Cyclic Sieving and Cluster Duality for Grassmannian

We introduce a decorated configuration space $\mathscr{C}onf_n^\times(a)$ with a potential function $\mathcal{W}$. We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of $(\mathscr{C}onf_n^\times(a), \mathcal{W})$ canonically parametrizes a linear basis of the homogenous coordinate ring of the Grassmannian ${\rm Gr}_a(n)$. We prove that $(\mathscr{C}onf_n^\times(a), \mathcal{W})$ is equivalent to the mirror Landau-Ginzburg model of Grassmannian considered by Marsh-Rietsch and Rietsch-Williams. As an application, we show a cyclic sieving phenomenon involving plane partitions under a sequence of piecewise-linear toggles.

For example, applying η to 3 2 2 3 1 0 ∈ P (2, 3, 6) yields Our first main result is as follows.  Theorem 1.5. The representation V cωa admits a natural basis Θ(a, b, c) which is • permuted under the action of C a , and • compatible with the weight decomposition of V cωa .
There is an equivarient bijection between Θ(a, b, c) under C a and the set P (a, b, c) under η.

Cluster Duality for Grassmannian
Cluster algebras are a class of commutative algebras introduced by Fomin and Zelevinsky [FZ02]. Their geometric counterparts form a family of log Calabi-Yau varieties called cluster varieties. A cluster ensemble 1 is a pair (A , X ) of cluster varieties associated to an equivalent class of skewsymmetric matrices (or quivers) introduced by Fock and Goncharov [FG09]. The variety A is equipped with an exceptional class {α} of coordinate charts called K 2 clusters. A rational function of A is called a universal Laurent polynomial if it can be expressed as a Laurent polynomial in every α. The ring up(A ) of universal Laurent polynomials of A coincides with the upper cluster algebra of [BFZ05]. The variety X is equipped with an exceptional class {χ} of coordinate charts called Poisson clusters. Let up(X ) be the ring of universal Laurent polynomials in every χ. The cluster modular group G is a discrete group acting on A and X that respects the cluster structures.
The Duality Conjecture of Fock and Goncharov [FG09] asserts that the ring up(A ) admits a natural basis G-equivariently paramatrized by the Z-tropical points of X , and vice versa. Cluster duality can be viewed as a manifestation of mirror symmetry between A and X . For example, the cluster duality for moduli spaces of local systems has been investigated in [GS15,GS18].
The present paper focuses on the cluster duality for Grassmannians. In detail, we introduce a pair of spaces, i.e., the decorated Grassmannian G r a (n) and the decorated configuration space C onf n (a), both of which are variants of the Grassmanian Gr a (n).
The decorated Grassmannian G r a (n) is essentially the affine cone over Gr a (n). Its coordinate ring O(G r a (n)) coincides with the homogeneous coordinate ring of Gr a (n). The G r a (n) admits a particular divisor, whose compliment is denoted by G r × a (n). A result of Scott [Sco06] implied that G r × a (n) is equipped with a cluster K 2 structure, and the coordinate ring O(G r × a (n)) coincides with an upper cluster algebra up(A ).
The decorated configuration space C onf n (a) parametrizes PGL(V )-orbits of n-many lines in an a-dimensional vector space V together with a linear isomorphism between every pair of cyclic neighboring lines. After imposing a consecutive general position condition, we obtain a smooth subvariety C onf × n (a). We prove that Theorem 1.6 (Theorem 2.36). The variety C onf × n (a) is equipped with a cluster Poisson structure. Its coordinate ring O(C onf × n (a)) coincides with the algebra up(X ) of universal Laurent polynomials.
Combining with work of Gross, Hacking, Keel, and Kontsevich [GHKK18], we prove the following Theorem in Section 5.1.
Theorem 1.7. The pair (G r × a (n), C onf × n (a)) admits a natural structure of cluster ensemble. The Duality Conjecture of Fock-Goncharov is true in this case, that is, the coordinate ring O (G r × a (n)) admits a cluster modular group equivariant parametrization by the Z-tropical set of C onf × n (a), and vice versa.
In Section 2, we introduce several natural functions and maps on G r × a (n) and C onf × n (a), which are summarized as follows:            decorated grassmannian G r × a (n); free recaling G m action; boundary divisor D = i D i ; action by a maximal torus T ⊂ GL n ; twisted cyclic rotation C a .
decorated configuration space C onf × n (a); twisted monodromy P ; potential function W = i ϑ i ; weight map M : C onf × n (a) → T ∨ ; cyclic rotation R.
We investigate the natural cluster correspondence between the ingredients in the above dictionary. In particular, the potential W exhibits an explicit cyclic symmetry. It is essentially equivalent to the potential of Grassmannian considered in [EHX97,MR13,RW17]. We identify W with the sum of theta functions associated to frozen vertices under the framework of [GHKK18]. As a consequence, we prove Theorem 1.5. Marsh and Rietsch constructed a B-model for the Grassmannian in [MR13], which is of the form Gr × a (n) × G m , W q . Rietsch and Williams proved in [RW17] that this B-model is an example of a cluster dual space of the Grassmannian. In contrast to their approach, our approach is more geometric and is purely motivated by the associated cluster structures. We include a section in the appendix describing the connection between our version of cluster duality and the version considered by Rietsch and Williams.

