{'fraction_non_alphanumeric': 0.050434306283746774, 'fraction_numerical': 0.020228499882619922, 'mean_word_length': 4.134391321815991, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 27, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the lattice-geometry and birational group of the six-point multi-ratio equation

The inherent self-consistency properties of the six-point multi-ratio equation allow it to be considered on a domain associated with a T-shaped Coxeter-Dynkin diagram. This extends the KP lattice, which has A_N symmetry, and incorporates also KdV-type dynamics on a sub-domain with D_N symmetry, and Painleve dynamics on a sub-domain with affine-E8 symmetry. More generally, it can be seen as a distinguished representation of Coble's Cremona group associated with invariants of point sets in projective space.


Introduction
The six-point multi-ratio equation, is best known in the theory of integrable systems as a discrete analogue of the Kadomtsev-Petviashvili (KP) equation. The multi-ratio form emerges naturally as superposition principle for the Schwarzian KP equation [1], and in the incidence-geometry approach to the KP hierarchy [2]. It is directly related to earlier forms [3,4]. A lattice with A N symmetry has played a central role in several contemporary developments in the theory of the discrete KP systems [5,6,7,8]. One reason to consider the A N lattice for (1) is based purely on its symmetry properties, and the purpose of this paper is to present an extension which is natural in the same sense. The extended domain is encoded in the Coxeter-graph of Figure 1. The situation is that sub-groups correspond to sub-domains, and for instance the A N KP lattice corresponds to case k = 0. Truncations satisfying the condition [9] i + j + k + 1 ≥ ijk − 1, imply the group is finite or affine, corresponding to inequality or equality respectively. Such truncation is the simplest way to pick out sub-domains which support integrable dynamics in a familiar sense. Besides the KP case, two other sub-domains are of particular significance. In the case k = j = 1, the system is equivalent to the discrete Schwarzian Korteweg-de Vries (KdV) system [10] in multi-dimensions [11,12]. In the case k = 1, j = 2, i = 6, it gives a representation of the elliptic Painlevé system [13,14].  [9]. The graph encodes a particular figure, the ringed node distinguishes it from other figures having the same symmetry.
These cases establish the basic significance of the considered domain: within it, the prototypes of integrable systems in one, two and three dimensions, fit together on an equal footing. It is therefore an example that allows to understand a natural unity between the dimensions encoded in a generalised lattice geometry. It is complementary to the usual hierarchical view; instead of placing the lower dimensional integrable systems as reductions of those in higher dimension, they are seen as co-operating parts of a larger whole.
This extends the similar relation between quadrirational Yang-Baxter maps (KdV-type dynamics) and Painlevé equations [15,16] to also include the KP case.
The instance k = 1 of the full system, formulated as a birational group, constitutes a representation of Coble's group of Cremona transformations associated with invariants of general point sets in projective space [17,18]. This group is the remarkable object, characterised geometrically, that takes a central role in the geometric framework of the Painlevé equations [13,19,20]. The representation which emerges here takes the form of an isotropic lattice system governed by (1). It is a representation which therefore establishes a rather direct contact between the infinite-degree-of-freedom integrable systems and the geometric framework of the Painlevé equations.
The structure of the paper is as follows. In Section 2 the domain and a canonical initial-value-problem are defined in terms of the T-shaped Coxeter graph with distinguished node. In Section 3, two local properties are singledout as sufficient for the global consistency. One is a well-known property connected with Desargues' theorem in the geometric setting for (1). The other is discussed in Section 4, it can be understood naturally by interpretation in the Möbius group, which establishes also a broader context of Caley-Bezout-type correspondences. The symmetry properties are the original motivation to extend previously known domains for (1), this is described in Section 5 under the title Isotropy. Identification with the discrete KP system in the A N case is made in Section 6, and the D N case is connected with the KdV equation in Section 7, this involves a change of coordinates, and in the latter case also a partial integration. In Section 8, connection with Coble's Cremona group is established by a change of variables.

