Generalized Color Orderings: CEGM Integrands and Decoupling Identities

In a recent paper, we defined generalized color orderings (GCO) and Feynman diagrams (GFD) to compute color-dressed generalized biadjoint amplitudes. In this work, we study the Cachazo-Early-Guevara-Mizera (CEGM) representation of generalized partial amplitudes and ``decoupling"identities. This representation is a generalization of the Cachazo-He-Yuan (CHY) formulation as an integral over the configuration space $X(k,n)$ of $n$ points on $\mathbb{CP}^{k-1}$ in generic position. Unlike the $k=2$ case, Parke-Taylor-like integrands are not enough to compute all partial amplitudes for $k>2$. Here we give a set of constraints that integrands associated with GCOs must satisfy and use them to construct all $(3,n<9)$ integrands, all $(3,9)$ integrands up to four undetermined constants, and $95 \%$ of $(4,8)$ integrands up to 24 undetermined constants. $k=2$ partial amplitudes are known to satisfy identities. Among them, the so-called $U(1)$ decoupling identities are the simplest ones. These are characterized by a label $i$ and a color ordering in $X(2,|[n]\setminus \{i\}|)$. Here we introduce decoupling identities for $k>2$ determined combinatorially using GCOs. Moreover, we identify the natural analog of $U(1)$ identities as those characterized by a pair of labels $i\neq j$, and a pair of GCOs, one in $X(k,|[n]\setminus \{i\}|)$ and the other in $X(k-1,|[n]\setminus \{j\}|)$. We call them {\it double extension} identities. We also provide explicit connections among different ways of representing GCOs, such as configurations of lines, configurations of points, and reorientation classes of uniform oriented matroids (chirotopes).