Decorated Grassmannian
Let V be an n-dimensional vector space and V * be its dual. The Grassmannian Gr a (V * ) parametrizes a-dimensional subspaces in V * . The decorated Grassmannian G r a (V * ) parametrizes pairs (W * , f * ), where W * is an a-dimensional subspace of V * and f * ∈ a W * is a nonzero a-form of W .
There is a free right G m -action on G r a (V * ) defined via rescaling the a-forms It induces a left G m -action on O (G r a (V * )). Irreducible representations of G m are 1-dimensional and are classified by integers. Denote by O (G r a (V * )) c the eigenspace in O (G r a (V * )) that is of weight c with respect to the G m -action. By the natural pairing between a V * and a V , every g ∈ a V gives rise to a regular function The ring O (G r a (V * )) is generated by ∆ g . Under the G m -action, we get Therefore we conclude the following statement.
Proposition 2.3. The map g → ∆ g is an isomorphism between a V and O (G r a (V * )) 1 , and The group GL(V ) acts on V as well as its dual space V * , and hence on the decorated Grassmannian G r a (V * ) and on the ring O (G r a (V * )). The representation V ωa of GL(V ) is by definition a V and therefore is isomorphic to O (G r a (V * )) 1 by Proposition 2.3. In general, (2.4) decomposes From now on, we fix a basis {e 1 , . . . , e n } of V and identify V with the vector space k n . We abbreviate Gr a (V * ) to Gr a (n) and G r a (V * ) to G r a (n). Every a-element subset I = {i 1 , . . . , i a } ∈ [n] a gives rise to a regular function ∆ I := ∆ e I where e I denotes the wedge product of e i 1 , . . . , e ia taken in ascending order (e.g., ∆ {5,7,4} := ∆ e 4 ∧e 5 ∧e 7 ). The functions ∆ I are called Plücker coordinates.
Let D i be the vanishing locus of the Plücker coordinate ∆ {i,i+1,...,i+a−1} . Let G r × a (n) denote the complement of D : . The image of D under the projection from G r a (n) to Gr a (n) is an anticanonical divisor of Gr a (n), of which the complement is denoted by Gr × a (n). Let Mat × a,n be the space of a × n matrices with column vectors v i such that every collection {v i , v i+1 , . . . , v i+a−1 } of a-many cyclic consecutive column vectors is linearly independent. The group SL a acts freely on Mat × a,n by matrix multiplication on the left. Lemma 2.6. The space G r × n (a) is canonically isomorphic to the quotient space SL a Mat × a,n . Remark 2.7. Under the above isomorphism, the coordinates ∆ I are identified with the minors of I-columns in an a × n matrix.
Proof. Let (W * , f * ) ∈ G r × n (a). Every W * ⊂ (k n ) * naturally induces a surjection π : k n → W . Let v i := π(e i ) be the image of the basis element e i under π. The coordinate ∆ i+1,...,i+a = 0 is equivalent to the linear independence of {v i+1 , . . . , v i+a }. Up to the action of SL a , there is a unique choice of linear isomorphisms from W to k a mapping the a-form f * to the standard a-form on k a . Hence we get a configuration in SL a Mat × a,n . It is easy to see that such a map is bijective.
Let T = (G m ) n be the maximal torus of GL n consisting of invertible diagonal matrices. It acts on the right of Mat × a,n by rescaling the column vectors v 1 , . . . , v n . Since G r × a (n) ∼ = SL a Mat × a,n , the T -action on Mat × a,n descends to a T -action on the decorated Grassmannian G r × a (n). Define the linear transformation C a on V such that It induces a twisted cyclic rotation on G r a (n) still denoted by C a .
To summarize, we obtain the following data            the decorated grassmannian G r × a (n); the free G m -action on G r × a (n) by rescaling the a-form; the boundary divisor D = i D i ; the T -action on G r × a (n); the twisted cyclic rotation C a on G r × a (n).

Decorated Configuration Space
Let W be a vector space of dimension a. The configuration space Conf n (W ) parametrizes the PGL(W )-orbits of n many lines in W , i.e., Conf n (W ) := PGL(W ) n PW .
As on Figure 1, a decorated configuration 2 is a PGL(W )-orbit of n lines in W together with linear isomorphisms φ i : l i → l i−1 for each pair of neighboring lines. The decorated configuration space is C onf n (W ) := PGL(W ) 1-dimensional subspaces l 1 , . . . , l n ⊂ W and linear isomorphisms φ i : l i → l i−1 .
Two vector spaces of the same dimension are isomorphic up to choices of bases. Since the action of PGL(W ) has been quotient out, the configuration spaces of n lines in vector spaces of the same dimension are canonically isomorphic and so are the decorated configuration spaces. Therefore we may abbreviate Conf n (W ) and C onf n (W ) to Conf n (a) and C onf n (a) respectively. Let Conf × n (a) be the subspace of Conf n (a) consisting of configurations [l 1 , . . . , l n ] such that every collection {l i+1 , . . . , l i+a } of a-many cyclic consecutive lines is linearly independent. The subspace C onf × n (a) of C onf n (a) is defined in the same way. Let [φ 1 , l 1 , . . . , φ n , l n ] ∈ C onf × n (a). Let us compose φ in anti-clockwise order as on Figure 1. Let (−1) a−1 P be the rescaling factor of the isomorphism φ i+1 • · · · • φ n • φ 1 • . . . φ i of l i . Note that P is independent of the initial index i chosen. We get a regular projection called twisted monodromy Pick a non-zero vector v i ∈ l i for each 1 ≤ i ≤ n . Let ϑ i be the scalar such that Note that ϑ i is independent of the choices of v i . We define the potential function on C onf × n (a) to be the regular function Again pick a non-zero vector v i ∈ l i for each 1 ≤ i ≤ n; fix a volume form ω of the vector space W . For each 1 ≤ k ≤ n, we define It is not hard to see that M k does not depend on the choices of ω and v i . Therefore we obtain a weight map where T ∨ ∼ = (G m ) n is the dual torus of the maximal torus T ⊂ GL n . Lastly, there is an order n biregular map (2.14) To summarize, we get the following data

Maps among the Decorated Spaces
Recall the n-dimensional vector space V with a basis {e 1 , . . . , e n }. Letl i be the line containing e i and letφ i :l i →l i−1 be the linear isomorphism such that Every W * ∈ Gr × a (n) ∼ = Gr × a (V * ) induces a projection π from V to W . The lines l i := π(l i ) in W satisfy the consecutive general position condition, andφ i descend to isomorphisms φ i : l i → l i+1 . Hence we obtain [φ 1 , l 1 , . . . , φ n , l n ] ∈ C onf × n (W ) It defines a natural injective map whose image consists of decorated configurations of twisted monodromy P = 1.
There is a surjective map G r × a (n) → Gr × a (n) defined by forgetting the a-form.
Definition 2.17. Denote by C onf × n (a) the space of SL a -orbits of where v i are vectors in k a satisfying consecutive general position condition, and each φ i is a linear isomorphism from the line spanning v i to the line spanning v i−1 .
By Lemma 2.6, the space G r × a (n) consist of configurations [v 1 , . . . , v n ] satisfying cyclic general position condition. Define the following map with the isomorphisms φ defined by φ 1 (v 1 ) := −v n and φ i (v i ) := v i−1 for the other i's. It is not hard to see that this map is injective.
There is a surjective map C onf × n (a) → C onf × n (a) by replacing each vector v i by its spanning line l i . There is also a surjective map C onf × n (a) → Conf × n (a) defined by forgetting the isomorphisms φ i .
Combining the aforementioned maps, we get the following commutative diagram.

