The Kashaev Equation and Related Recurrences

The hexahedron recurrence was introduced by R. Kenyon and R. Pemantle in the study of the double-dimer model in statistical mechanics. It describes a relationship among certain minors of a square matrix. This recurrence is closely related to the Kashaev equation, which has its roots in the Ising model and in the study of relations among principal minors of a symmetric matrix. Certain solutions of the hexahedron recurrence restrict to solutions of the Kashaev equation. We characterize the solutions of the Kashaev equation that can be obtained by such a restriction. This characterization leads to new results about principal minors of symmetric matrices. We describe and study other recurrences whose behavior is similar to that of the Kashaev equation and hexahedron recurrence. These include equations that appear in the study of s-holomorphicity, as well as other recurrences which, like the hexahedron recurrence, can be related to cluster algebras.


Introduction
The Kashaev equation is a polynomial equation involving 8 numbers indexed by the vertices of a cube; this equation is invariant under the symmetries of the cube. It originally appeared in the study of the star-triangle move in the Ising model [3]; it also arises as a relation among principal minors of a symmetric matrix [4].
We say that a C-valued array indexed by Z 3 satisfies the Kashaev equation if for every unit cube C in Z 3 , the 8 numbers indexed by the vertices of C satisfy the Kashaev equation. The Kashaev equation is quadratic in each of its variables, so we in general have two choices in solving for one value in terms of the remaining seven. If these seven values are all positive, then both solutions are real, and the larger solution is positive. This leads to a recurrence on positive-valued arrays on Z 3 that we call the positive Kashaev recurrence; it expresses the value at the "top vertex" of each unit cube in terms of the 7 values underneath it.
Our first observation is that solutions of this positive recurrence satisfy an additional algebraic constraint not implied by the Kashaev equation alone. This constraint involves the values indexed by the 27 vertices of a 2 × 2 × 2 cube in Z 3 . A solution of the Kashaev equation that satisfies this constraint is called coherent.
The hexahedron recurrence is a birational recurrence satisfied by an array indexed by the vertices and (centers of) two-dimensional faces of the standard tiling of R 3 with unit cubes. This recurrence was introduced by Kenyon and Pemantle [5] in the context of statistical mechanics as a way to count "taut double-dimer configurations" of certain graphs. It also describes a relationship among principal and "almost principal" minors of a square matrix [4].
A key observation of Kenyon and Pemantle [5] was that restricting an array satisfying the hexahedron recurrence to the vertices of the standard tiling of R 3 with cubes (i.e., to Z 3 ) yields an array satisfying the Kashaev equation. However, not all solutions of the Kashaev equation can be obtained this way. Our main result (Theorem 1.22) states that, modulo some natural technical conditions, a solution of the Kashaev equation can be extended to a solution of the hexahedron recurrence if and only if it is coherent.
We then generalize this result to a certain subclass of 3-dimensional cubical complexes. We show that a suitable generalization of Theorem 1.22 holds for these complexes (Proposition 7.3 and Theorem 7.10), but that the corresponding statement can be false for cubical complexes outside this subclass (Theorem 7.11).
We use this generalization to study the relations among principal minors of symmetric matrices. Given a symmetric matrix M, we associate principal minors of M to the vertices of a cubical complex, so that the resulting array is a coherent solution of the Kashaev equation. Conversely, for any generic coherent solution of the Kashaev equation, there exists a symmetric matrix whose principal minors appear as the entries of the given array. This leads to Theorem 3.26, which provides a simple test for whether a 2 n -tuple of complex numbers (satisfying certain genericity conditions) arises as a collection of principal minors of an n × n symmetric matrix. An alternative criterion was given by L. Oeding [10].
Going in another direction, we develop an axiomatic setup for pairs of recurrences whose behavior is similar to that of the Kashaev equation and the hexahedron recurrence, respectively. Theorem 9.24 generalizes Theorem 1.22 to this class of recurrences.
Among the applications of this generalization, we study a set of equations that appear in the context of s-holomorphicity in discrete complex analysis. We introduce an equation (4.1), similar to the Kashaev equation for arrays indexed by Z 2 , along with equations (4.12)-(4.14), similar to the hexahedron recurrence for arrays indexed by the edges and vertices of the standard tiling of R 2 with unit squares. The equations (4.13)-(4.14) for the edge values are independent of the values on the vertices, and can be used (with small modifications) to define s-holomorphic functions on the tiling of R 2 with unit squares. While the equations (4.1) and (4.12)-(4.14) have been studied before (cf. [1]), our main novelty is the notion of coherence similar to that for the Kashaev equation.
As another application, we introduce additional recurrences exhibiting hexahedron-like behavior that have their origins in the theory of cluster algebras. Whereas the connections with cluster algebras are to be discussed elsewhere, the definitions of coherence for these recurrences are provided herein.
The paper is organized as summarized in the We next review the content of each section of the paper. Section 1 introduces the basic concepts. Its main result is Theorem 1.22, which has been discussed above. The results from Section 1 are proved in Section 6.
While Sections 1 and 6 are necessary for the rest of the paper, Sections 2, 3, 7, 8 are independent of Sections 4, 5, 9, and vice versa. In Section 2, we discuss some combinatorial tools involving cubical complexes and zonotopal tilings that we use in Sections 3, 7, and 8. In Section 3, we review the background from Kenyon and Pemantle [4] on the use of the hexahedron recurrence and the Kashaev equation in the study of principal and almost principal minors. In that section, we also state a version of Theorem 1.22 for certain cubical complexes, and then apply this result to the study of principal minors of symmetric matrices. In Section 7, we extend Theorem 1.22 to the setting of cubical complexes, and in the process prove some results from Section 3. In Section 8, we prove the remaining results from Section 3.
In Section 4, we discuss a condition similar to the Kashaev equation that arises in the context of s-holomorphicity. In Section 5, we discuss some additional recurrences with behavior similar to the Kashaev equation and hexahedron recurrence, which are related to cluster algebras. Sections 4 and 5 can be read independently of each other. In Section 9, we describe an axiomatic setup for equations with properties similar to those of the Kashaev equation, and prove a more general version of Theorem 1.22. In the process, we prove all of the results from  This paper is a slightly edited version of the author's Ph.D. thesis [8].

The Kashaev Equation in Z 3
In this section, we introduce the Kashaev equation, the hexahedron recurrence, and the K-hexahedron equations. We then state our main results (Theorems 1.22-1.23) about the Kashaev equation for arrays indexed by Z 3 . Definition 1.1. Let z 000 , . . . , z 111 ∈ C be 8 numbers indexed by the vertices of a cube, as shown in Figure 1. We say that these 8 numbers satisfy the Kashaev equation if 2(a 2 + b 2 + c 2 + d 2 ) − (a + b + c + d) 2 + 4(s + t) = 0, (1.1) Definition 1.2. We say that a 3-dimensional array x ∈ C Z 3 satisfies the Kashaev equation if its components labeled by the vertices of any unit cube in Z 3 satisfy (1.1). More formally, given a unit cube C in Z 3 , define K C : C Z 3 → C by where a, b, c, d, s, t are the monomials in the components of x at the vertices of C, defined as in Figure 1. We then say that x satisfies the Kashaev equation if K C (x) = 0 for every unit cube C in Z 3 .
The Kashaev equation was originally introduced by R. Kashaev [5] in the study of the star-triangle move in the Ising model. It also appears as an identity involving principal minors of a symmetric matrix [4]; this connection is discussed in Section 3. Furthermore, up to changes of sign, the Kashaev equation can be interpreted as the vanishing of Cayley's hyperdeterminant of a 2×2×2 hypermatrix; this connection is also discussed in Section 3. The Kashaev equation is also related to the theory of cluster algebras and to Descartes's formula for Apollonian circles, connections that we will explore in later work. where A = 2z 100 z 010 z 001 + z 000 (z 100 z 011 + z 010 z 101 + z 001 z 110 ) D = (z 000 z 011 + z 010 z 001 )(z 000 z 101 + z 100 z 001 )(z 000 z 110 + z 100 z 010 ). (1.4) and √ D denotes any of the two square roots of D. Notice that if all 7 values z ijk contributing to the right-hand side of (1.3) are positive, then D > 0, so both solutions for z 111 in (1.3) are real; moreover, the larger of these two solutions is positive. This observation suggests the following definition. Definition 1. 4. We say that a 3-dimensional array x ∈ (R >0 ) Z 3 satisfies the positive Kashaev recurrence if for every (v 1 , v 2 , v 3 ) ∈ Z 3 , we have , (1.5) where z ijk denotes the component of x at (v 1 + i, v 2 + j, v 3 + k), for i, j, k ∈ {0, 1}, and we use the notation introduced in (1.4), with the conventional meaning of the square root. Definition 1. 6. Let x ∈ C Z 3 . Let v, w be two opposite vertices in a unit cube C in Z 3 . We set (z 111 z 2 000 − z 000 (z 100 z 011 + z 010 z 101 + z 001 z 110 )) − z 100 z 010 z 001 , where we use a labeling of the components of x on the vertices of C as in Figure 1, with z 000 corresponding to the component of x at v. Definition 1.7. Given v ∈ Z 3 and i 1 , i 2 , i 3 ∈ {−1, 1}, define C v (i 1 , i 2 , i 3 ) to be the unique unit cube containing the vertices v and v + (i 1 , i 2 , i 3 ).