(1.1) Here the T α i , Tã j are generators of SU (N ) and SU (Ñ ), respectively, α and β are called color orderings, m n (α, β) are partial amplitudes while M({k i , a i ,ã i }) is the color-dressed amplitude.
In [2], color-dressed generalized amplitudes were introduced. This was done for any (k, n) by defining generalized color orderings (GCO) using arrangements of (k−2)-planes (see e.g. [3]) in RP k−1 and generalized Feynman diagrams (GFD) using arrangements of metric trees [4][5][6]. The result is expressed as c(Σ I )c(Σ J ) m (k) n (Σ I , Σ J ), (1.2) where the sum is over all (k, n) generalized color orderings, Σ I denotes a generalized color ordering, while c(Σ I ) is the corresponding color factor. Here as in [2], c(Σ I ) are treated as formal variables. The (2, n) biadjoint amplitudes, m n (α, β), admit a Cachazo-He-Yuan (CHY) formulation as an integral over X(2, n), the configuration space of n points in CP 1 , with integrand given by the so-called Parke-Taylor factors [7,8]. For later convenience, we denote the set of all (k, n) GCOs as CO k,n .
Only when both Σ I and Σ J belong to the special type of (k, n) color orderings called type 0, the partial amplitude is computed using higher k Parke-Taylor (PT) factors, where type 0 corresponds to GCOs obtained from (2, n) color orderings by "pruning" several labels at a time (i.e. they are descendants of a (2, n) color orderings).
In this work, we present a series of constraints that CEGM integrands, associated with (k, n) generalized color ordering, must satisfy and which we implement in an algorithm in section 2. In a nutshell, a (k, n) color ordering, Σ, is a collection of n k−2 orders of type (2, n − k + 2), satisfying a certain compatibility conditions. The integrand associated with it, I Σ , is a rational function of the Plucker coordinates ∆ i 1 i 2 ···i k associated with X(k, n), with only simple poles determined by the simplices in the arrangement of hyperplanes defined by Σ. The numerator is constrained by the torus weight and by the condition that under a simplex flip (see section 3) that connects Σ to Σ ′ , the residues of I Σ and I Σ ′ coincide. This simple observation determines all integrands for k = 3 and n < 9. Modulo relabeling, there are 4 for n = 6, 11 for n = 7, and 135 for n = 8. For n = 9 we find that all 4381 types are fixed up to four constants, which are independent of the coordinates on X (3,9). We also determine 2469 of the 2604 types of (4,8) integrands up to 24 undetermined constants.
In the second part of this work, we study the generalization of decoupling identities. In (2, n) amplitudes, if one of the two generators assigned to, say particle n, commutes with those of the other particles, then the color-dressed amplitude can be shown to vanish. This implies identities among the partial amplitudes m n (α, β) known as U (1) decoupling identities 1 .
The way partial amplitudes organize is by the particular (2, n − 1) color ordering a given (2, n) ordering descends to after projecting out label n.
The CHY formulation gives a simple geometric interpretation of this identity in terms of a fibration of X(2, n) over X(2, n − 1). Consider the real model of X(2, n − 1) as the boundary of a disk with n − 1 points on it. The points split the boundary into n − 1 segments. Placing point n on a given segment corresponds to a given color ordering and hence to a Parke-Taylor factor. The poles in the Parke-Taylor factor that depend on n correspond to the points defining the segment. An identity is obtained by adding all Parke-Taylor factors corresponding to all color orderings obtained while point n traverses the circle. The (3, n) analog of this procedure reveals a rich structure which we fully explore for n ≤ 9.
We identify the analog of the U (1) decoupling identities, which we call "fundamental" identities, and make a proposal for all k and n (see section 6). Fundamental identities are characterized by the choice of two labels i ̸ = j and two generalized color orderings, one is a (k, n − 1) GCO, Σ 1 , on the set [n] \ {i} and the other is a (k − 1, n − 1) GCO, Σ 2 , on the set [n] \ {j}. In other words, one constructs a, possibly empty, set of (k, n) GCOs that "project" to Σ 1 under a k-preserving projection and to Σ 2 under a k-decreasing projection. If the set is not empty then the integrands corresponding to the GCOs in it satisfy a fundamental identity.
We also discuss how the duality between (k, n) and (n − k, n) reveals even more identities and apply it to the case k = n − 2 for which we identify all identities as shuffle identities in the dual (2, n) system.
The rest of this paper is organized as follows: A brief review of the CEGM formulation is given in section 2. Then in section 3, we show how to construct integrands that geometrize the combinatorial properties of GCOs, and in section 4, we discuss how to derive higher k irreducible decoupling identities in a purely combinatorial way using GCOs. In section 5, we point out ways to relate configurations of points to GCOs and to chirotopes. In section 6, we introduce double extension identities and argue that they are fundamental and are the natural analog of U (1) identities. We end with a discussion of future directions in section 7. Most data is presented either in the appendices or in an ancillary file.