Cluster Structures
The pair (G r × a (n), C onf × n (a)) admits a natural structure of cluster ensemble associated to minimal bipartite graphs of rank a on a disk with n marked points on the boundary. 3 In this section we focus on one particular minimal bipartite graph Γ a,n for each pair of parameters (a, n) as follows From a minimal bipartite graph Γ we obtain a quiver Q Γ by the following four-step procedure.
3 A rapid review on minimal bipartite graphs has been included in the appendix.
• Assign a vertex to each face of Γ.
• For each black vertex of Γ, draw a clockwise cycle of arrows as follows. • Freeze the vertices corresponding to boundary faces of Γ.
For example, the minimal bipartite graph Γ a,n gives rise to a quiver Q a,n as follows.
Here the vertex assigned to the top-left face is indexed by (0, 0). The other faces of Γ a,n form an a × b grid. Their corresponding vertices of Q a,n are indexed in the same way as matrix entries. The gray vertices are frozen. Denote by I the set of vertices of Q a,n and by I uf the set of unfrozen vertices of Q a,n . The exchange matrix ε of Q a,n is defined to be an I × I matrix with entries For simplicity, we will also use an integer i ∈ {1, . . . , n} to denote the frozen vertex corresponding to the boundary face lying between i and i + 1. In other words, Let (A a,n , X a,n ) be the cluster ensemble associated to Q a,n . See (5.31) for its rigid definition. Let {A i,j } (i,j)∈I be the K 2 cluster of A a,n associated to the quiver Q a,n and let {X i,j } (i,j)∈I be the Poisson cluster of X a,n associated to Q a,b . Abusing notation, we will frequently write f instead of (i, j) ∈ I with f being the face of Γ a,n corresponding to the vertex (i, j) of Q a,n .
Cluster K 2 structure on G r × a (n). Associate to each vertex (i, j) of Q a,n is an a-element set 4 Recall the Plücker coordinates ∆ I of G r × a (n). By defining we get a birational map Theorem 3] showed that the pull-back map ψ * gives an algebra isomorphism between O (G r a (n)) and the ordinary cluster algebra defined by the quiver Q a,n ; by allowing ourselves to invert the frozen variables we generalize his result to the following Theorem.
Theorem 2.23. The pull-back map ψ * is an algebra isomorphism between the upper cluster algebra up (A a,n ) := O (A a,n ) and O(G r × a (n)).
Cluster Poisson structure on Conf × n (a). Let Q uf a,n denote the full subquiver of Q a,n spanned by vertices in I uf . Let (A uf a,n , X uf a,n ) be the cluster varieties associated to Q uf a,n . There is a canonical regular map p : A a,n → X uf a,n defined on the cluster coordinate charts associated to Q a,n such that We define a rational map by first taking a lift from Conf × n (a) to G r × a (n), mapping over to A a,n via the birational equivalence (2.22), and then mapping it down to X uf a,n by the canonical p map. The map (2.25) is well-defined and does not depend on the lift, because ψ * (X g ) for g ∈ I uf is a ratio of Plücker coordintes with the same collection of indices (counted with multiplicity) in the numerator and in the denominator. It is known that (2.25) is a birational equivalence (see for example [Wen18, Cor. 5.1.9]). Lemma 2.26. The restricted exchange matrix ε| I×I uf of Q a,n is of full-rank.
Proof. It suffices to show that the pull-back map p * via (2.24) is injective, which is equivalent to proving that the corresponding map p : A a,n → X uf a,n is surjective. By definition, we have the following commutative diagram with the map p on the left as defined in (2.19). Since both of the horizontal maps ψ are birational and the map p on the left is surjective, we know that the map p on the right is dominant. Note that the restriction of the map p on the right to each seed torus is a dominant morphism induced by a linear map between their character lattices, which forces it to be surjective. Therefore the map p on the right is surjective.
Proposition 2.28. The birational equivalence ψ : Conf × n (a) X uf a,n induces an algebra isomorphism ψ * between up X uf a,n := O X uf a,n and O Conf × n (a) .
Proof. Since ψ map is birational, its pull-back map ψ * is an isomorphism between fields of rational functions on Conf × n (a) and X uf a,n . Let F be a rational function on on X uf a,n . It suffices to show that Let us make use of the commutative diagram (2.27) again. Note that both of vertical regular maps p are surjective. A rational function on a space downstairs is regular if and only if its pull-back is a regular function on the corresponding space upstairs. Therefore it suffices to show that which is a direct consequence of Theorem 2.23.
Cluster K 2 structure on C onf × n (a). The quiver Q a,n is obtained from Q a,n by adding, for each frozen vertex i, a new frozen vertex i ′ and a new arrow from i to i ′ . Denote the set of vertices of Q a,n byĨ and the exchange matrix of Q a,n byε. For instance, the quiver Q 3,6 is as follows: Let A a,n be the cluster K 2 variety associated to the quiver Q a,n . (2.30) We define a rational mapψ by identifying cluster variables of A a,n with functions on C onf × n (a) as follows A a,n is birational. Its pull-back mapψ * is an algebra isomorphism between the upper cluster algebra up A a,n := O A a,n and O C onf × n (a) .
Proof. Let H = (G m ) n be the split algebraic torus with coordinates (A 1 ′ , . . . , A n ′ ). Note that there is no arrow between the vertices i ′ and the unfrozen vertices in Q a,n . Therefore the variables A ′ i will not affect the cluster mutations. Hence we get Every configuration in C onf × n (a) consists of two pieces of data: a SL a -orbit of vectors [v 1 , . . . , v n ] satisfying the cyclic general position condition, and the isomorphisms φ i between the lines l i spanned by the vectors v i . By Lemma 2.6, the data [v 1 , . . . , v n ] is captured by a point in G r × a (n). The isomorphisms φ i is captured by the scaling factors λ i , which by (2.31) is computed from A i ′ together with the frozen variables of the corresponding point in G r × a (n). Therefore it is easy to see that After the above identifications, the mapψ = ψ × Id. The Corollary follows from Theorem 2.23.
Cluster Poisson structure on C onf × n (a). Analogous to the map (2.24), there is a canonical map p : A a,n → X a,n defined on the cluster coordinate charts associated to Q a,n by Let [φ 1 , l 1 , . . . , φ n , l n ] ∈ C onf × n (a). Let us lift it to a configuration in C onf × n (a) via picking a nonzero vector v i ∈ l i for each i. Composing with the map p •ψ : C onf × n (a) A a,n −→X a,n , we get a rational map The map ψ does not depend on the choices of v i ∈ l i . Indeed, if the vertex (i, j) is unfrozen, then X i,j coincides with the cluster Poisson coordinates X i,j on X uf a,n . For a frozen vertex i, one gets from which one can verify that X i are independent of the choices of v i ∈ l i . Proposition 2.35. The map ψ : C onf × n (a) X a,n is a birational equivalence.
Proof. Let U be an open subset of C onf × n (a) consisting of [φ 1 , l 1 , . . . , φ n , l n ] such that • every collection {l i 1 , . . . , l ia } of a-many lines is linearly independent.
Note that the Plücker coordinates on any lift of U are nonzero. Let T Qa,n ⊂ X a,n be the algebraic torus corresponding to the cluster chart associated to Q a,n . By definition, the map ψ restricted on U is a regular map to T Qa,n . Let (X f ) f ∈I be a generic point in T Qa,n . It suffices to show that it has a unique pre-image in U .
Recall that the map ψ : Conf × n (a) X uf a,n is birational. After imposing the generic condition, by using the unfrozen part (X f ) f ∈I uf , one can uniquely reconstruct a configuration of lines [l 1 , . . . , l n ] satisfying the • condition. Take a representative v i ∈ l i for each i. One can use the frozen part (X i ) n i=1 to uniquely reconstruct the isomorphisms φ i : l i → l i−1 , since the frozen variables X i contains the information of λ i−a in (2.30). It is easy to see that the isomorphisms φ i are independent of v i chosen. Therefore we obtain a unique configuration in U .
Theorem 2.36. The birational equivalence ψ : C onf × n (a) X a,n induces an algebra isomorphism between up (X a,n ) := O (X a,n ) and O (C onf × n (a)).
Proof. Note that we have the following commutative diagram The Theorem follows by running the same argument as in the proof of Proposition 2.28.

