , p , p , p , p ) or in Young diagrams, ( p , p , p , p , p , p ). M isthe following matrix: M = LUSTER DUALITY FOR LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS 13
This matrix is unimodular.4.1.
Unimodularity.Lemma 4.2.
The upper left n × n block of M n has the form: · · · · · · ... . . . ... · · · · · · = i = n j ≤ n − i, i = n j > n − i, i = n Proof.
By our choice of ordering for Pl¨ucker coordinates indexing the columns, the k th column(1 ≤ k ≤ n ) is the valuation of the Pl¨ucker coordinate p I k , where I k = [ n ] \ ( n − k + 1) ∪ { n } ,corresponding to the pair (0 , n − k ) or the Young diagram λ k = ( n n − k , ( n − k − ). The k th row corresponds to the k × n rectangle for 1 ≤ k ≤ n −
1, and the hook ( n, n − ) for k = n ,and we will call this diagram µ k . Using the maxdiag formula:val co-rect ( p λ j ) µ i = ( maxdiag(( n, n − ) \ λ j ) = 1 i = n maxdiag(( i n ) \ λ j ) + maxdiag(( n i ) \ λ j ) i = n Let i = n . Since ( n − × ( n − ⊂ λ j , and λ j has only n − i n ) \ λ j ) = 1.On the other hand, ( n i ) \ λ j = ( ∅ n − j ≥ i (1 i − ( n − j ) ) n − j < i In particular,val co-rect ( p λ j ) µ i = ( i = n n i ) \ λ j ) i = n = i = n n − j ≥ i, i = n n − j < i, i = n (cid:3) Applying the matrix with diagonal entries all 1, super-diagonal entries −
1, and all otherentries 0, followed by the matrix with diagonal entries all 1, all non-diagonal entries in thefirst column −
1, and all other entries 0, both on the right, sends the upper left n × n blockof M n to a permutation matrix (corresponding to the longest word). Each of these is aunimodular transformation, so the upper left n × n block of M n is unimodularly equivalentto Id n . We denote this sequence of unimodular transformations by T n . In particular, theupper left block of M n is unimodular. Lemma 4.3.
The lower right (cid:0) n (cid:1) × (cid:0) n (cid:1) block of M n is M n − .Proof. By our choice of ordering for the Pl¨ucker coordinates indexing the columns, the corre-sponding Young diagrams (indexing the columns in this block) all contain the hook partition( n, n − ) in the upper left. Similarly, by our choice of coordinates for the valuations, thecorresponding Young diagrams (indexing the rows in this block) also all contain the hook( n, n − ) in the upper left.Hence, to compute the valuations val co-rect ( p λ ) µ (e.g. using the maxdiag formula) in thisblock, we can remove all the upper left hooks ( n, n − ) from the corresponding λ and µ . Theresulting matrix is exactly a copy of M n − . (cid:3) Furthermore, since all of the Pl¨ucker coordinates indexing the last (cid:0) n (cid:1) columns of M n contain the hook ( n, n − ), then in particular val co-rect ( p I j ) ( n, n − ) = 0 for any I j in thesecolumns. In other words, the bottom row of the upper right n × (cid:0) n (cid:1) submatrix of M n is the0 vector. Lemma 4.4.