Proposition 1.8. Suppose that x = (x s ) ∈ C Z 3 satisfies the Kashaev equation. Then for any v ∈ Z 3 , where • the first product is over the 8 unit cubes C incident to the vertex v, • the second product is over the 12 unit squares S incident to v (cf. Figure 2), and • v, v 1 , v 2 , v 3 are the vertices of such a unit square S listed in cyclic order. Moreover, the following strengthening of (1.7) holds:       C=Cv (i 1 ,i 2 ,i 3 ) (1.8) where the rightmost product is the same as in (1.7).
v Figure 2. The 12 unit squares incident to v ∈ Z 3 . Theorem 1.9. Suppose that x = (x s ) ∈ (R >0 ) Z 3 satisfies the positive Kashaev recurrence. Then for any v ∈ Z 3 , (1.9) where the notational conventions are the same as in equation (1.7). Proposition 1.8 asserts that the expressions being squared in equation (1.7) are equal up to sign; in the case of the positive Kashaev recurrence, Theorem 1.9 states that the signs must match. Definition 1. 10. We say that a solution x of the Kashaev equation is coherent if it satisfies (1.9) for every v ∈ Z 3 . Equivalently, x is coherent if  Coherent solutions of the Kashaev equation are closely related to (a special case of) the hexahedron recurrence, introduced and studied by Kenyon and Pemantle [5]. We next discuss this important construction, which plays a central role in this paper. Definition 1.12. Let L be the subset of ( 1 2 Z) 3 defined by (1.11) Thus, L contains Z 3 , together with the centers of unit squares with vertices in Z 3 .
Kenyon and Pemantle [5] made the following important observation, which can be verified by direct computation. Proposition 1.13 ([5, Proposition 6.6]). (a) Let x = (x s ) ∈ (R >0 ) Z 3 satisfy the positive Kashaev recurrence. Extend x to an arraỹ x = (x s ) ∈ (R >0 ) L by setting appear in cyclic order along the unit square corresponding to s; see Figure 3. In other words, for all v ∈ Z 3 , where z ijk denotes the component ofx at v + (i, j, k), and A and D are given by (1.4).
Then the restriction ofx to Z 3 satisfies the positive Kashaev recurrence. Definition 1.14 ( [5]). We say that an arrayx ∈ (C * ) L satisfies the hexahedron recurrence if for any v ∈ Z 3 ,x satisfies equations (1.16)- (1.19). Notice that equations (1.16)-(1.19) involve the components ofx at the 14 points in L located at the boundary of the unit cube in Z 3 with the vertices v + (i, j, k), for i, j, k ∈ {0, 1}, namely the 8 vertices of the cube, and the 6 centers of its faces.
The hexahedron recurrence was introduced in [5] in the context of statistical mechanics as a way to count "taut double-dimer configurations" of certain graphs. This recurrence also describes a relationship among principal and "almost principal" minors of a square matrix [4], a connection we will discuss in Section 3.
Remark 1.15. The equations for the hexahedron recurrence, like those for the positive Kashaev recurrence above (and unlike the original Kashaev equation (1.1)), have a "preferred direction," viz., the direction of increase of all three coordinates. While replacing the direction (1, 1, 1) by the opposite direction (−1, −1, −1) does not change these equations, using any of the six remaining directions (±1, ±1, ±1) yields a different recurrence. See Remark 6.4.
We now extend Proposition 1.13 to complex-valued solutions of the hexahedron recurrence.
for all v ∈ Z 3 and all distinct i, j ∈ {1, 2, 3}. Then x can be extended to an arraỹ x = (x s ) ∈ (C * ) L satisfying the hexahedron recurrence along with (1.12). (b) Conversely, supposex = (x s ) ∈ (C * ) L satisfies the hexahedron recurrence along with (1.12). Then the restriction ofx to Z 3 is a coherent solution of the Kashaev equation and satisfies condition (1.20). Remark 1.17. Theorem 1.9 follows from Theorem 1.16(b), because a solution of the positive Kashaev recurrence can be extended to a solution of the hexahedron recurrence that satisfies (1.12) (by Proposition 1.13).
Remark 1.18. If x doesn't satisfy condition (1.20), and an arrayx extending x ∈ C L satisfies (1.12), then at least one of the face variables forx equals 0, requiring us to divide by 0 when we apply the hexahedron recurrence. On the other hand, ifx ∈ C L satisfies equations (1.16)-(1.19) with the denominators multiplied out (so that the denominators can equal 0), then the restriction ofx to Z 3 doesn't necessarily satisfy the Kashaev equation.
The following statement is straightforward to check.
Proposition 1.19. Letx = (x s ) ∈ (C * ) L be an array satisfying (1.12), for any s ∈ L−Z 3 . Then the following are equivalent: •x satisfies the hexahedron recurrence; • for any v ∈ Z 3 , we have where, as before, z ijk denotes the component ofx at v + (i, j, k), and A is given by (1.4). Definition 1.20. Letx = (x s ) ∈ C L be an array with x s = 0 for s ∈ Z 3 . We say thatx satisfies the K-hexahedron equations ifx satisfies equation (1.12) for all s ∈ L − Z 3 , and satisfies equations (1.21)-(1.24) for all v ∈ Z 3 . Remark 1.21. By Proposition 1.19, ifx ∈ (C * ) L , i.e.,x has all nonzero components, then the following are equivalent: •x satisfies the K-hexahedron equations; •x satisfies the hexahedron recurrence, along with equation (1.12) for s ∈ L − Z 3 .
We next restate Theorem 1.16 (and slightly strengthen part (b) thereof) using the notion of the K-hexahedron equations. (b) Conversely, suppose thatx = (x s ) ∈ C L (with x s = 0 for all s ∈ Z 3 ) satisfies the K-hexahedron equations. Then the restriction ofx to Z 3 is a coherent solution of the Kashaev equation.
The extension from x tox in Theorem 1.22(a) is not unique. The theorem below clarifies the relationship between different extensions. Theorem 1.23. Letx = (x s ) ∈ (C * ) L be an array satisfying the K-hexahedron equations.

Combinatorial Preliminaries on Cubical Complexes and Zonotopes
In this section, we review some standard combinatorial background on cubical complexes and zonotopes. This section introduces some unconventional terminology which later sections will use.
m such that {v j : j ∈ I} is linearly independent. Cubical tilings of zonotopes are examples of cubical complexes.
Definition 2.4. We denote by P n the regular (2n)-gon Z e 1 ,...,en where e j = e πi(j−1)/n ∈ C ∼ = R 2 for j = 1, . . . , n using the standard identification between C and R 2 (see Figure 4). We denote by v 0 the vertex of P n corresponding to the origin. We define a ♦tiling of P n to be a cubical tiling of Z e 1 ,...,en , i.e., a tiling of P n with the n 2 rhombi given by the translations of the Minkowski sums [0, e i ] + [0, e j ] for 1 ≤ i < j ≤ n (see Figure 5). We label the vertices of a ♦-tiling of P n by subsets of [n] as follows: we label a vertex v by I ⊆ [n] if we can reach v from v 0 by following the edges of the tiling corresponding to the vectors e j for j ∈ I (see Figure 5).   Figure 5. A ♦-tiling of P 4 . The vertex v 0 corresponding to the origin is labeled. In red, we label each rhombus that is a translation of the Minkowski sum [0, e i ] × [0, e j ] by ij. In blue, we label each vertex of the tiling by its corresponding subset of [4].
Definition 2.5. Two quadrangulations T 1 , T 2 of a polygon are connected by a flip if they are related by a single local move of the form pictured in Figure 6. Note that we can picture a flip as placing a cube on top of the hexagon where the flip occurs. It will be helpful for us to think about quadrangulations through the dual language of divides.
Definition 2.6. A divide D in a polygon R in R 2 is an immersion of a finite set of closed intervals and circles, called branches, in R, such that • the immersed circles do not intersect the boundary of R, • the immersed intervals have pairwise distinct endpoints on the boundary of R, and are otherwise disjoint from the boundary, • all intersections and self-intersections of the branches are transversal, and • no three branches intersect at a point, all considered up to isotopy. For further details, see [2, Definition 2.1]. Given a quadrangulation T of R, the divide associated to T is the divide in R such that for every tile Q in T , branches connect the 2 pairs of opposite edges in Q, and there is a single branch intersection in the interior of Q (see Figure 7). (All divides considered in the remainder of this paper are associated to quadrangulations.) A braid move is a local transformation of divides shown in Figure 8. A flip in a quadrangulation corresponds to a braid move in its associated divide.  Definition 2.7. A divide is called a pseudoline arrangement if all of its branches are immersed intervals with no self-intersections, and, moreover, each pair of branches intersects at most once. Note that the class of pseudoline arrangements is closed under braid moves.
The following fact is well known. Proposition 2.8. Let D be a divide in a polygon in R 2 . Then the following are equivalent: • D is a pseudoline arrangement in which every pair of branches intersects exactly once; • D is topologically equivalent to the divide associated to a ♦-tiling of P n . Remark 2.9. Pseudoline arrangements of n branches, each pair of which intersects exactly once, are in bijection with commutation-equivalence classes of reduced words for the longest element w 0 ∈ S n in the symmetric group. A braid move on the pseudoline arrangement corresponds to a braid move on the reduced word. Definition 2.10. Let T be a ♦-tiling of P n , and let D be the pseudoline arrangement associated to T . We call the connected components of the complement of D the chambers of D. Note that the chambers of D correspond to the vertices of T , and the crossings of D correspond to the tiles of T . Label the branches 1, . . . , n as in Figure 9, by starting at v 0 and traveling counterclockwise along the boundary of P n (so that branch j intersects ∆-crossing ∇-crossing Figure 10. A ∆-crossing and a ∇-crossing. Note that in the ∆-crossing, the triangle formed by the 3 intersecting branches points up and away from v 0 , while in the ∇-crossing, the triangle formed by the 3 intersecting branches points down and towards v 0 . the boundary of P n at the edges parallel to e j = e πi(j−1)/n ). Note that the label I ⊆ [n] for a vertex of T is precisely the set of labels for the branches in between the chamber and v 0 .