Cluster Nature of Decorated Configuration Space
In Section 2.4, we prove that O (G r × a (n)) ∼ = up (A a,n ) and O (C onf × n (a)) ∼ = up (X a,n ) as algebras. From now on, we will identify these algebras of regular functions and their corresponding field of rational functions, i.e., we will think of {A f } as regular functions on G r × a (n) and think of {X f } as rational functions on C onf × n (a). In particular, the cluster {X f } gives rise to a canonical log Poisson structure on C onf × n (a) such that This section is devoted to studying the cluster natural of three ingredients of C onf × n (a), i.e., the twisted monodromy P , the cyclic rotation R, and the potential function W, in terms of the cluster {X f } f ∈I associated to the quiver Q a,n .

Twisted Monodromy and Casimir
Proposition 3.2. The twisted monodromy The function P is a Casimir element with respect to the Poisson bracket (3.1), that is, {P, F } = 0 for any rational function F on C onf × n (a).
Note that Q a,n is made of cycles. Therefore for every f ∈ I, one has g∈I ε gf = 0. (3.4) Hence the only factors contributing to the product (3.3) are from the extra frozen vertices that expand Q a,n to Q a,n . Therefore The last equality follows by comparing (2.30) with the definition of P in (2.17). The statement that P is Casimir is a direct consequence of (3.4) and (3.1).

Cyclic Rotation and Cluster Transformation
Recall the twisted cyclic rotation C a on G r × a (n). Gekhtman, Shapiro, and Vainshtein proved that that C a is a cluster K 2 automorphism that can be realized by a mutation sequence ρ (see [GSV10,Page 90]). In this section, we briefly recall the definition of ρ. We show that the cyclic rotation R on C onf × n (a) is a cluster Poisson automorphism realized by the same mutation sequence ρ. The vertices of the quiver Q a,n are indexed by (i, j) ∈ I. Let ρ be a mutation sequence hitting in the unfrozen part of Q a,n along every column from bottom to top, starting at the leftmost unfrozen column and go all the way to the rightmost unfrozen column, that is, Equivalently, the sequence ρ may be realized by a sequence of 2-by-2 moves 5 on the minimal bipartite graph Γ a,n . Its resulting minimal bipartite graph Γ ′ a,n is identical to Γ a,n with the non-boundary faces remain in the same places and with all the boundary faces rotated to the neighboring one in the clockwise direction.
(3.6) One advantage of using minimal bipartite graph is that one may make use of the zig-zag strands to assign an a-element subset of {1, . . . , n} to each face of the graph (see Definition 5.36). For example, the a-element subsets assigned to faces of Γ a,n in (2.20) arise precisely in this way, which in turn determine Plücker coordinates to these faces. Moreover, it is not hard to see that a 2by-2 move combined with the cluster K 2 mutation formula yields precisely a Plücker relation (see (5.37) for more details); hence we can conclude that the Plücker coordinates on any bipartite graph obtained from Γ a,n via a sequence of 2-by-2 moves can be computed by using zig-zag strands as well.
Let {A ′ f } be the K 2 cluster associated to Γ ′ a,n after applying the mutation sequence ρ. According the above discussion, the cluster {A ′ f } is defined by the a-element sets (2.20) assigned to faces of the minimal bipartitie graphs Γ ′ a,n . Using the zig-zag strands on Γ ′ a,n , we find that which yields A ′ i,j = ∆ I ′ (i,j) . As a consequence, we show that the twisted rotation C a is realized by the mutation sequence ρ.
Let us apply the mutation sequence ρ to the extended quiver Q a,n . It is easy to see that the obtained quiver ρ Q a,n is the same as Q a,n up to rotations of frozen vertices. For example, if we start with Q 3,6 as in (2.29), then ρ Q 3,6 is as follows.
3 ′ Letε ′ f g be the exchange matrix of the quiver ρ Q a,n . The cluster K 2 frozen variables remain intact under mutations. Therefore i,j } be the Poisson cluster associated to ρQ a,n . We get As a consequence, the cyclic rotation R on C onf × n (a) is a cluster Poisson automorphism realized by the mutation sequence ρ.