Definition of the domain
The T-shaped graph in Figure 1 was considered by Coxeter in [9] (see Chapter XI as well as the epilogue), it emerges from the framework allowing a unified description of the Gosset polytopes and tessellations [21]. In that setting, the graph encodes a figure, with vertices, edges, faces, and so on.
Elements of the domain for (1) will be defined no differently than the elements of the figure, the association between the two is as follows: variables ↔ vertices, pair-triples ↔ octahedral cells. ( The pair-triple is the natural stencil for (1) because it shares its invariance under permutations of the variables.
Definition 2.1. Let integers i, j be positive, and k be non-negative. Denote by G the finitely presented group consisting of generators and relations encoded in Figure 1, where generators are identified with nodes as follows: Let H denote the subgroup generated by S \ {r k }, corresponding to deletion of the distinguished node in Figure 1. The variables of the domain are assigned to the left cosets of H: variables ↔ gH : g ∈ G.
A basic pair-triple is formed by the action of subgroup s 1 , r 0 , t 1 on the coset X = r 0 · · · r k H, The remaining pair-triples are obtained from (8) by the action of G: pair-triples ↔ {{gr 0 X, gr 0 s 1 t 1 X}, {gs 1 X, gt 1 X}, {gX, gs 1 t 1 X}} : g ∈ G. (9) This definition only involves the vertices and octahedral cells of the associated figure. Other sub-figures are specified in a similar way, for example, edges correspond to pairs of cosets obtained by the action of G from the basic pair {H, r k H}.

Remark 2.2.
When the group is finite, the number N = i + j + k + 1 gives dimension of the Euclidean space in which the associated polytope is found. More generally, the intuitive understanding of this domain as abstract polytope has the caveat of there being three natural ways in which to increment dimension, corresponding to the triple of integers (i, j, k).
Remark 2.4. Amongst other admissible choices, this initial data set is highly symmetric: it is an orbit of the subgroup generated by S \ {r 0 }. It constitutes a third relevant sub-figure extending the associations (3). This is called a compound figure because the associated Coxeter-graph is not connected, it is obtained by deleting the node labeled 0 in Figure 1.
It is the first intention to demonstrate that Definition 2.3 does indeed give a consistent initial-value-problem for equation (1). To proceed, some further definitions are useful. The analysis benefits by considering, not equation (1) itself, but a generic equation with the same stencil. Consistency of the initial-value-problem will mean the following. These definitions establish the setting in which local conditions for the global consistency will be considered.