Definition 2.11. Let T be a ♦-tiling of P n , and let D be the pseudoline arrangement associated to T . Label the branches as in Definition 2.10. Given three branches labeled i < j < k, we say that i, j, k have a ∆-crossing if the pairs of i, j, k intersect in the following counterclockwise order along the boundary of the triangle these branches form: (i, j), (i, k), (j, k). We say that i, j, k have a ∇-crossing if the pairs of i, j, k intersect in the following counterclockwise order along the boundary of the triangle these branches form: (j, k), (i, k), (i, j). Note that i, j, k must either have a ∆-crossing or a ∇-crossing. See Figure 10 for pictures which make clear the reasoning behind this choice of terminology. Note that when a braid move is performed with i, j, k, the triple i, j, k switches between having a ∆-crossing and having a ∇-crossing. T min, 4 T max,4 Figure 11. The ♦-tilings T min,4 and T max,4 .
Definition 2.12. Define T min,n to be the unique ♦-tiling of P n in which every vertex is labeled by consecutive subsets I ⊆ [n], and define T max,n to be the unique ♦-tiling of P n in which every vertex is labeled by a subset I ⊆ [n] whose complement [n] − I is consecutive (see Figure 11). Equivalently, T min,n is the ♦-tiling of P n in which every triple i < j < k has a ∆-crossing in its associated pseudoline arrangement, while T max,n is the ♦-tiling of P n in which every triple i < j < k has a ∇-crossing in its associated pseudoline arrangement.
Definition 2.13. We say that T = (T 0 , . . . , T ℓ ) is a pile of quadrangulations of a polygon if T i−1 and T i are connected by a flip for i = 1, . . . , ℓ.
Note that T 0 = T 8 . For each tiling T i , there are two possible flips that we can perform; applying one gives us T i−1 , and applying the other gives us T i+1 (with indices taken mod 8).
Definition 2.15. Define C(n) to be the set of all piles T = (T 0 , . . . , T ( n 3 ) ) with T 0 = T min,n and T ( n 3 ) = T max,n . Note that the shortest length of a pile starting with T min,n and ending with T max,n is n 3 , corresponding to switching from ∆-crossings to ∇-crossings for each of the n 3 triples. Every ♦-tiling T of P n appears in at least one pile in C(n). Remark 2.16. One can put a poset structure on the set of ♦-tilings of P n , called the second higher Bruhat order B(n, 2) [9], as follows: given ♦-tilings T 1 , T 2 with associated pseudoline arrangements D 1 , D 2 , we say that T 1 ≤ T 2 if i, j, k having a ∆-crossing in T 2 implies that i, j, k have a ∆-crossing in T 1 for all i < j < k. Note that T min,n is the minimum element of B(n, 2), and T max,n is the maximum element of B(n, 2).   with the 2-dimensional cubical complex corresponding to T 0 . For i = 1, . . . , ℓ, given T i−1 and T i labeled as in Figure 13, add a new vertex v to the complex corresponding to the new vertex in T i , and add the 3-dimensional cube labeled as in Figure 13, with v as its top vertex. Note that a flip cannot be centered at a vertex on the boundary of R, so each vertex on the boundary of R corresponds to a single vertex in κ. In the special case where R = P n and the quadrangulations are ♦-tilings, we denote the vertex of κ corresponding to v 0 as v 0 , by an abuse of notation.
Definition 2.20. Given a cubical complex κ, we denote by κ i the set of i-dimensional faces of κ, and set κ 02 = κ 0 ∪ κ 2 . Similarly, for a pile T of a quadrangulations, we denote by κ i (T) the set of i-dimensional faces of κ(T), and κ 02 (T) = κ 0 (T) ∪ κ 2 (T). • T 1,j = T 2,j for j = 1, . . . , i and j = i + 4, . . . , ℓ, and • T 1,j and T 2,j for j = i + 1, i + 2, i + 3 are related by the local moves shown in Figure 14.    [3] 2 . The inversion set of an admissible permutation σ of [n] k is the subset of [n] k+1 consisting of those I ∈ [n] k+1 for which the elements of I k appear in σ in the reversed lexicographic order. 3 . The following are equivalent: • σ is an admissible permutation of [n] 3 ; • there exists a pile T ∈ C(n) whose corresponding permutation of [n] 3 is σ.
The following are equivalent: • κ(T 1 ) and κ(T 2 ) are isomorphic directed cubical complexes; • the inversion sets of the permutations of [n] 3 associated to T 1 and T 2 coincide.

The Coherence Condition and Principal Minors of Symmetric Matrices
We begin this section by reviewing the earlier work of Kenyon and Pemantle [4] concerning the occurence of the hexahedron recurrence as a determinantal identity. We then formulate new criteria for the existence of symmetric matrices with prescribed values of certain principal minors. See in particular Corollary 3.20, Corollary 3.23, and Theorem 3.26. The proofs of these results are given later in Sections 7-8.
We begin by extending the definitions of the Kashaev equation, positive Kashaev recurrence, hexahedron recurrence, and K-hexahedron equations to arrays indexed on directed cubical complexes in the obvious way. For those readers who skipped Section 2, it may be helpful to review Definitions 2.1, 2.18, and 2.20 before processing the following definition.  Figure 1 and z 111 corresponding to the component of x at the top vertex of C. We say that an arrayx indexed by κ 02 satisfies the hexahedron recurrence (resp., K-hexahedron equations) if for all 3-dimensional cubes C of κ,x satisfies equations (1.16)-(1.19) (resp., equations (1.21)-(1.24), along with equation (1.12) for all s ∈ κ 2 ) with the components ofx labeled on the vertices of C as in Figure 1, labeled on the 2-dimensional faces of C by averaging the indices of the vertices on the boundary, and with z 111 corresponding to the component ofx on the top vertex of C. Remark 3.2. As we saw in Section 1, the Kashaev equation is independent of a choice of direction on each cube, and hence can be defined for arbitrary 3-dimensional cubical complexes. On the other hand, the positive Kashaev recurrence, hexahedron recurrence, and K-hexahedron equations depend on a choice of a pair of opposite distinguished vertices in each cube, and hence are defined on directed cubical complexes.  More generally, if T = (T 0 , . . . , T ℓ ) is a pile of ♦-tilings of P n , define x κ(T ) (M) to be the array indexed by κ 0 (T) whose restriction to κ 0 (T i ) is x T i (M), and definex κ(T ) (M) to be the array indexed by κ 02 (T) whose restriction to κ 02 (T i ) isx T i (M).
Remark 3.5. For any ♦-tiling T of P n , the vertex v 0 is labeled by ∅ ⊂ [n]. In Definition 3.4, because of the convention that M ∅ ∅ = 1, x v 0 = 1 independent of the matrix M. Definition 3.6. Given a ♦-tiling T of P n , we say that a complex-valued arrayx = (x s ) s∈κ 02 (T ) is standard if x v 0 = 1. Furthermore, given a pile of ♦-tilings of P n , T = (T 0 , . . . , T ℓ ), and κ = κ(T), we say that a complex-valued arrayx = (x s ) s∈κ 02 is standard with respect to T if x v 0 = 1.
Definition 3.7. Given a ♦-tiling T of P n , we say that a complex-valued arrayx indexed by κ 02 (T ) is generic if for any sequence of flips applied to T accompanied by applications of the hexahedron recurrence tox, the resulting coefficients are all nonzero.
Definition 3.8. We say that a square matrix is generic if all of its principal minors and odd almost-principal minors are non-zero. Let M * n (C) denote the set of n × n generic complex-valued matrices.
We can now provide some important results of Kenyon and Pemantle [4]. . Given a ♦-tiling T of P n , the mapx T (·) establishes a bijective correspondence between M * n (C) and standard, generic, complex-valued arrays on κ 02 (T ).
Before proceeding with the following theorem, the reader may want to review Definition 2.12.
Remark 3.13. Theorem 3.12 is not stated explicitly in [4], but follows immediately from the cited theorem. The original theorem concerns Hermitian matrices, and a slightly modified version of the hexahedron recurrence in which some complex conjugates are taken.
The next result follows from Proposition 3.11 and Theorem 3.12: Corollary 3.14. Let T be a pile of ♦-tilings of P n , with κ = κ(T). Let M be an n × n symmetric matrix with nonzero principal minors. Then x κ(T) (M) satisfies the K-hexahedron equations.
The next corollary follows immediately from Theorem 3.12 and the fact that the entries of M are Laurent polynomials in the components ofx κ(T min,n ) (M): Hence, the following is immediate from Proposition 3.11 and Corollary 3.15: Corollary 3.16. Let T be a pile of ♦-tilings of P n containing T min,n , with κ = κ(T). Let x = (x s ) ∈ (C * ) κ 0 be a standard array satisfying the property that for all 2-dimensional faces of κ with vertices v 1 , v 2 , v 3 , v 4 in cyclic order. Then the following are equivalent: • x can be extended to a standard array indexed by κ 02 satisfying the K-hexahedron equations; • there exists a symmetric matrix M such that x = x κ(T) (M).
We want a set of equations that tell us whether an array x indexed by κ 0 (T) can be extended to an array indexed by κ 02 (T) satisfying the K-hexahedron equations. Below, we define a notion of coherence generalizing the notion of coherence from Section 1.