Potential Function
Proposition 3.8. In terms of the Poisson cluster {X i,j } associated to Q a,n , the theta functions ϑ i in (2.10) are ϑ n = X 0,0 , ϑ a = X a,b , (3.9)

11)
Remark 3.12. For (3.10), the terms in ϑ i are in bijection with rectangles of all possible lengths across the ith row of the quiver Q a,n that ends at the vertex (i, b). For (3.11), the terms in ϑ i are in bijection with rectangles of all possible heights across the (n − i)th column of Q an that rises from the vertex (a, n − i). For instance, the formulas of ϑ 2 and ϑ 5 with a = 3 and n = 7 are as follows: (1, 1) Therefore the potential function W in (2.11) can be expressed as (3.14) Similarly we prove that ϑ a = X a,b . Now let us prove (3.11). The proof of (3.10) goes along the same line. For a < i < n, let k = n − i. Then 0 < k < b = n − a. For 1 ≤ j ≤ a, define the a-element set Recall the a-element set I(j, k). One has the Plücker relation Dividing by ∆ I(j,k−1) ∆ I(j+1,k+1) on both sides, we get . (3.16) Let us set Y j,k := 1 + X j,k (1 + X j−1,k (. . . (1 + X 1,k ) . . . )) Let us fix k. We prove by induction on j that Y j,k = ∆ I(j,k) ∆ J(j+1,k) ∆ I(j,k−1) ∆ I(j+1,k+1) . (3.17) Note that I(1, k + 1) = J(1, k). Therefore for j = 1, we have .
If 1 < j < a and (3.17) is true for j − 1, then it is true for j because k+1) .
where the last equality is similar to (3.14). On the other hand, we have X a,k X a−1,k . . . X j,k .

Tropicalization and Plane Partitions
In this section, we relate C onf × n (a) to plane partitions via tropicalization. See [FG09] for more details on tropicalization of positive spaces.

Tropicalization
Let X be a cluster variety (either K 2 type or Poisson type). A positive rational function on X is a nonzero function that can be expressed as a ratio of two polynomials with non-negative integer coefficients in one (and hence all) cluster(s) of X . The set P (X ) of positive rational functions is a semifield, i.e., a set closed under addition, multiplication, and division. The pair (X , P (X )) is called a positive space.
Let Z t = (Z, min, +) be the semifield of tropical integers. We define the tropicalization of (X , P (X )) to be the set X (Z t ) := Hom semifield P (X ), Z t .
Let (X , P (X )) and (Y , P (Y )) be a pair of cluster varieties. A rational map f : X Y is positive if the pull back f * P (Y ) is contained in P (X ). Every positive rational map f admits a tropicalization f t : X Z t → Y Z t defined by precomposing with the pull-back map f * . In particular cluster mutations are subtraction-free. Therefore every cluster automorphism is positive and can be tropicalized. It induces a natural action of the cluster modular group G on X (Z t ).
The m-tuple (x 1 , . . . , x m ) is called the tropical coordinates on X (Z t ) in terms of the cluster χ. Tautologically, every positive function f ∈ P (X ) gives rise to a Z-valued function which can be expressed as a piecewise linear function in terms of the tropical coordinates (x 1 , . . . , x m ) by the following procedure: 1. Change every addition into taking minimum.
2. Change every multiplication into addition and every division into subtraction.
3. Change the constants that are coefficients of the original expression into 0.

Change the variables
For example, the tropicalization of the positive rational function f = Let us tropicalize the cluster variety C onf × n (a). Recall that the potential W and the twisted monodromy P are positive functions on C onf × n (a). Define the subset The rotation R : C onf × n (a) → C onf × n (a) is a cluster automorphism and therefore can be tropicalized into R t : C onf × n (a)(Z t ) → C onf × n (a)(Z t ). By definition W and P are invariant under R. Hence R t acts on (4.1). The remainder of the Section is devoted to proving the following Theorem.

Gelfand-Zetlin Coordinates
Recall the cluster coordinates {X i,j } on C onf × n (a) associated to Q a,n . To each vertex (i, j) of Q a,n is associated With the convention that L i,j = 1 for i > a or j > b, we have for (i, j) = (0, 0).
Therefore {L i,j } forms a coordinate system on C onf × n (a), called Gelfand-Zetlin coordinate system. The following Lemma is a direct consequence of Proposition 3.2 and (3.13).
Lemma 4.3. The twisted monodromy P = L 0,0 . The potential Lemma 4.5. Recall λ k in (2.30) and A i,j in (2.21). Set A i,0 = A 0,j = A 0,0 . Then where the indices of λ k are defined modulo n.
Proof. We prove (4.6) by a double induction on the indices (i, j) in the descending order. It is certainly true for (a, b) since Take (a, j) with j < b. The cluster coordinate X a,j is associated to a boundary face. By definition If (4.6) is true for (a, j + 1), then Similarly, for (i, b) with i < a, by induction we get Take (i, j) with i < a and j < b. If (4.6) is true for (i + 1, j), (i, j + 1) and (i + 1, j + 1), then Recall the rotation R on C onf × n (a). Set L i,0 = L 0,j = P and L a+1,j = L i,n−a+1 = 1. Let Note that R * P = P . Therefore L ′ i,0 = L ′ 0,j = P and L ′ a+1,j = L ′ i,n−a+1 = 1.
Lemma 4.8. We have It is easy to check that the coordinates satisfy the Plücker relations Let us take the ratio of them and multiply by b−j k=i−a λ k λ k−1 on both sides. The right hand side is The left hand side becomes , which is precisely the right hand side.
Lemma 4.8 allows us to compute L ′ i,j recursively. Let Π = (Π ij ) be a matrix such that its rows are numbered 0, . . . , a + 1 and its columns are numbered 0, . . . , b + 1. For 1 ≤ i ≤ a and 1 ≤ j ≤ b, we define a birational toggling action τ i,j sending Π to the matrix τ i,j Π such that Recall the toggling sequence η in (1.2).
Lemma 4.9. Let us apply the toggling sequence η to the initial (a + 2) × (b + 2) matrix Then the final matrix applying η becomes In other words, the pull-back of the Gelfand-Zetlin coordinates via the rotation R is given by η.
Proof. Note that the sequence η toggles at each (internal) entry exactly once. It suffices to prove that the step τ i,j within the sequence η changes L i,j to L ′ i,j . This follows from the fact that the toggling sequence η goes from bottom to top within each column and from left to right through all columns. So when we toggle at the entry (i, j), the matrix entry to the left and and the matrix entry below have already been changed to L ′ i,j−1 and L ′ i+1,j respectively. The rest of the proof is just a straightforward comparison between the toggling formula and Lemma 4.8.