Review of CEGM Formulation
In this section, we review the (k, n) CEGM formalism [9] which is a generalization of the CHY formalism obtained for k = 2. The construction is based on the configuration space of n points on CP k−1 , usually denoted by X(k, n), and defined as X(k, n) := SL(k, C)\M * (k, n)/T n , (2.1) where M * (k, n) is the space of k × n matrices with non-vanishing maximal minors and T n = (C * ) n is the algebraic torus that acts on each column by rescalings.
The key ingredient in the CEGM formulation is the scattering equations. These are the equations which determine the critical points of S (k) n := a 1 <a 2 <...<a k s a 1 a 2 ...a k log |∆ a 1 a 2 ...a k | with ∆ a 1 a 2 ...a k := det (2.2) Here s a 1 a 2 ...a k are the generalized Mandelstam invariants and partial amplitudes are rational functions of them. They are assumed to be entries of a generic completely symmetric rank k tensor subject only to a "masslessness" condition, s a 1 a 2 ...a k−2 ,b,b = 0 and "momentum conservation", n a 1 ,a 2 ,...,a k−1 =1 s a 1 a 2 ...a k−1 ,b = 0 for all b. A CEGM integral has the form X(k,n) dµ k,n I L I R , (2.3) with dµ k,n a measure that localizes the integral to the solutions to the scattering equations.
The simplest way to write down the measure is by choosing a chart of X(k, n). We postpone its definition to (2.7) and instead discuss its transformation properties under the torus action, In order to have a well-defined integral it must be that the full integrand in (2.3), I L I R , transforms opposite to dµ k,n . It is standard to assign the same weight to I L and to I R and so This property is one of the keys to finding integrands in the next section. Let us introduce an explicit parameterization of X(k, n) given by 1 0 · · · 0 1 y 1,k+1 y 1,k+2 · · · y 1,n 0 1 · · · 0 1 y 2,k+1 y 2,k+2 · · · y 2,n . . . Some advanced techniques to solve the scattering equations are developed in [13,17,18]  n (Σ,Σ) as a sum over generalized Feynman diagrams (up to an overall sign). Moreover, in analogy with the CHY construction, once the integrands in the CEGM formula (2.9) are fixed, then m (k) n (Σ,Σ) is also fixed and the sign needed to make the sum over generalized Feynman diagrams agree with it can be uniquely determined for all partial amplitudes. We have verified this proposal for (3,6), (3,7), (4,7) and (3,8) using the integrands constructed in section 3.
In the original CEGM paper [9], most of the integrands studied were combinations of Parke-Taylor factors, which are defined as PT (k) (α) := 1 ∆ α 1 ,α 2 ,...,α k ∆ α 2 ,α 3 ,...,α k+1 · · · ∆ αn,α 1 ,··· ,α k−1 . (2.10) These integrands are associated with a special type of (k, n) color orderings known as descendants of a (2, n) ordering. Given α, a (2, n) color ordering, one constructs a (k, n) ordering by computing the array of dimension n k−2 with component l 1 , l 2 , . . . , l k−2 obtained by deleting those labels from α to get a (2, n − k + 2) color ordering. These are called type 0 GCOs and they are in bijection with (2, n) orderings. Therefore, there are (n − 1)!/2 of them. The rest of this work is devoted to developing techniques for computing the integrand associated with generalized color orderings of other types and to the study of the linear relations they satisfy.

CEGM Integrands from GCOs
This section is devoted to the computation of CEGM integrands associated with generalized color orderings. We present constraints that integrands must satisfy and use them to develop an algorithm that aids in their computation. For (3, n < 9) GCOs, the algorithm completely determines all integrands.
In a nutshell, the connection between generalized color orderings and CEGM integrands is found by identifying the poles in the integrands with the (k − 1)-simplices in the GCOs and by imposing that whenever two GCOs are connected via a simplex-flip, the residues of the corresponding integrands must agree up a sign.

Review of Generalized Color Orderings (GCOs)
Let us start by recalling the appropriate definitions given in [2] regarding generalized color ordering as well as a short review of their most important properties.
is a (2, n − k + 2) color ordering constructed as follows. Let {H 1 , H 2 , . . . , H n } be an arrangement of n projective (k−2)-planes in generic position in RP (k−1) . Intersecting any (k−2) such H's, . The line so defined intersects the remaining (n−k+2) H's each on a point, resulting in a sequence of points on the line which defines a (2, n−k+2) color ordering In the definition we chose to construct Σ [k] out of (2, n−k+2) color orderings. However, it is sometimes convenient to note that since each This means that we have an arrangement of n − 1 (k−3)-planes in RP k−1 , i.e. a (k−1, n − 1) color ordering, Clearly, we have 2 By definition, removing a (k−2)-plane, say H i , from the arrangement with n > k + 2 must result in another arrangement but with (n − 1) (k−2)-planes. Therefore, the operation must give a (k, n − 1) color ordering, on the set [n] \ {i}, i.e. label i does not participate. This operation is called a (k-preserving) projection [2], The projection π i acting on a k = 2 ordering σ (i 1 ,i 2 ,··· ,i k−2 ) just means to remove the label i from the set regardless of its position. The projection operation is the key to finding the dual of a GCO [2] under the X(k, n) ≃ X(n − k, n) duality. More important for us here is that it is also the key to defining pseudo-GCOs recursively. The reason the "pseudo" qualifier has to be added is that it is known that the recursive procedure can give rise to arrangements that are not realizable 3 (see [3] for details).
Definition 3.2. A n k−2 -tuple of standard color orderings with n > k +2 is said to be a (k, n) pseudo-GCO if all its projections are (k, n − 1) pseudo-GCOs, while (k, k + 2) pseudo-GCOs are all descendants of (2, k + 2) color orderings.
A GCO must be a pseudo-GCO and a pseudo-GCO which is not a GCO is called a non-realizable pseudo-GCO.
Let us now generalize the notion of a triangle flip introduced for (3, n) color orderings in [2] to all k.
A (combinatorial) simplex flip of a GCO Σ [k] is an operation that produces a new pseudo-GCO (potentially a GCO) as follows. Given a simplex {i 1 , i 2 , . . . , i k } of Σ [k] as above, send . . , i k } and leave all other k = 2 orderings invariant.