Birational group and consistency
In this section the initial-value-problem (Definition 2.3) for a generic pair-triple equation (Definition 2.5) is formulated as a birational group. This formulation is used to establish basic conditions for the consistency. The analysis is formal (in essence, combinatorial), singularities of the actions are not considered.
It is natural to think of the initial data set, i.e., those variables of Definition 2.1 associated with cosets (10), as being arranged into arrays: Remark 3.1. The initial data array is finite whenever i, j and k are all finite. The dynamical systems with an infinite number of degrees of freedom correspond to the case when at least one of these integers is considered to be infinite, and in that case the corresponding arrays will have unbounded extent.
The actions are defined on these arrays. (11) to be associated with generators, i.e., elements of S (5), are as follows:
The consistency is characterised in terms of these actions. Proof. This can, in essence, be considered a re-statement of the definition, because the actions of Definition 3.2 are induced by the initial-value-problem. To be convinced requires consideration of the equations Here w(Y ) denotes the variable associated with coset Y in Definition 2.1, so in particular y n = w(Y n ) and y mn = w(Y mn ) for variables (11) associated with cosets (10). What is meant by g in this equation is distinguished by context. On the left-hand-side of (14) g denotes the action associated with elements of S (5) that have been specified in Definition 3.2, whereas on the right-hand-side g is considered as an element of the finitely presented group of Definition 2.1. Equation (14) is easily checked as consistent when g ∈ S by comparing actions in Definition 3.2 with left action on the cosets (10). Consider G to be the finitely presented group as in Definitions 2.1 and 2.3. Any g ∈ G permutes variables, sending variables of the initial data set to some similar subset of variables in the domain. If the pair-triple equation is assumed to be consistent in the sense of Definition 2.6, then the new variables can be expressed as composed rational functions of variables from the initial data set. Through equation (14) this induces an action associated with g on variables (11), and, by construction, the actions induced by the generators are exactly those in Definition 3.2. Associating actions this way gives a representation of G, and in particular the elements of S satisfy the desired relations. Now assume that actions in Definition 3.2 satisfy the desired relations. Through associations (14) this allows to consistently obtain all remaining variables in the domain as composed rational functions of those in (11). And because G acts transitively on pair-triples by construction (Definition 2.1), it also means the imposed equation (1)  Proof. According to Lemma 3.3, the global consistency is demonstrated by showing that actions (12) satisfy the relations encoded in the Coxeter-graph (6). Relations not involving generator r 0 are inherently satisfied. Commutativity of the actions r 2 , . . . , r k , s 2 , . . . , s j , and t 2 , . . . , t i , with r 0 , rely only on the assumed pair-triple symmetry of the imposed equation. Similarly, relation r 2 0 = id is a consequence of the assumed symmetry. It therefore remains to consider the relations (r 0 s 1 ) 3 = (r 0 t 1 ) 3 = (r 0 r 1 ) 3 = id. Consider first the mapping (r 0 r 1 ) 3 . It is clear that it inherently fixes all entries of the array on the left in (11), and so it remains to consider action on the array on the right. Assuming the relations hold in the case (i, j, k) = (1, 1, 1) implies that (r 0 r 1 ) 3 fixes the 2 × 2 sub-array at the intersection of the first two rows and columns. Because s 2 , . . . , s j , and t 2 , . . . , t i inherently commute with r 0 and r 1 , they also commute with (r 0 r 1 ) 3 . But s 2 , . . . , s j , t 2 , . . . , t i permute the remaining entries of the array with entries on row 2 and column 2, and this implies that if (r 0 r 1 ) 3 fixes entries of the 2 × 2 sub-array, it must fix all remaining entries as well. Thus (r 0 r 1 ) 3 = id in general.
Similar arguments apply to the composed mapping (r 0 s 1 ) 3 . First, this composed mapping inherently fixes entries of the array on the left in (11). Second, assuming the relations hold when (i, j, k) = (2, 1, 0), implies that (r 0 s 1 ) 3 also fixes the 3 × 2 sub-array at the intersection of the first three rows and two columns of the array on the right in (11). Finally, the inherent commutativity between the actions s 3 , . . . , s j , t 2 , . . . , t i , and (r 0 s 1 ) 3 , implies that all remaining rows and columns of the array must also be fixed by (r 0 s 1 ) 3 , and therefore (r 0 s 1 ) 3 = id in general.
Clearly if relation (r 0 s 1 ) 3 = id holds for any (i, j, k), it means it also holds when the value of i or j is increased so that i = j. In such case, an additional mapping ω, which fixes the array on the left in (11), but transposes the square array on the right, can also be considered. This corresponds to the diagram automorphism present when i = j, it interchanges the two undistinguished branches of the graph, ωs n = t n ω, n ∈ {1, . . . , i}, it clearly commutes with r 1 , . . . , r k , and commutativity with r 0 follows from the assumed symmetry of the imposed equation. Thus, conjugation by ω shows that (r 0 t 1 ) 3 = id is a consequence of (r 0 s 1 ) 3 = id, and this completes the proof. Theorem 3.4 singles out two local properties required for the global consistency. The first case (i, j, k) = (1, 1, 1) corresponds to the figure of the four-dimensional cross-polytope, it has eight vertices and four octahedral cells. The consistency required is that any two of the imposed equations imply the remaining two. The figure associated with the second kind of consistency, (i, j, k) = (2, 1, 0), is another four-dimensional polytope called the rectified 4simplex, it has ten vertices and five octahedral cells. The consistency requires that any three of the imposed equations imply the remaining two.
For a given pair-triple equation, the two properties are easily investigated by calculation, but for (1) there is a deeper understanding available. For the (2, 1, 0)-consistency this is a well-studied fact inherent from the geometric interpretation [2,22,23], and is discussed in depth in [5,8]. In the following section the (1, 1, 1) consistency will be discussed further.

Connection with Möbius involutions
Unlike the (2, 1, 0) case, the (1, 1, 1) consistency property of (1) is not well known in the context of integrable systems. An algebraic interpretation in terms of the Möbius group offers the desirable intuition about it.
A Möbius involution will transpose any given point in the extended complex plane with a unique other point. Calling such pair of points an orbit of the transformation, the following can be verified.