Proof of Theorem 4.2
The Gelfand-Zetlin coordinates L i,j are clearly positive functions on C onf × n (a). Denote by l i,j the tropicalization of L i,j . By Lemma 4.3, we get P t = l 0,0 , (4.10) The condition that W t ≥ 0 is equivalent to the conditions After imposing the conditions that P t = l 0,0 = c, we obtain a natural bijection between (4.1) and P (a, b, c) by identifying l i,j with entries π i,j in plane partitions. It remains to show that this bijection is equivariant with respect to the actions of R t and η. Let us denote l ′ i,j := (L ′ i,j ) t . By Lemma 4.9, with the convention that l 0,j = l i,0 = l 0,0 = c and l a+1,j = l i,b+1 = 0, the l ′ i,j can be computed recursively: l ′ 0,0 = l 0,0 = c and for (i, j) = (0, 0), (4.12) In the process of computing l ′ i,j , the coordinates below and to the left of l i,j has been toggled in the way the mutation sequence is constructed. Therefore the formula (4.12) recovers the toggling formula (1.2). It concludes the proof of Theorem 4.2.

Duality Conjecture and Canonical Basis
Let A |Q| , X |Q| be the cluster ensemble associated to a quiver Q. Recall the algebras up A |Q| and up X |Q| and the cluster modular group G |Q| .
Duality Conjecture of Fock-Goncharov asserts that Conjecture 5.1 has been proved in [GHKK18] under two combinatorial assumptions.
• The non-frozen part of the quiver Q exists a maximal green (or reddening) sequence 6 .
• The exchange matrix ε = (ε ij ) of Q, with i running through the non-frozen vertices and j running through all the vertices, is of full rank.
Remark 5.3. Let us assume that the quiver Q satisfy the above combinatorial conditions. Let us denote the bases of up(A |Q| ) and up(X |Q| ) as follows The bases elements θ q and ϑ p satisfy many remarkable properties. One of them is that every θ q ∈ up A |Q| in terms of the K 2 cluster {A j } associated to Q is expressed as where (x j ) is the tropical coordinates of q ∈ X (Z t ) in terms of the Poisson cluster associated to Q, and F is a polynomial with constant term 1 and variables of the form j A ε kj j as k ranges through all unfrozen vertices. The elements ϑ p admit similar formulas.
One may notice that, on the one hand, we define p * (X i ) = j A ε ji j in the definition of the p map, and on the other hand, the above polynomial F depends on j A ε ij j = p * X −1 i . The reason this happens is that the cluster Poisson variables used in this paper are inverses of those used by Gross, Hacking, Keel, and Kontsevich in [GHKK18]. Such a switch frees us from considering tropicalization with taking maximum.
Proof of Theorem 1.7. For the quiver Q a,n , a maximal green sequence was found by Marsh and Scott in [MS16], and the existence of a cluster Donaldson-Thomas transformation (which is equivalent to a reddening sequence) was proved by Weng in [Wen16]. By Lemma 2.26, the second combinatorial condition holds. Hence Conjecture 5.1 holds for the cluster ensemble (A a,n , X a,n ). By Theorems 2.23 and 2.36 we know that up (A a,n ) ∼ = O (G r × a (n)) and up (X a,n ) ∼ = O (C onf × n (a)), which concludes the proof Theorem 1.7.