Constructing Integrands
Now we are ready to start building the connection between GCOs and CEGM integrands.
Having defined the poles, the final step is to obtain information regarding the residues of I(Σ [k] ) to find its numerator.
A preliminary step is to note that the number of poles of I(Σ [k] ) is n + p, with p ≥ 0, and therefore by using the torus transformation of I(Σ [k] ) given in (2.5) with all t i = t, one has that the numerator must be a polynomial of degree p in the Plucker variables. This means that whenever the number of poles in I(Σ [k] ) is n or n + 1, we can completely determine it using the known torus transformation of I(Σ [k] ). When p = 0 the numerator is 1 and when p = 1 the numerator must be a single Plucker variable, ∆ a 1 ,a 2 ,...,a k , with {a 1 , a 2 , . . . , a k } the labels that participate in k + 1 Plucker variables in the denominator.
Let us call the GCOs integrands with p = 0, 1 basic integrands. All GCOs of type 0 and type I give rise to basic integrands.
The algorithm for computing I(Σ [k] ) given a GCO is based on the observation that if Σ [k] andΣ [k] are related by a simplex flip, then they share a pole and their residues at the pole must agree (up to a sign).

Algorithm I: Building a System of Integrands
Input: All (k, n) GCOs. Output: Rational functions associated with each GCO with poles in correspondence with simplices of the GCOs and matching residues whenever there exists a simplex flip.
1. Construct a list L with all basic integrands with their corresponding GCOs.
2. Consider a GCO in L and perform all possible simplex flips. Select one of the GCO generated in the process for which the integrand has the smallest p and is not already in L. Call it Σ test .

Construct an ansatz for
(3.5) whereQ := Q/∆ flip , i.e. the polynomial Q with ∆ flip removed. Note that the above equations are sign agnostic, so we can solve them (for all choices of ∆ flip and ∆ ′ ) simultaneously and simplify the ansatz P test (∆). 5. By construction, some GCOs generated by simplex flips are already in L, say Σ old , and we can get more constraints on the ansatz P test (∆) than (3.5) by actually matching the residues, P test (∆)Q old (∆) In practice, step 4 provides a simpler ansatz that makes the matching in this step easier.
6. If using all constraints completely determines P test (∆) up to an overall sign then add Σ test to the list L. Otherwise, store it in another list and reconsider it once the list L has grown further and includes more GCOs that can impose constraints on Σ test .
Step 6': Using all available constraints, determine as many unknown coefficients in P test (∆) as possible in terms of the rest. Add Σ test to the list L and treat it as if it were known.
Note that by adding a partially known integrand, say I test ′ (∆), to the list L, Step 5 can produce more equations for a new "test" integrand, I test (∆). The new constrains may not only fix I test (∆) but also I test ′ (∆).
It is important to mention that there are further subtleties for (3,9), (4,8), and beyond as they have pseudo-GCOs that are not GCOs. This means some combinatorial simplex flips of a valid GCO may fail to lead to valid GCOs. In this work, we choose to discard such simplex flips when we apply Algorithm I. This way, using the modified version of the algorithm, we computed all 4381 types of (3,9) integrands in terms of only four unknown parameters. We also computed 2469 types of (4,8) integrands in terms of 24 parameters, while the remaining 2604 − 2469 = 135 types of integrands are left for future research since their computation requires a significant amount of computing resources.
All our results for integrands are present in an ancillary file. Our findings for (3,9) show that starting at n = 9 one has to combine the algorithm with some explicit evaluations of the CEGM formula to fix the undetermined coefficients by comparing the results to a sum over generalized Feynman diagrams. We expect that a similar phenomenon should also happen to (4,8). It is tempting to relate the presence of unfixed parameters to that of non-realizable pseudo-GCOs. However, at this point, we do not have a direct way to link them.