Definition 3.17. Let κ be a 3-dimensional cubical complex. We say that x = (x s ) s∈κ is a coherent solution of the Kashaev equation if x satisfies the Kashaev equation (i.e., K C (x) = 0 for every 3-dimensional cube C of κ), and for any interior vertex v of κ: where • the first product is over 3-dimensional cubes C incident to the vertex v, • the second product is over 2-dimensional faces S incident to the vertex v, and • v, v 1 , v 2 , v 3 are the vertices of such a face S listed in cyclic order.
Remark 3.18. The property of being coherent solution of the Kashaev equation is defined for 3-dimensional cubical complexes, not only for all 3-dimensional directed cubical complexes, as no choice of direction needs to be made in each cube.
Theorem 3.19 is proved in Section 7, where we obtain results (Proposition 7.3 and Theorem 7.10) generalizing both Theorem 3.19 and Theorem 1.22.
As an immediate corollary of Corollary 3.16 and Theorem 3.19, we obtain the following: Next, we consider the problem of checking whether an array of 2 n numbers could correspond to the principal minors of some symmetric matrix.
where ∆ denotes the symmetric difference. Although Corollary 3.23 can be deduced from Theorem 3.9, Theorem 3.12, Corollary 3.14, and Theorem 3.19, we provide a proof in Section 8.
where C is a 3-dimensional cube, and x is the array indexed by the vertices of C shown in Figure 15, and v is the vertex of C at which x has entry x I .
x I x I∆{j} x I∆{k} 3 ) Furthermore, it is straightforward to check that ifx =x(M) for some 4 × 4 symmetric matrix M, then equation (3.13) holds for all I ⊆ [4]. Hence, the result below follows from Corollary 3.23. and A ∈ [n] 4 , We prove Theorem 3.26 in Section 8. He considers the natural action of (SL 2 (C) ×n ) ⋉ S n (where S n is the symmetric group on n elements) on C 2 [n] , and proves that forx = (x I ) I⊆[n] the following are equivalent: • there exists a symmetric matrix M such that The left-hand side of equation (3.15) can be identified as Cayley's 2 × 2 × 2 hyperdeterminant. Equivalently, equation (3.15) is just equation (3.12) for I = ∅ and {i, j, k} = {1, 2, 3} with the appropriate changes of sign (because we don't put additional signs on the principal minors in this setting). For the above equivalence, Oeding does not impose an assumption of genericity onx. Consider the subgroup H ⊆ SL 2 (C) defined by In this language of [10], our condition that a (signed) arrayx satisfies equation (3.12) for all I ⊆ [n] and distinct i, j, k ∈ [n] can be restated as the condition that all images of the "unsigned version" ofx, under the action of the group (H ×n )⋉S n , satisfy equation (3.15). Thus, Theorem 3.26 requires an additional assumption of genericity and an additional equation (equation (3.14)) compared to Oeding's criterion, but uses a weaker version of the second requirement above.

S-Holomorphicity in Z 2
In this section, we discuss a certain equation (see (4.3)) which shares many properties with the Kashaev equation. We also study a related system of equations (see (4.12)-(4.16)) which plays the role analogous to the K-hexahedron equations. The equations studied herein arise in discrete complex analysis and in the study of the Ising model (see [1] and Remark 4.11). The presentation of results in this section follows a plan similar to that of Section 1. The results in this section are proved in Section 9 as special cases of a general axiomatic framework.
Definition 4.1. Given a unit square C with vertices in Z 2 and an array x ∈ C Z 2 , define Q C (x) = z 2 00 +z 2 10 +z 2 01 +z 2 11 −2(z 00 z 10 +z 10 z 11 +z 11 z 01 +z 01 z 00 )−6(z 00 z 11 +z 10 z 01 ), (4.1) where z 00 , z 10 , z 01 , z 11 denote the components of x at the vertices of C, as shown in Fig Let v and w be two opposite vertices in a unit square C in Z 2 . We set where we use a labeling of the components of x on the vertices of C as in Figure 16, with z 00 corresponding to the component of x at v.
to be the unique unit square containing the vertices v and v + (i 1 , i 2 ).
where • the first product is over the 4 unit squares C incident to the vertex v, • the second product is over the 4 edges S incident to v, and • v and v 1 are the vertices of such an edge S. Moreover, the following strengthening of (4.4) holds: where the rightmost product is the same as in (4.4).
Remark 4.5. The right-hand side of equation (4.1) is a quadratic polynomial in each of the variables z ij . Setting this expression equal to zero and solving for z 11 in terms of z 00 , z 10 , z 10 , we obtain where (z 00 + z 10 )(z 00 + z 01 ) denotes either of the two square roots. Notice that if z 00 , z 10 , z 01 > 0, then both solutions for z 11 in (4.6) are real; moreover, the larger of these two solutions is positive. However, solving the equation for z 00 , with z 10 , z 01 , z 11 > 0, may result in a unique negative solution. (For example, take z 10 = z 01 = z 11 = 1.) Hence, unlike the positive Kashaev recurrence, the equation (4.7) only defines a recurrence on R >0 -valued arrays in one direction.
where the notational conventions are the same as in equation (4.4).
In order words, E is the set of centers of edges in the tiling of R 2 with unit squares.
We next state the analogue of Theorem 1.22.
Then x can be extended to an arrayx = (x s ) ∈ C Z 2 ∪E satisfying the recurrence together with the conditions Then the restriction x ofx to Z 2 satisfies Q C (x) = 0 for all unit squares C, and satisfies (4.9) for all v ∈ Z 2 . Remark 4.10. Comparing Theorem 1.22 to Theorem 4.9, we see that equations (4.12)-(4.16) play a role analogous to that of the K-hexahedron equations.
Remark 4.11. The equations (4.12)-(4.16) appear in discrete complex analysis in the context of s-holomorphicity [1]. Consider the labeling ℓ : E → C described in Figure 17. If we orient the edge with midpoint s ∈ E from the even height vertex to the odd height vertex, then ℓ(s) is a square root of the complex number associated with the directed edge. An s-holomorphic function on the tiling of R 2 with unit squares is a complexvalued function F on the faces of the tiling such that for any two faces f 1 , f 2 sharing an edge with midpoint s, we have Hence, given an s-holomorphic function F , we can define x ′ = (x s ) ∈ R E by setting for either of the faces f using the edge corresponding to s. It is straightforward to check that x ′ = (x s ) ∈ R E corresponds to an s-holomorphic function F by (4.18) if and only if x ′ satisfies (4.13)-(4.14). If we extend x ′ tox = (x s ) ∈ R Z 2 ∪E satisfying (4.12)-(4.16), the function H : corresponds to a certain discrete integral. For more on this recurrence and its connections to discrete complex analysis and the Ising model, see [1].

Further Generalizations of the Kashaev Equation
In this section, we provide two additional examples of equations with behavior similar to the Kashaev equation and its analogue (4.1). In Section 9, we shall develop a general framework which will allow us to prove all of the results in this section (as well as the results in Section 4).
Both recurrences considered in this section come with complex parameters that one can choose arbitrarily. For certain values of these parameters, the corresponding recurrences have cluster algebra-like behavior. We will explore the cluster algebra nature of these recurrences in future work [7].
We begin with the following, relatively simple, one-dimensional example: be an array such that  .2) is the partial derivative of the left-hand side of (5.1) with respect to z 3 . This expression plays the role of K C v in Proposition 1.8.
where, as before, we denote z i = x v+i . Moreover, assume that for all v ∈ Z, the number z 3 = x v+3 is the larger of the two real solutions of (5.1): where D is given by (5.3). Then or equivalently, We next show that an array x satisfying (5.1) satisfies condition (5.6) if and only if it can be extended to an array on a larger index set satisfying conditions resembling the K-hexahedron equations.
(a) For any array x = (x s ) ∈ (C * ) Z satisfying (5.1) and (5.6), there exists an array y = (y s ) s∈Z such that x and y together satisfy the recurrence together with the condition where D is given by (5.3), and we use the notation z i = x v+i and w i = y v+i . (b) Conversely, suppose x ∈ (C * ) Z and y ∈ C Z satisfy (5.7)-(5.9). Then x satisfies (5.1) and (5.6).
Remark 5.5. The components of y are most naturally indexed by intervals of length 2 in Z. Here, we index the components of y by the midpoints of those intervals.
Next, we consider the following two-dimensional example: . Given a 1 × 2 rectangle B with vertices in Z 2 , a vertex w ∈ Z 2 at a corner of B, and the components of x at the 6 points of Z 2 in B labeled as in Figure 18, define R B,0 w (x) ∈ C by R B,0 w (x) = z 2 00 z 12 − α 1 z 10 z 2 01 − 2z 00 z 10 z 02 − α 2 z 00 z 01 z 11 . (5.14) Given a 0 × 2 rectangle (line segment) S with vertices in Z 2 , and the components of x at the 3 points of Z 2 in S labeled as in Figure 18, define R S,1 (x) ∈ C by R S,1 (x) = α 1 z 2 01 + 4z 00 z 02 . (5.15) Given a 1 × 1 square C with vertices in Z 2 , and the components of x at the 4 points of Z 2 in C labeled as in Figure 18, define R C,2 (x) ∈ C by R C,2 (x) = α 1 (z 2 00 z 2 11 + z 2 01 z 2 10 ) + 2α 2 z 00 z 01 z 10 z 11 . (5.16) are the four 1×2 rectangle/corner pairs shown in Figure 19, • S 1 , S 2 are the two 0 × 2 rectangles (line segments) shown in Figure 19, and • C 1 , C 2 are the two 1 × 1 squares shown in Figure 19.