The lower left (cid:0) n (cid:1) × n block of M n has all columns equal and nonzero.Proof. By choice of ordering on the valuation coordinates, the coordinates µ i indexing theserows are all complements of rectangles i × j with both i, j < n . Since the Pl¨ucker coordinates p λ j indexing the columns all contain ( n − × ( n −
1) and have only n − µ i \ λ j ) = 1 independent of the Pl¨ucker coordinate p λ j indexing the column.In particular, when we compute the valuation, we getval co-rect ( p λ j ) µ i = ( maxdiag( µ i \ λ j ) = 1 µ i = µ Ti maxdiag( µ i \ λ j ) + maxdiag( µ Ti \ λ j ) = 1 + 1 = 2 µ i = µ Ti independently of j . Hence all columns of this block are equal. (cid:3) Note in particular that the T n described above applied on the right to the lower left (cid:0) n (cid:1) × n block of M n only gives nonzero entries in the last column, and furthermore the entries aregiven by 1 if the row index is invariant under transpose and 2 otherwise. Proposition 4.5. M n is unimodular.Proof. We proceed by induction, proving a slightly stronger statement: M n is unimodularlyequivalent by only column operations to a lower triangular matrix with 1’s on the maindiagonal. When n = 1, M n is a (cid:0) (cid:1) × (cid:0) (cid:1) = 1 × p ∅ = p . The valuation is in the coordinate p = p . By either the flow model orthe max-diag formula, we see that val( p ∅ ) = 1. Hence M = 1.Now let n >
1. By induction, we have some unimodular transformation T consisting ofonly column operations such that M n − T is lower triangular with 1’s on the main diagonal.Now, apply the block diagonal (hence unimodular, since both of the blocks are unimodular)transformation with T n , which only used column operations, in the upper left n × n blockand T in the lower right (cid:0) n (cid:1) × (cid:0) n (cid:1) block to M n on the right to get M ′ = M n diag( T n , T ).The upper left n × n block of M ′ is Id n by the discussion immediately following 4.2.The lower left (cid:0) n (cid:1) × n block of M ′ is nonzero in only the last column and zero everywhereelse by the discussion immediately following 4.4. Finally, the lower right (cid:0) n (cid:1) × (cid:0) n (cid:1) block of M ′ is lower triangular with 1’s on the main diagonal by induction and 4.3. Furthermore,since we have only used column operations on the first n columns and the last (cid:0) n (cid:1) columnsindependently, the 0 bottom row of the upper right n × (cid:0) n (cid:1) submatrix (see the discussionimmediately following 4.3) is preserved. Hence, we can perform further column operationsto zero out the upper right n × (cid:0) n (cid:1) submatrix without affecting the lower right (cid:0) n (cid:1) × (cid:0) n (cid:1) submatrix. Composing these gives us the desired unimodular transformation. (cid:3) LUSTER DUALITY FOR LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS 15
Example 4.6. M and M are: M = , M = Note first that the lower right (cid:0) (cid:1) × (cid:0) (cid:1) matrix of M is M , the lower left (cid:0) (cid:1) × × (cid:0) (cid:1) matrix is 0. The verifications ofthese facts exactly follow the proofs of the lemmas. Analyzing the upper left 3 × A −→ B −→ C −→ where the matrices A, B, C are given by A = − −
10 0 1 , B = − − , C = and their product is: ABC = − − − The matrix sending M to the desired form constructed in the proposition is: − − = Putting these together, we apply first the following transformation to M : − − − − − = and then we finish by zeroing out the remaining nonzero entry above the diagonal usingthe second column. Hence M is unimodular.4.2. Surjectivity.
It remains to prove that M n (Γ) = ∆ co-rect . To aid in our proof, we definean auxiliary polytope δ G in the same ambient space as ∆ co-rect : Definition 4.7. δ G = conv( { val co-rect ( p λ ) | λ ∈ (cid:0) [2 n ] n (cid:1) } ).A priori δ G ⊂ ∆ co-rect (for our choice of G and D , δ G = conv(val co-rect ( H ( X , O ( D ))))is the r = 1 part of the Newton-Okounkov body). For general G , δ G ( ∆ co-rect (i.e. the Pl¨ucker coordinates may not form a Khovanskii basis), but in our case, we will show that δ G = ∆ co-rect by showing that M n (Γ) = δ G , and computing volumes.Recall that the vertices of Γ are characteristic functions of antichains of P n . For eachsingleton antichain { a ij } of P n , we associate the vertex v ij ∈ Γ, and the hook partition ν ij = ( n + 1 − i, j − i ). Because j ≤ n , then n + 1 − i > j − i . Lemma 4.8.