(3, 6) Integrands
We give a full explanation on how to construct the (3,6) integrands. There are four types of GCOs, with representatives, type III Figure 1. Examples of GCOs corresponding to the four types of (3, 6) GCOs. All triangles in each arrangement of lines have been shaded. There are 6,6,7, and 10 triangles respectively. To recognize the triangle bounded by lines L 2 , L 3 , L 4 in the first graph requires using that points on the boundary of the disk marked with the same label are identified.
Using the combinatorial technique explained in section 3.1 (or in appendix A for general polygons), one finds that they have 6,6,7 and 10 triangles respectively, All triangles in the arrangements of lines are shown explicitly in fig. 1. Even though the figures are intuitive, the combinatorial way of finding all triangles is very efficient even for large values of n.
Obviously, type 0 and type I integrands are of basic type with p = 0. So they have trivial numerators, Type II is also of the basic type with p = 1. Hence a single minor is needed in the numerator. We see lines L 2 , L 4 and L 5 participate in four triangles while L 1 , L 3 and L 6 in three, which means the minor is ∆ 235 , The last of the (3, 6) GCOs is type III. This GCO has ten triangles. This means that the numerator of I(Σ III ) has to be a degree four polynomial in the minors. Moreover, it easy to see that each line participates in five triangles and therefore the numerator must be homogeneous of degree two in each label. The most general form of the numerator is where AB is the line that contains points 3, 4, 5 in the configuration of points in RP 2 . This implies that P 4 (∆) must vanish if ∆ 345 = ∆ 126 = 0. Imposing this condition to the ansatz (3.18) fixes 9 of the 16 parameters, Repeating the computation but with the pole at ∆ 256 = 0 gives where AB is the line that contains points 2, 4, 6. This implies that P 4 (∆) must vanish if ∆ 256 = ∆ 135 = 0. This time, the condition fixes three parameters, Repeating the computation for all the remaining 8 poles in (3.19), we fix all 16 parameters up to an overall rescaling, We see how powerful the constraints (3.5) are and P ′ 4 (∆) can be thought of as a simplified version of the ansatz. At this point, it is much easier to match (up to a sign) the residue of (3.19) with that of its neighbours such as (3.20) using (3.6). Doing so, we finally get x 1 → 1.
This gives For this particular example, there is a more beautiful explanation for the numerator of (3.26). Note that the structure of (3.21) and (3.23) is the same. In fact, the same is true for all ten poles and we conclude that P 4 (∆) is a polynomial that must vanish whenever the six points are on any of the 6 3 /2 = 10 configurations of two lines with three points in each. All these configurations are degenerate conics and it is well known that the Veronese polynomial vanishes on all of them, It is surprising at first that V cannot do what is needed since it manifestly vanishes at only four of the ten configurations, e.g. when ∆ 123 = ∆ 456 = 0. However, one can check that under any permutation of labels, V either stays the same or becomes −V .
Let us mention that these four functions give rise to canonical forms in the sense of [20,21] for the regions closely related to the chambers of X (3,6). In fact, they were independently obtained in [22] using "triangulations" to build canonical forms. More relevant discussions are put in section 5.2.
There are eleven types of (3, 7) integrands, with seven of them of basic type: 4 with p = 0 and 3 with p = 1. Here we list a representative for each of the eleven types: where the 11 (3, 7) GCOs Σ • of different types are put in appendix D.1. Similarly, we can get the eleven types of (4, 7) integrands using Algorithm I. As expected, they are dual to the (3, 7) integrands by sending ∆ i,j,k → sign(i, j, k, o, p, q, r)∆ o,p,q,r where the sign is determined by an ordering. For example, we have where the (4,7) GCO Σ ′ V II is dual to the (3,7) GCO Σ V II used in (3.35).
Here we present two examples, one with p = 2 and one with p = 3 and provide a complete list of integrands as well as their corresponding GCOs in the ancillary file, The subscripts in Σ 43 and Σ 73 are the locations in the list of 135 pairs of integrands and GCOs in the ancillary file. 5 These (3,8) GCOs are also shown in Appendix B of [2]. We comment that starting at (3,7), there are different GCOs whose corresponding arrangements of lines have the same set of triangles. For example, it is easy to check the following two (3,7) GCOs of type VII respectively. This means that the two GCOs are indeed different.
In the appendix, we generalize the method introduced in [2] for finding triangles to one for finding all polygons based on a simple combinatorial rule applied to the k = 2 color orderings in the collection that make the k = 3 GCO. Having an efficient algorithm for determining all polygons is very useful for identifying GCOs.
Back to the two (3, 7) GCOs, the corresponding integrands of type VII share the same denominators but have different numerators, The two GCOs are related by 2 ↔ 7, 3 ↔ 5 as are their integrands. We see that for k > 2, integrands are not always uniquely characterized by their explicit sets of poles.