Moreover, the following strengthening of (5.17) holds:  Figure 19. On the top row, the rectangle/corner pairs (B i , w i ) for i = 1, . . . , 4, and on the bottom row, the line segments S 1 , S 2 and the squares C 1 , C 2 that appear in (5.17).
Remark 5.11. In Theorem 5.10, the components of y 1 are most naturally associated to 0 × 2 rectangles (line segments) with vertices in Z 2 (although we index it by the center of the line segment in the theorem), and the components of y 2 are most naturally associated to 1 × 1 unit squares with vertices in Z 2 (although we index it by the center of the strip in the theorem). If we think about the recurrence (5.23)-(5.25) in this way, each step of the recurrence uses the six vertices, two 0 × 2 rectangles, and two 1 × 1 unit squares contained in the 1 × 2 rectangle v + {0, 1} × {0, 1, 2}, as is pictured in Figure 20.

Proofs of Results from Section 1
In this section, we prove Proposition 1.8 and Theorems 1.22-1.23 (of which all other results in Section 1 are corollaries). Lemma 6.1. Let C be a cube with vertices V (C) labeled as in Figure 21, and let x = (x s ) ∈ C V (C) . Then Figure 21. Labels for the vertices of a cube C.
Proof. The proof follows from a straightforward computation.
Proof of Proposition 1.8. Let x = (x s ) ∈ C Z 3 satisfy the Kashaev equation. Given a unit cube C of Z 3 labeled as in Figure 21, by Lemma 6.1, we have Taking the product over unit cubes C of Z 3 containing v, we obtain here we use that the double product counts each unit square containing v twice. Similarly, because each double product counts each unit square containing v twice.
For the proof of Theorem 1.22(b), we will need the following lemma.
Becausex satisfies the K-hexahedron equations, it follows that if g ∈ I, then specializing z ijk = x v ′ +(i,j,k) in g yields 0. Given j ℓ = min(i ℓ , 0) + 1 and k ℓ = 1 − j ℓ for ℓ = 1, 2, 3, let . It can be checked that Proof of Theorem 1.22(b). Let x ∈ (C * ) Z 3 be the restriction ofx to Z 3 . Applying Lemma 6.2 for the 8 cubes incident to a vertex v ∈ Z 3 , we get: Proof. The proof follows from a straightforward computation. Proposition 6.3 enables us to run the K-hexahedron equations "in reverse." We note that the property that we show for the K-hexahedron equations in Proposition 6.3 holds for the original hexahedron recurrence.
Remark 6.4. Here, we address the comments in Remark 1.15. Letx be an array of 14 numbers indexed by the vertices and 2-dimensional faces of a cube C with a distinguished "top" vertex v, where the components ofx indexed by the 8 vertices of the cube are nonzero. Supposex satisfies the K-hexahedron equations. Proposition 6.3 tells us thatx would satisfy the K-hexahedron equations if we took the vertex w opposite v to be the "top" vertex of C. On the other hand, Lemma 6.2 tells us that if the components of x indexed by the faces are nonzero, and w is a vertex of C other than v or the vertex opposite it, thenx would not satisfy the K-hexahedron equations if we place the vertex w at the "top" of C. This argument also implies the analogous statement for the hexahedron recurrence.
We next work toward a proof of Theorem 1.22(a). Lemma 6.5. Fix v ∈ Z 3 andx = (x s ) ∈ C L satisfying the equations (1.21)-(1.24), with x s = 0 for s ∈ Z 3 . Then the following are equivalent: • the following equations hold: (6.14) • the following equations hold: Proof. This is a straightforward verification.
Definition 6.6. For U ⊆ Z, we denote In other words, L U,V contains the integer points at heights in U, and the half-integer points of L at heights in V . In particular, we will be interested in Proof. This follows directly from Lemma 6.5.
Remark 6.8. By Proposition 6.3, givenx init = (x s ) ∈ C L init with x s = 0 for all s ∈ Z 3 init , there exists at most one extensionx = (x s ) ∈ C L ofx init to L (with x s = 0 for all s ∈ Z 3 ) satisfying the K-hexahedron equations. We say "at most one" instead of "one" because in the course of running the recurrence (1.21)-(1.24), we might get a zero value at an integer point. Definition 6.9. We say that an arrayx init indexed by L init that satisfies equation (1.12) for all s ∈ L init − Z 3 init is generic ifx init can be extended to an arrayx indexed by L satisfying the K-hexahedron equations, with all components ofx nonzero. Similarly, we say that an array x init indexed by Z 3 init is generic if every extension of x init to an arraỹ x init indexed by L init satisfying equation (1.12) for all s ∈ L init − Z 3 init is generic. Definition 6.10. Letx init be a generic array indexed by L init that satisfies equation (1.12) for s ∈ L init − Z 3 init . We denote by (x init ) ↑L the unique extension ofx init to L satisfying the K-hexahedron equations. Proof. Note that  given (6.26) Remark 6.14. Assume that x init is a generic array indexed by Z 3 init . Letx init be any (generic) array indexed by L init that restricts to x init and satisfies equation (1.12) for s ∈ L init − Z 3 init . Note that is the set of arrays indexed by L which satisfy the K-hexahedron equations and restrict to x init . Proof. Suppose there exist constants α i , β i , γ i ∈ {−1, 1} for i ∈ Z such that t satisfies equations (6.29)-(6.31). Let u = (u s ) = ψ(t). Then for any (a, b, c) ∈ Z 3 , u (a,b,c)+( 1 2 , 1 2 , 1 2 ) = α 2 a β 2 b γ 2 c = 1, as desired. Next, suppose that t is in the kernel of ψ. It is straightforward to check that the following identities for t for all (a, b, c) ∈ Z 3 :    where the second product is over the lines s ∈ Z 3 determined by the edges of C.
Next, suppose that condition (6.38) holds. It is clear that u is uniquely determined by its components at if v 3 ∈ Z and either v 1 = 1 2 or v 2 = 1 2 .
We can now prove a weaker version of Theorem 1.22(a), under the additional constraint of genericity. Corollary 6.22. Let x ∈ (C * ) Z 3 be a coherent solution of the Kashaev equation, whose restriction to Z 3 init is generic. Then x can be extended tox ∈ (C * ) L satisfying the Khexahedron equations.
Proof. Let x ∈ (C * ) Z 3 be a coherent solution of the Kashaev equation, whose restriction to Z 3 init is generic. By Lemma 6.21, there exists an arrayx ∈ (C * ) L satisfying the Khexahedron equations that agrees with x on Z 3 {0,1,2,3,4,5} . Let x ′ be the restriction ofx to (As x is generic, x must satisfy condition (1.20), and so K C v (x) = 0 for all unit cubes C in Z 3 and vertices v ∈ C. Hence, the denominator of this rational expression is nonzero.) Hence, x ′ = x, as desired.
Proof of Theorem 1.22(a). We need to loosen the genericity condition in Corollary 6.22 to the conditions that x satisfies (1.20) and has nonzero entries.
Let x ∈ (C * ) Z 3 be a coherent solution of the Kashaev equation satisfying (1.20). Let A j = [−j, j] 3 ∩ Z 3 and B j = [−j, j] 3 ∩ L for j ∈ Z ≥0 . We claim that if there exist x j ∈ (C * ) B j satisfying the K-hexahedron equations that agree with x on A j for all j, then there existsx ∈ (C * ) L satisfying the K-hexahedron equations that agrees with x on Z 3 . Construct an infinite tree T as follows: • The vertices of T are solutions of the K-hexahedron equation indexed by B j that agree with x on A j (over j ∈ Z ≥0 ). • Add an edge betweenx j ∈ (C * ) B j andx j+1 ∈ (C * ) B j+1 ifx j+1 restricts tox j . Thus, T is an infinite tree in which every vertex has finite degree. By König's infinity lemma, there exists an infinite pathx 0 ,x 1 , . . . in T withx j ∈ (C * ) B j . Thus, there exists x ∈ (C * ) L restricting tox j for all j ∈ Z ≥0 , sox is a solution of the K-hexahedron equations that agrees with x on Z 3 .
Given j ∈ Z ≥0 , we claim that there existsx ∈ (C * ) B j satisfying the K-hexahedron equations that agree with x on A j . It is straightforward to show that there exists a sequence x 1 , x 2 , · · · ∈ (C * ) Z 3 of coherent solutions of the Kashaev equation that converge pointwise to x whose restrictions to Z 3 init are generic. By Corollary 6.22, there existx 1 ,x 2 , · · · ∈ (C * ) L satisfying the K-hexahedron equations such thatx i restricts to x i . However, the sequencex 1 ,x 2 , . . . does not necessarily converge (see Proposition 1.23 below). Letx ′ 1 ,x ′ 2 , · · · ∈ (C * ) B j be the restrictions ofx 1 ,x 2 , . . . to B j . There exists a subsequence ofx ′ 1 ,x ′ 2 , . . . that converges to somex ∈ (C * ) B j . (For each s ∈ B j \ A j , we can partition the sequencex ′ 1 ,x ′ 2 , . . . into two sequences, each of which converges at s. Because B j is finite, the claim follows.) The arrayx must satisfy the K-hexahedron equations and agree with x on A j , so we are done.
We shall now work towards a proof of Theorem 1.23.