Proposition 4.1 ([24]). (i) A freely chosen disjoint pair of orbits uniquely determines a Möbius involution. (ii) A Möbius involution with the three orbits (4) exists, if and only if the points in these orbits satisfy constraint (1).
Proof. This follows by considering the linear system for the coefficients of a Möbius involution. It leads to a constraint on three orbits in a determinant form, which is equivalent to (1). Note that in this form the pair-triple symmetry of the equation is visible.
More generally, consider the constraint on N orbits of a Möbius involution for any N > 2. It can be written as N !/3! distinct 3-orbit constraints like (15), but it is clear that all must be satisfied as a consequence of just N − 2 of them. Identifying N = k + 3, this is equivalent to the consistency property corresponding to case i = j = 1 of Definition 2.6, whose associated figure is the cross-polytope in N dimensions. In particular, the (1, 1, 1) consistency of (1) is associated with the constraint on four orbits of a Möbius involution, Remark 4.2. The connection between equation (1) and the Clifford configuration [22,25] has led to the conformal version of the Desargues theorem in relation to the (2, 1, 0)-consistency [8]. Clifford configurations are discussed in terms of Möbius involutions in [26], in that context the Möbius involutions are called half-turns of the inversive plane, and points of an orbit have been called mates. Although a C4 Clifford configuration determines four orbits of a Möbius involution, they are not generic orbits, indeed, only three orbits are needed to determine a unique C4 Clifford configuration. Therefore, despite having the same combinatorics, the (1, 1, 1) consistency of (1) is not itself directly equivalent to existence of the C4 Clifford configuration. It is logical that this can be resolved in a suitably generalised geometric setting.
Here r 1 and r 2 are polynomials of degree n with no common roots, the correspondence is (n − 1)-valued, but its orbits inherently close, forming n-tuples. The roots of r 1 and r 2 constitute two of the orbits, which may therefore be chosen freely, but there is a constraint on any set of three orbits. The present work relates to the case n = 2, and as such is a prequel to [15,16] which investigated the n = 3 family of correspondences. That work was in fact restricted to members of the family that share a single orbit, furthermore the next case n = 4 is the highest value of n for which the three-orbit constraint is rational, and its significance is un-known at this time.

Isotropy
The purpose of this section is to establish a completeness property for the introduced domain, it is a combinatorial notion based on symmetry introduced in [15]. This property is not connected to integrability, instead it is a notion to help decide if the domain of a system has been prematurely truncated.
Recall that in general the domain of a pair-triple equation (Definition 2.5) is considered as a set of variables arranged into pair-triples, and a symmetry of the domain is a permutation of the variables that also permutes the pair-triples. In the present situation the symmetry considerations single out the particular case i = j. Proposition 5.1. When i = j, the pair-triple arrangement of Definition 2.1 is symmetry-complete. That is, every symmetry of every pair-triple is the restriction of a symmetry of the whole arrangement.
Proof. Consider Definition 2.1. Inherent from the way the arrangement is defined, only a single pair-triple needs to be considered, and it is natural to focus on the basic one (8). It is clear that subgroup s 1 , r 0 , t 1 stabilizes this pairtriple, however, this subgroup does not give the full pair-triple (in other words, octahedral) symmetry. This is rectified in the case i = j when the arrangement has an additional symmetry corresponding to the diagram automorphism. Specifically, the finitely presented group G is complemented by an additional generator, ω, where relations involving ω are ω 2 = (r n ω) 2 = id, n ∈ {0, . . . , k}, and ωt n = s n ω, n ∈ {1, . . . , i}. Both G and H are normalized by ω, and therefore the action of conjugation by ω permutes the cosets of H in G. It is straightforward to verify that this action also permutes the pair-triples (9), and therefore gives a symmetry of the arrangement. Furthermore it stabilizes the basic pair-triple (8). Unlike s 1 , r 0 , t 1 , the group ω, r 0 , t 1 gives the complete pair-triple symmetry.

Remark 5.2.
A domain which is symmetry-complete can be called isotropic in a sense appropriate for discrete systems. It is the condition that, viewed from an elementary cell, the remainder of the domain looks the same in all directions.

The A N KP lattice
In this section the A N KP lattice is discussed. Its theory has been developed as the multidimensional extension of the face-centered-cubic lattice in [6,7,8]. Connection with instance k = 0 of Definition 2.1 is made simply by changing to standard coordinates, the canonical initial-value-problem (Definition 2.3) is also re-expressed in these coordinates. It is established that the A N lattice can be be viewed as a symmetry-complete extension of the 'hypercube' domain for (1).

Coordinatisation
In the case k = 0, the figure associated with Coxeter graph of Figure 1 is the jrectified N -simplex, where N = i + j + 1. A natural coordinatisation is available by associating variables with the subsets of size j + 1 in a set of size N + 1.
The symmetry group of this arrangement acts naturally by free permutation of indices 1, . . . , N + 1, which is the point of contact with the coordinatisation given in Definition 2.1. Proof. Consider case k = 0 of Definition 2.1. A consistent and faithful representation of G as free permutation of some set of indices 1, . . . , N + 1 can be introduced by associating actions with the generators as follows: s n : j + 1 − n ↔ j + 2 − n, n ∈ {1, . . . , j}, r 0 : j + 1 ↔ j + 2, t n : j + 1 + n ↔ j + 2 + n, n ∈ {1, . . . , i}. (20) In this representation, H is characterised as the subgroup which stabilises the index set {1, . . . , j + 1}. More generally, the image of this subset by the action of a left coset of H is single-valued, so there is a bijection This gives the coordinatisation of Definition 6.1.
Remark 6.3. It is clear, from the proof of Theorem 3.4, that the (1, 1, 1)consistency of (1) plays no role for consistency on the A N sub-lattice. Sufficiency of the (2, 1, 0)-consistency in this setting was established originally in [8]. It contrasts with the considerations in [7] where consistency on a larger stencil was required, but this is explained by a generalised setting involving no symmetry assumptions of the imposed equations. Also it can be compared with the approach in [5], where the consistency is presented as being inherent from the axioms of projective geometry, so the properties of equations are seen as inherited.