(3, 9) Integrands
In (3,9), there are 4381 types of GCOs and one type of non-realizable pseudo-GCOs. We focus on the triangle flips between GCOs first, match all residues using (3.6) when applying Algorithm I, and deal with the subtleties relevant to the non-realizable pseudo-GCOs later.
This way, though cumbersome, we get 3582 types of (3,9) integrands without ambiguities and 798 other types in terms of four free parameters. The GCO for the last missing type of integrands has 21 triangles, and we provide a more clever way to determine its integrand later in section 4.4.3 by making use of irreducible decoupling identities 7 . Among the 4381 types of (3,9) integrands, 369 are of basic type split as 48 with p = 0 and 321 with p = 1.
Here we present some examples with p = 2, 3, 4, . The following is an example of an integrand with one of the four undetermined coefficients, denoted y, It is remarkable that the system of 4381 types of integrands (for each type one must add integrands obtained by permutations of labels) can be determined up to only four coefficients. Once again, the list of representatives for all 4381 types is provided in the ancillary file.
Here we make some comments on non-realizable pseudo-GCOs defined below Definition 3.2. In (3,9), there is exactly one type of non-realizable pseudo-GCOs. A representative is given by  Combinatorially, a (3,9) GCO can be connected to a non-realizable pseudo-GCO via a triangle flip, but this cannot be realized geometrically. See more details in appendix B.1. The reader might wonder what happens to the algorithm once a triangle flip of a target GCO leads to a pseudo-GCO. We do not have a definite answer to this question and in our implementation, we have chosen to treat pseudo-GCOs as unknown. If we were to impose that the residue of an integrand vanishes when its triangle flip leads to a non-realizable pseudo-GCO we would have been able to fix one of the four coefficients. Since we expect that a direct computation of partial amplitudes using generalized Feynman diagrams will fix the value of all four coefficients, we leave its value undetermined for now. Let us comment on another yet new phenomenon starting at (3,9). In (3, 7) we found two distinct GCOs with the same triangles. Those two were not related by a triangle flip (3.42). In (3,9), there are two GCOs that are connected by a triangle flip and share the same set of triangles! We put some examples in appendix B.2.

(4, 8) Integrands
In (4,8), there are 2604 types of GCOs and 24 types of non-realizable pseudo-GCOs. Among the 2604 types of (4,8) integrands, 279 are of basic type split as 41 with p = 0 and 238 with p = 1. Based on them, we further fix 1711 integrands completely and find expressions for 479 other integrands in terms of 24 parameters. This leaves 135 integrands which we have yet not computed.
Here we present some examples with p = 2, 3, 4, , (3.57) Note that the last example contains one of the 24 undetermined coefficients, which we denoted as z. All 2469 = 279+1711+479 integrands we obtained are included in the ancillary file. They are dual to themselves up to a relabelling, which is a strong consistency check of our algorithm. We expect that the remaining 135 types of integrands can be worked out similarly but would require either large computing resources or improvements to the algorithm and we leave them for future research.
Note that 11 of 24 non-realizable pseudo-GCOs are connected to the 2469 integrands we obtained by combinatorial tetrahedron flips. Similar to the (3,9) case, if we were to impose that the residues of integrands vanish when their tetrahedron flips lead to non-realizable pseudo-GCOs, we would have been able to fix nine of the 24 parameters.