Lemma 6.23. Letx ∈ (C * ) L be a solution of the K-hexahedron equations. Letx init ∈ (C * ) L init denote the restriction ofx to L init . Let t = (t s ) ∈ {−1, 1} Z 3 be in the kernel of ψ. Then (t ·x init ) ↑L = (y s ) s∈L , where Proof. This follows from Lemma 6.18 and Proposition 6.3. Lemma 6.24. Letx ∈ (C * ) L be a solution of the K-hexahedron equations. Letx init ∈ (C * ) L init denote the restriction ofx to L init . For t ∈ {−1, 1} Z 3 , the following are equivalent: •x and (t ·x init ) ↑L agree on Z 3 ; • t is in the kernel of ψ (see Proposition 6.16).
Proof. If t is in the kernel of ψ, thenx and (t ·x init ) ↑L agree on Z 3 by Lemma 6.23.
If t is not in the kernel of ψ, let u = (u s ) s∈Z 3 +( Proof of Theorem 1.23. Letx init ∈ (C * ) L init denote the restriction ofx to L init .

Coherence for Cubical Complexes
In this section, we generalize Proposition 1.8 and Theorem 1.22 from Z 3 to certain classes of 3-dimensional cubical complexes. Proposition 1.8 generalizes to arbitrary 3dimensional cubical complexes embedded in R 3 (see Proposition 7.1), while Theorem 1.22(b) generalizes to directed cubical complexes corresponding to piles of quadrangulations of a polygon (see Proposition 7.3). Theorem 1.22(a) does not hold for arbitrary directed cubical complexes corresponding to piles of quadrangulations of a polygon. It turns out that an additional property of a cubical complex is required, which we call comfortable-ness. This property is satisfied by the standard tiling of R 3 with unit cubes, as well as by cubical complexes corresponding to piles of ♦-tilings of P n (see Proposition 7.8). Let κ be the directed cubical complex corresponding to a pile of quadrangulations of a polygon. In Theorems 7.10-7.11, we show that Theorem 1.22(a) holds for κ if and only if κ is comfortable. The proof of Theorem 7.10 is nearly identical to the proof of Theorem 1.22(a) in Section 6.
First, we note that Proposition 1.8 generalizes to arbitrary 3-dimensional cubical complexes embedded in R 3 as follows: Proposition 7.1. Let κ be a 3-dimensional cubical complex embedded in R 3 . Suppose that x = (x s ) s∈κ 0 satisfies the Kashaev equation. Then for any interior vertex v ∈ κ 0 (see • the first product is over 3-dimensional cubes C incident to the vertex v, • the second product is over 2-dimensional faces S incident to the vertex v, and • v, v 1 , v 2 , v 3 are the vertices of such a face S listed in cyclic order.
Proof. The proof is almost identical to the proof of Proposition 1.8 in Section 6.
Remark 7.2. With Proposition 7.1 in mind, we can think of the notion of coherence from Definition 3.17 as follows. Let T = (T 0 , . . . , T ℓ ) be a pile of quadrangulations of a polygon with κ = κ(T). Start with an arbitrary array x init indexed by κ 0 (T 0 ) whose entries are "sufficiently generic." We want to extend x init to an array x indexed by κ 0 that is a coherent solution of the Kashaev equation. Building x inductively, suppose we have defined the values of x at κ 0 (T 0 , . . . , T i−1 ), and we need to define the value x w of x at the new vertex w in T i . Let C ∈ κ 3 be the cube corresponding to the flip between T i−1 and T i , and let v be the bottom vertex of C, i.e., let v be the unique vertex in T i−1 but not T i . In order that x continue to satisfy the Kashaev equation, there are 2 possible values for x w , say a and b, so that K C (x) = 0. If the vertex v is in T 0 , i.e., v is not an interior vertex of κ, then we can either set x w = a or x w = b, and x will continue to be a coherent solution of the Kashaev equation. Now, suppose v is not in T 0 , i.e., v is an interior vertex of κ. Because we have chosen x init to be "sufficiently generic," the value of C∋v K C v (x) depends on whether we set x w = a or x w = b. Proposition 7.1 tells us that for one of the 2 possible values, say x w = a, equation (3.6) holds, while for the other value, x w = b, the following equation holds: Hence, the condition of coherence tells us which of the 2 solutions is the "correct" one when v is an interior vertex of κ.
We now prove the following generalization of Theorem 1.22(b). Proof. The proof follows almost exactly the same as the proof of Theorem 1.22(b). For an interior vertex v of κ, there is exactly one cube C for which v is the top vertex, and exactly one cube C for which v is the bottom vertex. Let x be the restriction ofx to κ 0 . By Lemma 6.2, taking the product over the cubes incident to v, so the restriction ofx to x is a coherent solution of the Kashaev equation.
The following statement generalizes Theorem 1.9: In order for a converse of Proposition 7.3 (equivalently, a generalization of Theorems 1.22(a) and 3.19(a)) to hold, one must impose an additional condition on the underlying cubical complexes; see Definition 7.6 below. Definition 7.6. Let κ be a three-dimensional cubical complex that can be embedded into R 3 , cf. Definition 2.2. (While this embeddability condition can be relaxed, it is satisfied in all subsequent applications. In fact, κ will always be the cubical complex associated to a pile of quadrangulations.) Let ∼ be the equivalence relation on κ 2 generated by the equivalences s 1 ∼ s 2 for all pairs (s 1 , s 2 ) involving opposite faces of some 3-dimensional cube in κ 3 . Let κ denote the set of equivalence classes under this equivalence relation. Denote by [s] ∈ κ the equivalence class of s ∈ κ 2 . By analogy with Definition 6.13, denote by ψ κ : {−1, 1} κ → {−1, 1} κ 3 the map sending an array t = (t [s] ) [s]∈κ to the array ψ κ (t) = (u C ) C∈κ 3 defined by , where a, b, c are representatives of the three pairs of opposite 2-dimensional faces of C. We say that the cubical complex κ is comfortable if the following statements are equivalent for every u = (u C ) ∈ {−1, 1} κ 3 : (C1) u is in the image of ψ κ ; (C2) for every interior vertex v ∈ κ 0 (cf. Definition 2.2), we have C∋v u C = 1, (7.4) the product over 3-dimensional cubes C ∈ κ 3 containing v. By Lemma 6.19, the standard tiling of R 3 by unit cubes is comfortable.
We next state four results (Propositions 7.8-7.9 and Theorems 7.10-7.11) which the rest of this section is dedicated to proving. The reader may want to review Definitions 2.6-2.7 before proceeding with the following proposition.
Proposition 7.8. Let T be a pile of quadrangulations of a polygon. Suppose that the divide associated to each quadrangulation in T is a pseudoline arrangement. Then κ = κ(T) is comfortable. In particular, if T is a pile of ♦-tilings of the polygon P n , then κ = κ(T) is comfortable.  Figure 22. The divide associated to each quadrangulation T i is a pseudoline arrangement. Hence, by Proposition 7.8, for any pile T i including T i , κ(T i ) is comfortable. Choose T i , so that we can associate the vertices of κ(T i ) with {−j, . . . , j} 3 , so that ∞ j=1 κ 0 (T i ) = Z 3 . Repeating the König's infinity lemma argument from the end of the proof of Theorem 1.22(a), Theorem 7.10 implies Theorem 1.22. quadrangulation T 1 of R 1 quadrangulation T 2 of R 2 Figure 22. The quadrangulations T j of regions R j described in Remark 7.12.
The rest of this section is dedicated to proving Propositions 7.8-7.9 and Theorems 7.10-7.11.
We can now prove Proposition 7.8 in the special case where T be a pile of ♦-tilings of P n . Lemma 7.14. Let T be a pile of ♦-tilings of P n . Then κ = κ(T) is comfortable.
Proof. Labeling the vertices of κ by subsets of [n] (as in Section 3), we can label the cubes in κ 3 by 3-element subsets of [n] by taking the symmetric difference of the labels of any opposite vertices in the cube. Note that we can extend T to a longer pile T ′ so that for every A ∈ [n] 3 , at least one cube of κ(T ′ ) is labeled by A. Hence, by Proposition 7.13, it suffices to prove the theorem under the additional assumption that each set in [n] 3 labels at least one cube in κ 3 .
Let A 1 be the set of u ∈ {−1, 1} κ 3 satisfying (C1), and A 2 be the set of u satisfying (C2). Because A 1 ⊆ A 2 , it suffices to show that |A 1 | ≥ |A 2 | in order to prove that A 1 = A 2 . We claim that both A and B have size 2 ( n−1 2 ) . First, we claim that |A 1 | ≥ 2 ( n−1 2 ) . Identify each element S ∈ κ with a 2-element subset of [n] by taking the symmetric difference of the labels of any pair of opposite vertices of any tile in S. Note that if u = ψ κ (t), and a cube C labeled by {i, j, k}, then u C = t {i,j} t {i,k} t {j,k} . Define a map of vector spaces f : If we fix t {1,2} = · · · = t {1,n} = 1, then u {1,j,k} = t {j,k} , so the rank of f is at least the number of 2-element subsets of {2, . . . , n}, i.e., n−1 2 . Hence, it follows that |A 1 | ≥ 2 ( n−1 2 ) . Thus, in order to prove the proposition, we must show that |A 2 | ≤ 2 ( n−1 2 ) . Note that there are n−1 2 vertices in the interior of any ♦-tiling of P n . In choosing u satisfying (C2), we can make an arbitrary choice of sign for any cube that shares its bottom vertex with T 0 , but the signs of the remaining cubes is determined by condition (C2). Hence, because at most n−1 2 cubes can share their bottom vertices with T 0 (the bottom of a cube cannot be on the boundary of T 0 ), there are at most 2 ( n−1 2 ) such u satisfying condition (C2), proving our claim.