Partial Compactification, Optimized Quiver, and Potential Function
Let Q = (I uf ⊂ I, ε) be a quiver satifying the combinatorial conditions in Theorem 5.2. Let I 0 := I − I uf be the set of frozen vertices. For i ∈ I 0 , let D i denote the (irreducible) boundary divisor of A |Q| defined by setting A i = 0. Let us glue A |Q| with these boundary divisors, obtaining the partial compactified space This section is devoted to studying the ring O A |Q| of regular functions on A |Q| .
Let f ∈ up A |Q| . Denote by ord D i (f ) the order of f along the boundary divisor D i . Note that f can be extended to a regular function on D i if and only if ord D i (f ) ≥ 0. Therefore Recall the canonical basis Θ A of O(A |Q| ). Consider the intersection (5.5) Conjecture 9.8 of [GHKK18] implies that the intersection Θ A descends to a linear basis of O A |Q| . The paper loc.cit. provides a sufficient condition under which the aforementioned conjecture holds.
Definition 5.6. Let i ∈ I 0 . If ε ki ≥ 0 for all unfrozen vertices k, then we say the quiver Q is optimized for i. If there exists a mutation sequence τ such that the mutated quiver τ Q is optimized for i, then we say that i admits an optimized quiver in the equivalence class |Q|.
Remark 5.7. Because of different conventions used in this paper and in [GHKK18], here we say Q is optimized for i if all arrows between i and unfrozen vertices point towards the unfrozen ones.
Proposition 5.8. If every frozen vertex i of Q admits an optmized quiver in |Q|, then the set Θ A forms a linear basis of O A |Q| .
Proof. The linear independence of Θ A is clear. Suppose that By definition, ord D i (f ) ≥ 0 for every frozen i. By [GHKK18, Proposition 9.7], if i admits an optmized quiver in |Q|, then ord D i (θ q ) ≥ 0 for all q with α q = 0. Therefore whenever the coefficient α q is nonzero, the function θ q ∈ Θ A . In other words, the set Θ A spans O A |Q| . Now it is natural to address the following question: • For which q ∈ X |Q| (Z t ) we have θ q ∈ Θ A ?
A criterion for recognizing such q's was suggested in [GS15, §12.2], and was proved in [GHKK18]. Let i ∈ I 0 . Let p i ∈ A |Q| (Z t ) be the tropical point such that its tropical coordinates A t j (p i ) = δ ij , where δ ij is the Kronecker delta symbol. The existence and uniqueness of p i is a direct consequence of the definition of cluster K 2 mutations. Let ϑ p i ∈ Θ X be the theta function parametrized by p i .
Proposition 5.10. Assume that every frozen vertex of Q admits an optmized quiver in |Q|. Let q ∈ X |Q| (Z t ). Then θ q ∈ Θ A if and only if W t (q) ≥ 0.
Proof. By the definition of tropicalization, we have Therefore W t (q) ≥ 0 if and only if every ϑ t p i (q) ≥ 0. It suffices to show that if i ∈ I 0 admits an optimized quiver, then ord D i (θ q ) = ϑ t p i (q). Without loss of generality, let us assume that Q is optimized for i, i.e., ε ki ≥ 0 for all unfrozen vertices k. Let {A j } be the cluster of A |Q| associated to Q. Note that θ q is of the form (5.4). Therefore ord D i (θ q ) = x i := X t i (q), where X i is the cluster variable of X associated to the vertex i in Q. By [GHKK18, Lemma 9.3], if Q is optimized for i, then ϑ p i = X i .
Let us apply the above results to the cases of Grassmannians. It boils down to finding optimized quivers for frozen vertices in the quiver Q a,n . As observed by L. Williams and appeared in [GHKK18,Proposition 9.4], the quiver Q a,n is optimized for the vertices (0, 0) and (a, b); since the mutation sequence ρ in (3.5) rotates the frozen vertices of Q a,n clockwise to their neighbors, by applying ρ repeatedly, each frozen vertex i has a chance to be at the position of (0, 0) and therefore admits an optimized quiver. Indeed, the quiver ρ n−i Q a,n is optimized for the frozen vertex i.
The next Proposition shows that in the Grassmannian case, the potential function W = i ϑ i in (2.11) coincides with the function W in (5.9).
Proposition 5.11. Under the algebra isomorphism up (X a,n ) ∼ = O (C onf × n (a)), the theta function ϑ p i is identified with the function ϑ i defined in (2.10).
For the other frozen vertices, let us apply the rotation mutation sequence ρ. Then By Theorem 2.23, or more precisely by the original version [Sco06, Theorem 3], the coordinate ring O(G r a (n)) is isomorphic to O(A a,n ). Combining Propositions 5.8, 5.10, and 5.11, we get Theorem 5.12. Under the isomorphisms O(G r a (n)) ∼ = O(A a,n ) and O (C onf × n (a)) ∼ = up (X a,n ), the coordinate ring O (G r a (n)) admits a natural basis

G m -action
Recall the G m -action on G r a (n) in (2.1). Let us restrict the G m -action to the open subset G r × a (n). It induces a G m -action on O (G r × a (n)) extending the one on O (G r a (n)). Recall the twisted monodromy function P on C onf × n (a).
Proposition 5.13. Let q ∈ C onf × n (a) Z t . Its corresponding theta function θ q ∈ O (G r × a (n)) is an eigenvector of the G m -action with weight P t (q): Proof. Let {A i,j } be the K 2 cluster of G r × a (n) associated to Q a,n . By (5.4), the function θ q can be expressed as a Laurent polynomial where (x i,j ) is the tropical coordinates of q with respect to the quiver Q a,n , and F is a polynomial with constant term 1 and variables of the form g A ε f g g for f ∈ I uf . By definition, every A f is a Plücker coordinate and therefore is of weight 1 with respect to the G m -action. Since f ε f g = 0 for all g ∈ I uf by construction, the whole factor F is invariant under the G m -action. It implies that θ q is eigenvector of weight (i,j) x i,j . By Proposition 3.2, we have P t (q) = (i,j) x i,j , which concludes the proof.
As a direct consequence, we get Corollary 5.15. The representation O (G r a (n)) c = V cωa has a canonical basis Combining with Theorem 4.2 we deduce that the basis Θ(a, b, c) is in natural bijection with the plane partitions P (a, b, c).

Torus Action and Weight Decomposition
By (2.6), G r × a (n) ∼ = SL a Mat × a,n . The group GL n acts on the right of Mat × a,n by matrix multiplication. The maximal torus T = (G m ) n ⊂ GL n of diagonal matrices acts by rescaling the column vectors v i of the matrices in Mat × a,n (v 1 , . . . , v n ) . (t 1 , . . . , t n ) := (t 1 v 1 , t 2 v 2 , . . . , t n v n ) .
Its induced left (G m ) n -action on O (G r × a (n)) gives rise to a weight decomposition where µ = (µ 1 , . . . , µ n ) ∈ Z n and O(µ) consists of the functions F such that In particular, if we restrict to the representation V cωa = O (G r a (n)) c , then we get the weight decomposition In this section, we show that the theta basis Θ(a, b, c) is compatible with the weight decomposition of the representation V cωa . Recall the following dual torus projection defined in (2.13) Let us tropicalize the map M , obtaining Proposition 5.17. Let q ∈ C onf × n (a). The theta function θ q is an eigenvector of the T -action on O (G r × a (n)) with weight M t (q), i.e., The proof of Proposition 5.17 will require a little preparation. Recall the a-element set I(i, j) assigned to each vertex (i, j) of the quiver Q a,n : For k ∈ {1, . . . n}, let F k denote the collection of vertices (i, j) in Q a,n such that k ∈ I(i, j). By the definition of I(i, j), it is easy to check that the sets F k are of two patterns. When 1 ≤ k ≤ b, the vertices in F k are enclosed in a stair-shape diagram, such that the difference between the consecutive steps is 1, until its height becomes 0 or until it touches the rightmost column.
Lemma 5.19. Recall the clusters {X i,j } of C onf × n (a) associated to Q a,n in (2.33). The function Proof. Recall the definition of M k in (2.12). We prove the Lemma for 1 ≤ k ≤ min{a, b}. The proof for the other cases goes along the same line. Recall the Gelfand-Zetlin coordinates of C onf × n (a). We get By Lemma 4.5, we get Proof of Proposition 5.17.
The proof makes use of the expression (5.14) of θ q again. For a nonfrozen vertex f , the product g A is independent of the rescaling of the column vectors v i due to the well-defined-ness of the unfrozen variable X f . Therefore the polynomial F in (5.14) is invariant under the rescaling T -action. For the Plücker coordinates A i,j , note that it is affected by the t k component of T if and only if (i, j) ∈ F k . Therefore By Lemma 5.19, we get (i,j)∈F k x i,j = M t k (q).
Combining Corollary 5.15 with Proposition 5.17, we get Corollary 5.22. The weight space V cωa (µ) has a canonical basis Note that Θ(a, b, c) is parametrized by the set P (a, b, c) of plane partitions. In the rest of this section, we present a concrete decomposition of P (a, b, c) compatible with the above decompostion of Θ(a, b, c).