Higher k Irreducible Decoupling Identities
Decoupling identities in physics refer to linear relations among partial amplitudes which are derived by using that color-dressed amplitude vanish whenever the color of a particle commutes with that of all the others. The vanishing of the color-dressed amplitude is manifest from the Lagrangian viewpoint as interaction vertices are proportional to the structure constants of the color group. From the viewpoint of partial amplitudes, it is not obvious that they must satisfy identities. One of the advantages of the Witten-Roiban-Spradlin-Volovich (Witten-RSV) formulation [23,24] of N = 4 SYM amplitudes and of the CHY formulation of Yang-Mills and biadjoint scalar amplitudes is that decoupling identities can be understood directly from partial amplitudes. In this section, we start with a review of how decoupling identities are understood in the CHY formulation, i.e., when k = 2, and then move on to the corresponding generalization for k = 3.
It turns out that k = 2 decoupling identities are always irreducible, i.e., no proper subset of terms participating in a certain identity adds up to zero. In [2], we found that a naive extension of the k = 2 procedure to higher k leads to decoupling identities but they are reducible. Here we find a natural explanation for it and propose an algorithm for k = 3 to generate various of irreducible decoupling identities. Moreover, in the process, we identify the analog of the U (1) decoupling identities for higher k in a procedure we call double extension, with section 6 devoted to them.

k = 2 Decoupling Identities
In the Witten-RSV and CHY formulations of partial amplitudes, the part of the integrand that carries the color information is the Parke-Taylor function, In our terminology, this is the integrand for type 0 color orderings. For k = 2 this is the only type of color orderings present. Decoupling particle n means that the position of the n th label in an ordering is irrelevant and therefore one should group all partial amplitudes according to the (2, n − 1) ordering they map to under the projection, π n , that is deleting n from the ordering. This means that given a (2, n − 1) ordering, sayσ := (1, 2, . . . , n − 1), then Here the sum is over all (2, n) orderings σ such that when label n is deleted they becomeσ. The proof of (4.2) is very simple. Consider the LHS of (4.2) as a rational function of the position of the n th point in CP 1 . Denote its inhomogeneous coordinate by z. Clearly, the function is O(1/z 2 ) as z → ∞ and can only have simple poles which are located where ∆ in = 0 for any i. By inspection, one can compute the residue at ∆ in = 0 noting that only two of the n − 1 terms in (4.2) contain the pole. Moreover, the residue becomes proportional to that of which clearly vanishes since This cancellation has a beautiful combinatorial interpretation. If we represent the (2, n − 1) color ordering as a circle with n − 1 points on it, then placing point n in any of the n − 1 intervals corresponds to a point in X(2, n). Each interval corresponds to a color ordering and crossing from one interval to another is an "interval" flip of the color ordering. The fact that the residues of the corresponding integrands agree up to a sign is built in the construction. Adding up overall integrands as we move along the circle guarantees that the resulting object has no poles.
Moreover, each identity involving n − 1 partial amplitudes is irreducible, i.e., no proper subset of the n − 1 partial amplitudes can be added with constant coefficients to get zero. This fact is also clear from the picture using the circle.