We can now prove Proposition 7.8 in its full generality.
Proof of Proposition 7.9. We describe a pile T = (T 0 , . . . , T 8 ) of quadrangulations of a square such that κ = κ(T) is not comfortable. Let T 0 be as in Figure 23. It is easier to understand this example by looking at the divides associated to T 0 , . . . , T 8 , displayed in Figure 24. Note that the divides associated to these quadranguations are not pseudoline arrangements. Note that κ has no interior vertices. Hence, every u ∈ {−1, 1} κ 3 satisfies (C2). However, it is not difficult to check that if u satisfies (C1), then the sign on a given cube is determined by the sign on the other 7. Hence, κ is not comfortable.
The rest of this section is dedicated to the proofs of Theorems 7.10-7.11. Definition 7.15. Let T = (T 0 , . . . , T ℓ ) be a pile of quadrangulations of a polygon, with κ = κ(T), and x = (x s ) s∈κ 0 . We say that x is generic if for all extensions of x init (the restriction of x to κ 0 (T 0 )) to an arrayx indexed by κ 02 (T) satisfying the K-hexahedron equations, the entries ofx are all nonzero.
Proof. The proof follows directly from Lemma 6.11.
Lemma 7.18. Let T = (T 0 , . . . , T ℓ ) be a pile of quadrangulations of a polygon, with κ = κ(T). Let x and x ′ be generic and distinct coherent solutions of the Kashaev equation, both indexed by κ 0 , such that x and x ′ agree at κ 0 (T 0 ). Let i be the minimum value such that x and x ′ do not agree at κ 0 (T i ). Then the cube C i shares its bottom vertex with T 0 .
Proof. Assume (for contradiction) that C i doesn't share its bottom vertex with T 0 . Then the bottom vertex of C i must be an interior vertex of κ. Hence, by the coherence and genericity of x and x ′ , the values of x and x ′ are uniquely determined by their values at κ 0 (T 0 ), . . . , κ 0 (T i−1 ), which are the same for x and x ′ . Hence, x and x ′ agree at the top vertex of C i , and thus agree at κ 0 (T i ), a contradiction.
We can now prove a weaker version of Theorem 7.10, under the additional constraint of genericity.
Corollary 7.19. Let T be a pile of quadrangulations of a polygon such that κ = κ(T) is comfortable. Any generic, coherent solution of the Kashaev equation x = (x s ) s∈κ 0 can be extended tox = (x s ) s∈κ 02 satisfying the K-hexahedron equations. Figure 24. The divides associated to the quadrangulations T 0 , . . . , T 8 from the proof of Proposition 7.9.
(For example, choose u so that u C i = −1, and u C = 1 for all other cubes C that share a bottom vertex with T 0 . Then the remaining values are determined by condition (C2).) Because κ is comfortable, there exists t i such that ψ κ (t i ) = u, as desired.
Proof of Theorem 7.10. We need to loosen the condition that x is generic from Corollary 7.19 to the conditions that x has nonzero components and satisfies condition (3.7). Let x ∈ (C * ) κ 0 be a coherent solution of the Kashaev equation with nonzero components that satisfies condition (3.7). It is straightforward to show that there exists a sequence x 1 , x 2 , · · · ∈ (C * ) κ 0 of generic, coherent solutions of the Kashaev equation that converge pointwise to x. By Corollary 7.19, there existx 1 ,x 2 , · · · ∈ (C * ) κ 02 satisfying the Khexahedron equations such thatx i restricts to x i . There exists a subsequence ofx 1 ,x 2 , . . . that converges to an arrayx. (For each s ∈ κ 2 (T), we can partition the sequencẽ x 1 ,x 2 , . . . into two sequences, each of which converges at s. Because κ 2 (T) is finite, the claim follows.) The arrayx must satisfy the K-hexahedron equations and restrict to x, so we are done.
In order to complete the proof of Theorem 7.11, we will need the following technical lemma.
Lemma 7.20. Let T = (T 0 , . . . , T ℓ ) be a pile of quadrangulations of a polygon such that κ(T) is not comfortable, but κ(T 0 , . . . , T ℓ−1 ) is comfortable. Let C ℓ be the cube of κ corresponding to the flip from T ℓ−1 to T ℓ . (a) Let v be the bottom vertex of the cube C ℓ , i.e., let v be the vertex of Proof. Let κ ′ = κ(T 0 , . . . , T ℓ−1 ). Let Because a 1 , a 2 , b 1 , b 2 enumerate the elements of vector fields over F 2 , all four quantities must be powers of 2. Because κ ′ is comfortable, a 1 = a 2 . Because κ is not comfortable, b 1 < b 2 . It is clear that b 1 ≤ 2a 1 and b 2 ≤ 2a 2 . Hence, it follows that a 1 = a 2 = b 1 = b 2 /2.
Assume (for contradiction) that v is not in T 0 , so v is in the interior of κ. But then if u = (u C ) C∈κ 3 satisfies (C2), so a 2 = b 2 , a contradiction. Hence, we have proved (a).
Proof of Theorem 7.11. Without loss of generality, we assume that κ(T 0 ,. . . ,T ℓ−1 ) is comfortable. (If not, let m be minimum so that κ(T 0 , . . . , T m ) is not comfortable, but κ(T 0 , . . . , T m−1 ) is comfortable. If we can prove the theorem for κ(T 0 , . . . , T m ), it follows that it holds for κ(T).) We now construct an array x satisfying the desired conditions. Let C be the cube of κ corresponding to the flip from T ℓ−1 to T ℓ , and let v be the top vertex of C (i.e., v is the new vertex in T i ). Choose arbitrary positive values for x init . Extend x init to x by the positive Kashaev recurrence until we reach v, where we choose the other value such that K C (x) = 0.
By construction, x restricted to κ(T 0 , . . . , T ℓ−1 ) satisfies the positive Kashaev recurrence, and hence is a coherent solution of the Kashaev equation. By Lemma 7.20(a), no vertices of C are in the interior of κ. Hence, x is a coherent solution of the Kashaev equation.
Next, we show that x cannot be extended to an array indexed by κ 02 satisfying the K-hexahedron equations. Let x pK be the array satisfying the positive Kashaev recurrence that restricts to x init at T 0 (so x pK agrees with x everywhere except v). Letx pK be an extension of x pK to κ 02 satisfying the K-hexahedron equations. Assume (for contradiction) that there exists t ∈ {−1, 1}κ 2 such thatx(t·(x pK ) 0 ) restricts to x. Hence, by Lemma 7.17, ψ κ (t) has value −1 at C, and value 1 everywhere else. But Lemma 7.20(b) says that array is not in the image of ψ κ , a contradiction. Hence, no such t exists, so x cannot be extended to an array indexed by κ 02 satisfying the K-hexahedron equations.

Proofs of Corollary 3.23 and Theorem 3.26
This section contains the proofs of Corollary 3.23 and Theorem 3.26. We use the following lemma in proving Corollary 3.23. Proof. Note that due to the homogeneity of the Kashaev equation and the coherence equations (equation (3.17)), we can rescale the components of x to obtain a standard array. Hence, we can assume that x is standard.
By Theorem 3.19(a), we can extend x to an arrayx indexed by κ 02 satisfying the K-hexahedron equations. Note that we can choose a sequencex 1 ,x 2 , . . . of standard arrays indexed by κ 02 satisfying the K-hexahedron equations converging tox such that the restriction ofx i to κ 02 (T ) for any tiling T in T is generic. By Theorems 3.9 and 3.12, there exist symmetric n × n matrices such thatx i =x κ(T) (M i ). Hence, the components ofx i at s 1 and s 2 must agree, so the components ofx at s 1 and s 2 must agree.
Proof of Corollary 3.23. The first bullet point implies the second two by Corollary 3.14, and it is obvious that the third implies the second. Thus, we need to show that the second bullet point implies the first.
Next, suppose T = (T 0 , . . . , T ℓ ) is a pile of ♦-tilings of P n in which every I ⊆ [n] labels at least one vertex of κ(T), and x = x κ(T) (x) is a coherent solution of the Kashaev equation. Let T ′ = (T 0 , . . . , T ℓ , . . . , T ℓ ′ ) be an extension of T where T ′ contains the tiling T min,n . By Lemma 8.1, x κ(T ′ ) (x) is the unique extension of x to κ 0 (T ′ ). By Theorem 3.19(a), there exists an arrayx indexed by κ 02 (T ′ ) extending x κ(T ′ ) (x) that satisfies the K-hexahedron equations. By Theorem 3.10, Proposition 3.11, and Corollary 3.15 (all of which are due to Kenyon and Pemantle [4]), there exists a unique symmetric matrix M such that x =x κ(T ′ ) (M), so M satisfies condition (3.8).
Next, we shall work towards a proof of Theorem 3.26. where the plus sign appears on the right-hand side of equation (8.2) if either • i, k ∈ I and j ∈ I, or • j ∈ I and i, k ∈ I, and the minus sign appears otherwise.