0
Consider the sums of l i,j along each diagonal: . . , δ n = al 0,0 Following the same argument as in (5.21), we get Comparing (5.26) with (5.24), the Proposition follows.

Proofs of Main Theorems
Proof of Theorem 1.5. Corollaries 5.15 and 5.22 show that V cωa admits a natural basis Θ(a, b, c) which is in bijection with the set of plane partitions P (a, b, c) and is compatible with the weight decomposition. It remains to show that the bijection is equivariant under C a -and the toggling ηactions.
As discussed in Section 3.2, the action C a on G r × a (n) and the action R on C onf × n (a) are cluster automorphisms realized by the same mutation sequence ρ. In other words, the actions C a and R correspond to the same element in the cluster modular group G |Qa,n| . By Conjecture 5.1 (proved in [GHKK18, Proposition 3.6]), the bijection between the theta basis of O(G r × a (n)) and C onf n × (a)(Z t ) is equivarient under C a -and the tropical R t -actions. By Theorem 4.2, the R t action coincides with the toggling η action. Theorem 1.5 is proved.
Proof of Theorem 1.3. Let D(p, q) be the diagonal matrix diag pq n−1 , . . . , pq, p . Let V λ be a representation of GL n . The character formula asserts that It has n-distinct roots ζ − a−1 2 , ζ − a−1 2 ζ, . . . ζ − a−1 2 ζ n−1 over C. Therefore C a is conjugate to D ζ − a−1 2 , ζ and C k a is conjugate to D ζ − (a−1)k 2 , ζ k . By Theorem 1.5, the number of plane partitions fixed by η k is equal to the number of basis vectors in θ Φ(π) π∈P (a,b,c) fixed by C k a . Therefore By Proposition 5.25, we have dimV cωa (µ) = #P (a, b, c)(µ).
Let π ∈ P (a, b, c). We see from (5.24) that Quiver mutations. Let Q = I uf ⊂ I, ε be a quiver without loops or 2-cycles: I is the set of vertices, I uf is the set of unfrozen vertices, and ε is an I × I skew-symmetric matrix called the exchange matrix encoding the data of number of arrows between vertices For the rest of this appendix we let m = #I and l = #I uf . Given a quiver Q, the quiver mutation µ k at a non-frozen vertex k ∈ I uf creates a new quiver µ k (Q) by the following procedure In general the map p is neither injective nor surjective.
The Cluster modular group. To each torus T t,α in (5.27) is associated a differential form This Ω t is compatible with the transition (5.29) and therefore can be lifted to a global differential form Ω on A |Q| . A cluster automorphism τ of A |Q| is a biregular isomorphism of A |Q| such that • It preserves the differential form: τ * Ω = Ω.
Locally, τ can be realized by a sequence of mutations that sends a quiver Q to itself up to permutations of vertices. The cluster modular group G A ,|Q| consists of cluster automorphisms of A |Q| .
To each torus T t,χ is associated a bi-vector which can be lifted to a global bi-vector B on X |Q| . A cluster automorphism of X |Q| is a biregular isomorphism of X |Q| that preserves the bi-vector B and permute its Poisson clusters. Denote by G X ,|Q| the group of cluster automorphisms of X |Q| .
From the tropical cluster duality proved by Nakanishi and Zelevinsky [NZ12] one can deduce that the cluster modular group G X ,|Q| = G A ,|Q| . Hence we will drop the subscripts A and X in the notation and denote it by G |Q| .
Quiver extensions. Let Q = (I uf ⊂ I, ε) be a quiver. Let Q = I uf ⊂Ĩ,ε be a quiver obtained from Q by adding frozen vertices labelled by I ′ = {1 ′ , . . . , f ′ } and arrows such that Q contains Q as a full subquiver. In other words,Ĩ = I ⊔ I ′ andε contains ε as a submatrix. Let (A | Q| , X | Q| ) be the cluster ensemble associated to Q.
Following [She14, (3.5)], we define the following map The mapp is a natural map as (5.32). The map j is a surjective map such that j * X i,t = X i,t for all i ∈ I. The map k is the composition ofp and j. It is surjective if and only if the submatrix ε|Ĩ ×I of the exchange matrixε is of full rank. In this case we get a natural injection k * : up X |Q| −→ up(A | Q| ).
The following easy Lemma generalizes one key part of the proof of Theorem 2.36. • Type II.

←→
A type II 2-by-2 move changes neither the quiver nor the dominating sets of faces.
A result of Thurston [Thu17, Theorem 6] can be restated in the minimal bipartite graph language as saying that any two rank a minimal bipartite graphs on D n can be transformed into one another via a sequence of 2-by-2 moves of the above two types. In conclusion, all the K 2 cluster structures on G r × a (n) defined by rank a minimal bipartite graphs on D n are equivalent.

C. Example of a Minimal Bipartite Graph Transforming under ρ.
Below is an example showing how a minimal bipartite graph transforms under the rotation mutation sequence ρ. This example can be easily generalized to other standard minimal bipartite graph Γ a,n with arbitrary parameters (a, n). The example we choose to do is with parameters (a = 3, n = 7). Here P is the twisted monodromy, and v, φ(v), . . . , φ n−1 (v) denote the matrix representative of the Grassmannian point (we abbreviate the subscripts of φ to simplify the notation). It is easy to see that Φ is a biregular isomorphism.
Therefore we can conclude that our potential function W is precisely Hence the map Φ : (C onf × n (a), W) → Gr × a (n) × G m , W q is an isomorphism between our version of cluster dual space and Rietsch-Williams' cluster dual space.