The map { a ij } → ν ij between singleton antichains of P n and nonempty hookpartitions ( a, b ) ⊂ n × n with a > b described above is a bijection.Proof. For the hook partition ( a, b ) ⊂ n × n where a > b , set i = n +1 − a and j = n +1 − a + b .We have i ≤ j and 1 ≤ i ≤ j ≤ n as required. Conversely, if ν ij = ν kl , then i = k and j = l .Hence the above map is a bijection. (cid:3) We first identify where these vertices are sent under M n , and then use that to identifywhich antichains correspond to which Pl¨ucker coordinate valuations. Lemma 4.9. M n ( v ij ) = val co-rect ( p λ ) , where λ is the complement of ν ij in the n × n square,and the complement is taken by right-justifying ν ij in the bottom right corner.Proof. Since the vertices corresponding to the singleton antichains are unit vectors in R ( n +12 ),their images under M n are the corresponding columns of M n . In particular, since the columnsof M n are totally ordered, M n provides an order-preserving bijection between the singletonantichains (lexicographic ordering) and the column labels. We will consider the singletonantichains as labelled by a ij , and the columns as labelled by pairs i, j with 0 ≤ i, j ≤ n − M n can be reduced to finding such a bijection. The map a ij ( i − , n − − ( j − i )) is a bijection from the singleton antichains to the pairs labellingthe columns of M n . It remains to check that this is order preserving. Suppose a ij ≤ a i ′ j ′ ,so that i < i ′ or i = i ′ and j ≤ j ′ . In the first case i − < i − ′ . In the case of equality, i − i − ′ , and n − − ( j − i ) ≥ n − − ( j ′ − i ′ ), so the map is order preserving.Hence M n must be the map sending v ij to the valuation of the Pl¨ucker coordinate indexedby the pair ( i − , n − − ( j − i )). Then this diagram is the complement of the hook( n − ( i − , n − − ( n − − ( j − i ))) = ( n − i + 1 , j − i ) = ν ij (right justified, in the bottomright corner) as described. (cid:3) Any partition λ ⊂ n × n has a right-justified complement partition λ c ⊂ n × n . Wedecompose λ c into a union of k nonempty (right-justified) hooks λ c = ν + · · · + ν k , wherethe decomposition comes from taking the hooks from the boxes of λ c along the main diagonal. Example 4.10.
For ⊂ , the complement is , which decomposes into the hooks + .For an asymmetric example, take ⊂ . The complement is , and decomposes into thehooks + . Lemma 4.11.
Let λ ⊂ n × n be a partition with at least as many boxes above the main diag-onal as below. Then the hook decomposition of the transpose of the complement correspondsto an antichain of P n under the bijection 4.8.Proof. Let λ c = ν + · · · + ν k be the hook decomposition, and set the notation ν i = ( a i , b i )for the hooks. By our assumption that λ has at least as many boxes above the diagonal asbelow, we have that a i > b i . Then ν i corresponds to the poset element x i = a n +1 − a i ,b i + n +1 − a i LUSTER DUALITY FOR LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS 17 (because b i + n + 1 − a i = n + 1 − ( a i − b i ) ≤ n , this is actually an element of P n ). Nowsuppose that there are i, i ′ such that the x i ≤ x i ′ . By definition of P n , this means: b i + n + 1 − a i ≥ b i ′ + n + 1 − a i ′ ⇐⇒ b i − a i ≥ b i ′ − a i ′ ≤ n + 1 − a i − ( n + 1 − a i ′ ) ≤ b i + n + 1 − a i − ( b i ′ + n + 1 − a i ′ ) ⇐⇒ ≤ a i ′ − a i ≤ b i − a i − b i ′ + a i ′ ⇐⇒ ≤ b i − b i ′ and 0 ≤ a i ′ − a i In order to come from a partition, the hooks ν i must satisfy a nesting condition: for any i , we must have a i +1 < a i and b i +1 < b i . Hence, for any i < j , we must have a j < a i and b j < b i . However, these conditions contradict the ones above, so it could not have beenthat x i ≤ x i ′ . Hence for all i, i ′ , the pair x i , x i ′ must be incomparable, so the set { x i } is anantichain of P n . (cid:3) Example 4.12.
For ⊂ , the complement is , which decomposes into the hooks + ,corresponding to the elements a and a , respectively. For ⊂ , the complement is ,and decomposes into the hooks + , corresponding to the elements a , a . Lemma 4.13.