• Writing σ i−1 = (β 1 , . . . , β ( n 3 ) ), β j = α j for j = i + n−1 2 , . . . , n 3 . Hence, the third bullet point holds. Because β i+( n−1 2 ) = α i+( n−1 2 ) and the flip between κ(T i−1 ) and κ(T i ) consists of the inclusion of {1} ∪ α i+( n−1 2 ) to the inversion set, the fourth bullet point follows. For i = 1, . . . , n 3 − n−1 2 , write σ i−1 = (β 1 , . . . , β ( n 3 ) ). We want to obtain σ i . Write it is straightforward to check that σ i is an admissible permutation with the desired properties. + 1 elements for a representative for the smallest element of the fourth higher Bruhat order. See [9] or [11] for further discussion of higher Bruhat orders. i.e., we "flip" the index of each variable in its ith coordinate. The action of π a,i extends from z [a] to the polynomial ring C[z [a] ]. Given an array x = (x s ) ∈ C Z d and integer vectors v ∈ Z d , a ∈ Z d ≥0 , and α ∈ {−1, 1} d , we denote by x v+[a⊙α] ∈ C [a] the array whose entries are Thus, given a polynomial f ∈ C[z [a] ], the number f (x v+[α⊙a] ) ∈ C is obtained by setting z i = x v+α⊙i for each variable z i for i ∈ [a]. We say that Proposition 9.2. Let a = (a 1 , . . . , a d ) ∈ Z d ≥1 , and a polynomial f ∈ C[z [a] ] satisfy the following conditions: (9.2.1) f is invariant under the action of π a,i for i = 1, . . . , d; (9.2.2) f has degree 2 with respect to the variable z a ; as a quadratic polynomial in z a , f has discriminant D which factors as a product is invariant under the action of π a−1 i ,j for j = 1, . . . , d. Then for any x = (x s ) ∈ C Z d satisfying f , we have, for all v ∈ Z d : ). (9.7) appearing on the left-hand side of (9.6) run over all boxes of size a 1 × · · · × a d containing v + [a − 1]. The subscripts v−(a−1)⊙α+[(1+2α)⊙a] appearing on the right-hand side of (9.6) run over i = 1, . . . , d and boxes of size In particular, when a 1 = · · · = a d = 1, all of these products are over boxes of a certain size containing the vertex v. For example, in the case a = (1, 2) (like in Proposition 5.7), the boxes we are considering on the left-hand side of (9.6) are given in the top row of Figure 19, while the boxes we are considering on the right-hand side of (9.6) are given in the bottom row of Figure 19.
Definition 9.9. Given a = (a 1 , . . . , a d ) ∈ Z d ≥1 and 1 ≤ i ≤ d, let denote the set of boxes of size a 1 × · · · × a i−1 × (a i − 1) × a i+1 × · · · × a d in Z d . Set We want to develop a generalization of the K-hexahedron equations for arrays indexed by Z d ∪F a . Suppose that f ∈ C[z [a] ] is a polynomial satisfying conditions (9.2.1)-(9.2.2), with the polynomials f 1 , . . . , f d from condition (9.2.2) fixed. Let g and h be the coefficients of z 2 a and z a in f , viewed as a polynomial in z a . We consider arraysx = (x s ) ∈ C Z d ∪F a such that for all v ∈ Z d , (9.25) where r 1 , . . . , r d are some rational functions in the variables z s for s ∈ [a * ]. Note that if x satisfies conditions (9.25) and (9.27), then by the quadratic formula, its restriction x = (x s ) s∈Z d satisfies f . In the following definition, we formulate the properties that our tuple of rational functions (r 1 , . . . , r d ) should have in order for the subsequent developments to follow.
Definition 9.11. Let a = (a 1 , . . . , a d ) ∈ Z d ≥1 , and let f ∈ C[z [a] ] be a polynomial satisfying conditions (9.2.1)-(9.2.2). Fix the polynomials f 1 , . . . , f d from condition (9.2.2). Let g be the coefficient of z 2 a in f , viewed as a polynomial in z a . For i = 1, . . . , d, let r i = p i q i be rational functions in the variables z s for s ∈ [a * ], with p i , q i polynomials in these variables. We say that (r 1 , . . . , r d ) is adapted to (f ; f 1 , . . . , f d ) if there exist signs β 1 , . . . , β d ∈ {−1, 1} such that the following properties hold for i = 1, . . . , d: • the denominator q i of r i is of the form • for all arraysx = (x s ) ∈ C Z d ∪F a satisfying (9.25), (9.27), and q i (x v+s : s ∈ [a * ]) = 0 for all v ∈ Z d ; (9.29) g(x v+[a] ) = 0 for all v ∈ Z d ; (9.30) the following condition holds: Note that one can obtain a tuple (r 1 , . . . , r d ) adapted to (f ; f 1 , . . . , f d ) by choosing the signs β 1 , . . . , β d ∈ {−1, 1} and using condition (9.25) to replace all instances of x v+a in (9.31).
The proof of Lemma 9.17 relies on the following lemma.
Example 9.20. Let us continue with Examples 9.4 and 9.13. Following the argument in the proof of Lemma 9.17, it can be shown that γ α = 1 if α = ±1, and γ α = −1 otherwise. Note that this fact is equivalent to Lemma 6.2.
Example 9.23. Let us continue with Examples 9.7 and 9.16. Following the argument in the proof of Lemma 9.17, it can be shown that γ α = 1 if α = ±1, and γ α = −1 otherwise.
We now state the main theorem of this section.
] be a polynomial that is irreducible over C and satisfies conditions (9.2.1)-(9.2.2). Fix the polynomials f 1 , . . . , f d from condition (9.2.2). Let g and h be the coefficients of z 2 a and z a in f , viewed as a polynomial in z a . Let (r 1 , . . . , r d ) be a tuple of rational functions in the variables z s for s ∈ [a * ] that is adapted to (f ; f 1 , . . . , f d ). Let (γ α ) α∈{−1,1} d be the propagation signs corresponding to (f ; f 1 , . . . , f d ; r 1 , . . . , r d ). Then x can be extended to an arrayx = (x s ) ∈ C Z d ∪F a satisfying (9.25)-(9.27). (b) Conversely, ifx = (x s ) ∈ C Z d ∪F a satisfies conditions (9.25)-(9.27) and (9.29)-(9.30), then the restriction ofx to Z d satisfies f and the condition (9.56).
Before we prove Theorem 9.24, we first prove Proposition 9.25. Proposition 9.25 follows from the lemma below.
Definition 9.33. We say that an arrayx init indexed by Z d a,init ∪ F a init satisfying condition (9.27) is generic if there exists an extension ofx init to an arrayx indexed by Z d ∪F a satisfying equations (9.25)-(9.27) where the restriction ofx to Z d satisfies conditions (9.54)-(9.55). Similarly, we say that an array x init indexed by Z d a,init is generic if every extension of x init to an arrayx init indexed by Z d a,init ∪ F a init satisfying condition (9.27) is generic.
Definition 9.34. Letx init be a generic array indexed by Z d a,init ∪ F a init satisfying condition (9.27). We denote by (x init ) ↑Z d ∪F a the unique extension ofx init to Z d ∪F a where (x init ) ↑Z d ∪F a satisfies equations (9.25)-(9.27).
The next lemma generalizes Lemma 6.11. Proof. Note that = 0, (9.69) so y a = x a . By (9.48),  The next lemma generalizes Lemma 6.18. Lemma 9.39. Letx init be a generic array indexed by Z d a,init ∪ F a init satisfying condition (9.27). Let t ∈ {−1, 1} F a , and u = (u s ) s∈Z d +a/2 = ψ(t). Let (x init ) ↑Z d ∪F a = (x s ) s∈Z d ∪F a , and (t ·x init ) ↑Z d ∪F a = (y s ) s∈Z d ∪F a . Suppose v ∈ Z d {a,a+1,... } satisfies the condition that u w−a/2 = 1 for all w ∈ Set t = (t s ) ∈ {−1, 1} F a . It is straightforward to check that ψ(t) agrees with u at S. Hence, because ψ(t) and u both satisfy condition (9.75), it follows that u = ψ(t).
The next lemma generalizes Lemma 6.21. Assume thatx satisfying f , and, moreover, its restriction to Z d a,init is generic. Then there exists an arrayx indexed by Z d ∪F a satisfying equations (9.25)-(9.27) and extendingx.
Proof. For i = a, . . . , a + d − 1, we will show by induction on i that there exists an arrayx init indexed by Z d a,init ∪ F a init satisfying (9.27) such that (x init ) ↑Z d ∪F a agrees witĥ x = (x s ) s∈Z d {0,...,a+d−1} on Z d {0,...,i} . Letx ′ init be an array indexed by Z d a,init ∪ F a init satisfying (9.27) such that (x ′ init ) ↑Z d ∪F a = (y s ) s∈Z d ∪F a agrees withx on Z d {0,...,i−1} . (For i = a, we can obtainx ′ init be taking an arbitrary extension of x init to Z d a,init ∪ F a init satisfying condition (9.27). For i > a, we have shown thatx ′ init exists by induction.) Choosẽ u = (u s ) ∈ {−1, 1} Z d {a,...,a+d−1} −a/2 so that • u s−a/2 = 1 if s ∈ Z d {i} and x s = y s ; • u s−a/2 = −1 if s ∈ Z d {i} and x s = y s ; • u s−a/2 = 1 if s ∈ Z d {j} for 0 ≤ j < i. Extendũ to u = (u s ) s∈Z d +a/2 by condition (9.27). By Lemma 9.40, there exists t ∈ {−1, 1} F a such that u = ψ(t). Setx init = t ·x ′ init . Then by Lemma 9.39, (x init ) ↑Z d ∪F a agrees withx on Z d {0,...,i} , as desired. We can now prove a weaker version of Theorem 9.24(a), under the additional constraint of genericity.
paper. I am also grateful to Dmitry Chelkak for pointing out the connection with sholomorphicity, and to Thomas Lam and John Stembridge for helpful discussions and editorial suggestions.