Restriction to hypercube
An important domain for (1) is the hypercube of arbitrary dimension, it is natural in the context of nonlinear superposition [4,1], where commuting Bäcklund transformations are associated with lattice edges. Multi-indexJ is associated with a vertex of the hypercube, and each pairtriple is associated with a cube. More precisely, each pair-triple is associated with a cube that has a distinguished diagonal. Such restriction of the cube does not have the full pair-triple symmetry, and it is therefore easily seen that the arrangement of Definition 6.4 is not symmetry-complete (recall Section 5). The purpose of this sub-section is to establish that the A N lattice is a symmetrycomplete extension of it: . . , 2M − 2}, the key restriction is to subsets whose intersection with A is sequential: It is clear that each such J is uniquely determined by the part not contained in A, J \ A. IdentificationJ = J \ A defines a correspondence between variables (18) restricted by (26), and the variables (23). The corresponding restriction on the set of pair-triples (19) is that participating subsets K and {a, b, c, d} satisfy the additional constraints Notice that this implies exactly one element of {a, b, c, d} is contained in A. It can then be verified, identifyingK = K \ A and {n, m, l} = {a, b, c, d} \ A, that there is a one-to-one correspondence between pair-triples (19) restricted by (27), (28), and the pair-triples (24).

The D N KdV lattice
This section considers a D N sub-lattice, corresponding to case j = k = 1 of Definition 2.1. The purpose is to establish that equation (1) imposed on this domain is equivalent to the discrete KdV equation [27,28] in its well-known Schwarzian guise [10,29,30] and multidimensional setting [11,12]. The associated figure is the demihypercube in dimension N = i + 3. The transition to natural demihypercube coordinates can be made as follows. where reduce to actions on I as follows: This gives explicit construction of the coordinate system, and it can be verified straightforwardly using the group relations. Note that in (33) the action of the composition s 1 r 1 = σ 1 is given instead of s 1 directly, σ n corresponds to the demihypercube reflection I → I ⊕ {0, n}, and multi-index sets I correspond to demihypercube vertices.
Introducing notation u I for variable associated with coset σ I H (defined in (32)), the basic pair-triple (8) corresponds to The pair-triple arrangement of Definition 2.1 in the case k = j = 1 is therefore written alternatively as the set of pair-triples generated from the basic one (34) by the actions (33), which is exactly (30).
Equations (37) allow the subset of variables in the initial data set (36), to be replaced by lattice parameters, α 0 , . . . , α i+2 , on which the group acts not rationally, but by pure permutation. The resulting partially-integrated system is generated by the group permutation action (Definition 2.1 and (38)) from the basic set of equations (37). Written in the demihypercube variables (Lemma 7.1), this is recognisable as the lattice Schwarzian KdV system: and α 0 , . . . , α i+2 are free parameters.
Remark 7.4. The system (39), (40) is in coordinates of the demihypercube in N = i + 3 dimensions. It is better-known in coordinates of the hypercube in one lower dimension: the hypercube coordinates are recovered by deleting all occurrences of the index 0 and setting α 0 = 0, which can be done without losing generality (it lifts the restriction that |I| is odd). However, this leaves two kinds of equations. If 0 ∈ {n, m, l}, then the equation is associated with a quad face of the hypercube, whereas if 0 ∈ {n, m, l}, then it is associated with a tetrahedron formed by four vertices of an embedded cube. The equations on tetrahedra have played a significant role in the theory of this and similar quad-equations [34,35,36]. In the demihypercube coordinates, both kinds of equations are accounted for together.

The Cremona system
Case k = 1 of Definition 3.2 gives generators for a representation of the group of Coble [17,18]. This group is a basic element in the geometric framework of the Painlevé equations [13,19,37], and is a canvas for several extensions [38,20,39]. The connection is made by a change of variables.