Let λ ⊂ n × n be any partition with at least as many boxes above the maindiagonal as below. Let λ c = ν + · · · + ν k be the hook decomposition of the complement. Then maxdiag( µ \ λ ) = P i maxdiag( µ \ λ i ) , where λ i is the complement of ν i (right-justified in thebottom right corner of n × n ), for any µ labelling a face of G co-rect .Proof. Since µ is a face label of G co-rect , it is the complement of a rectangle ρ ⊂ n × n in thebottom right corner. Then maxdiag( µ \ λ i ) = ( ν i ρ ν i ⊂ ρ Hence, the sum S = P i maxdiag( µ \ λ i ) is the number of i such that ν i ρ . Because ofthe nesting condition of the hooks { ν , . . . , ν k } , there is some 0 ≤ i ≤ k such that ν j ⊂ ρ for j > i and ν j ρ for j ≤ i . Hence S = i . Furthermore, for any box b of ν i not containedin ρ , then this guarantees the existence of a diagonal of µ \ λ of length i ending at b . (Since b ∈ ν i and b / ∈ ρ , then b ∈ µ . Since ν i is the i th hook of the complement of λ , then the i boxes diagonally above and to the left of b are also not in λ while being in µ .) Hencemaxdiag( µ \ λ ) ≥ P i maxdiag( µ \ λ i ).Conversely, start with a maximal diagonal of µ \ λ . Since µ is the complement of a rectangle,we can assume that the diagonal has its corner at one of the (at most) two corners of µ .(Since λ is a partition, if b ∈ ( µ \ λ ), then b ′ ∈ ( µ \ λ ) for any b ′ below or to the right of b in µ .In particular, since µ is the complement of a rectangle, this means that we can always eithermove the diagonal down or to the right unless its corner aligns with a corner of µ .) Since ν j meets the j th box of the diagonal (counting from the bottom-most box), then the translateof ν j to the bottom right meets the 1 st box of the diagonal, hence ν j is not contained in ρ ,so contributes 1 to S . Hence maxdiag( µ \ λ ) ≤ P i maxdiag( µ \ λ i ). (cid:3) Corollary 4.14.
Let λ ⊂ n × n be any partition with at least as many boxes above themain diagonal as below. Let λ c = ν + · · · + ν k be the hook decomposition with correspondingantichain { x i } , using the bijection from 4.11. Then M n sends the vertex of Γ correspondingto the antichain { x i } ki =1 to val co-rect ( p λ ) . Hence ∆ co-rect ∼ = Γ . Proof.
Applying 4.13 coordinatewise, and keeping the same notation, we get the equationval co-rect ( p λ ) = k X i =1 val co-rect ( p λ i )By 4.9 and 4.11, the right hand side is exactly M n acting on the antichain { x i } ki =1 . Henceevery valuation of a Pl¨ucker coordinate is obtained by M n acting on the vertices of Γ, and M n (Γ) = δ G . Since M n is unimodular, then vol( δ G ) = vol(Γ) = deg( X ) by 3.22. Since δ G ⊂ ∆ co-rect have the same volume and are both closed and convex, they must be equal.Hence M n (Γ) = ∆ co-rect . (cid:3) Example 4.15.
Adding the bottom two rows gives the top row, as desired. µ val co-rect ( p ) µ co-rect ( p ) µ co-rect ( p ) µ Future Work
The main theorem in [RW19] about the two polytopes ∆ G and Γ G is that they are equal. Inorder to obtain this theorem in our situation, we need an explicit cluster structure on the type B orthogonal Grassmannian OG( n, n + 1) to carry out the remainder of the proof strategy.In upcoming work [SW21], we will present a cluster structure for the type B orthogonalGrassmannians OG( n, n + 1). This will then allow us to use the cluster structure alongwith the result in this article to study cluster duality for X and X ∨ . References [FWZ16] Sergey Fomin, Lauren Williams, and Andrei Zelevinsky,
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LUSTER DUALITY FOR LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS 19
Department of Mathematics, Harvard University, Cambridge, MA, USA
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