k > 2 Decoupling Identities
In [2], we generalized the definition of decoupling a particle to higher k. n , the operation that identifies (up to a sign) any two (k, n)-color orderings, Σ andΣ, if their i th projections are the same, i.e., π i (Σ) = π i (Σ), is called decoupling the i th -particle.
Decoupling identities are obtained if one requires that the color-dressed amplitude M (k) n vanishes when a particle is decoupled.
This definition heavily relies on the projection operator defined in (3.4), π i : CO k,n → CO k,n−1 .
When k > 2 one can straightforwardly define the notion of a decoupling set of GCOs as done in [2], but unlike the k = 2 case, this does not lead to irreducible identities. The reason is that when k > 2 one can have lower k cancellations in the set! In this section, we present an algorithm for finding irreducible decoupling identities for k = 3. The algorithm is purely combinatorial and only uses two of the basic operations on generalized color orderings introduced in [2]: one is the projection π i and the other is the simplex flip reviewed at the end of section 3.1.
Projecting, say particle n, induces a partition of the set of all (k, n) GCOs in terms of the pre-image of π n . In other words, given a (k, n − 1) GCO Σ, then π −1 n (Σ) ⊂ CO k,n is called a decoupling set. For example, starting with (3, 6) one finds that the 372 GCOs split into 12 decoupling sets each with 31 GCOs. The 12 sets are labeled by the 12 (3, 5) GCOs. When (3, n > 6), decoupling sets can have a different number of elements.
It turns out that proper subsets of π −1 n (Σ) satisfy identities when n > 5 and k = 3. In fact, for every subset that generates an identity, its complement also generates one. We find that there are two types of irreducible identities distinguished by whether their complement is also irreducible or not.

Constructing Irreducible Decoupling Identities
Let us introduce the algorithm for finding irreducible identities in the (3, n) case and then provide examples.
An irreducible identity can be characterized by a given (3, n) GCO, Σ, and any pair of its triangles that share at least one label. We identify such a label as the one being decoupled.

Algorithm II: Building GCO Sets for Irreducible Decoupling Identities
Input: A (3, n) GCO Σ, a particle to decouple, say n, and a choice of two distinct triangles of Σ that contain n, say {i, j, n} and {p, q, n}.
Output: A set of (3, n) GCOs belonging to the n decoupling set that contains Σ and which combine to produce an irredubible decoupling identity.
1. Construct a list L which starts with Σ as its only element.
2. For every GCO in L, determine all its triangles that contain label n and discard any that coincide with either {i, j, n} or {p, q, n}. In the following, we call n the decoupling particle, Σ the original GCO that generates the identity, and {i, j}, {p, q} the "walls" as they constitute the forbidden triangles that cannot be flipped (or crossed).
Proposition 4.2. Any set of GCOs generated by using Algorithm II gives rise to an irreducible decoupling identity.
The proof is similar to that for k = 2 explained in section 4.1 and uses a residue theorem. The only new feature is that there are cancellations inside the region and on the boundary. The proof is presented in appendix C.

Geometric Intuition and Examples
Let us illustrate the use of algorithm II in several (3, n) cases. When n < 7 it is easy to develop a geometric picture for the decoupling identities since the real model of X(3, n) captures all the relevant structures.
The idea is to consider a representation of the fibration X(3, n) → X(3, n − 1). One starts by selecting a particular point in X(3, n − 1). The choice is equivalent to that of a decoupling set 9 . Having n − 1 points on RP 2 one can draw all n−1 2 lines defined by any pair of such points. So, a line is labeled by the pair that it contains, e.g. L 24 is the line that passes through points 2 and 4. The n−1 2 lines intersect partitioning RP 2 into polygons known as chambers.
Before continuing, let us explain the connection of this picture to that of the arrangement of lines used to describe GCOs. As it turns out, the two pictures are dual to each other. Lines in one picture correspond to points in the other and vice-versa. Now, the fibration of X(3, n) over X(3, n − 1) is realized by placing point n on the figure with the other n − 1 points fixed. Point n must be placed on one of the chambers and once it is in there, it cannot be moved to another chamber without making it become collinear with two others. For example, if point n is placed on a chamber bounded by lines L ab , L cd , L ef , then in order to make it move to another chamber one must have that at least one of ∆ abn , ∆ cdn , or ∆ ef n vanishes.
This means that a chamber is dual to a particular GCO in a decoupling set and crossing into another chamber is equivalent to performing a triangle flip on the GCO. Now all the elements in algorithm II come to life: • Selecting a label to decouple, say n, is equivalent to selecting a fibration, X(3, n) → X(3, n − 1).
• Selecting a GCO, say Σ, corresponds to selecting a point in X(3, n − 1), i.e. a configuration of n − 1 points in RP 2 , via the projection π n (Σ) and the duality between lines and points. Moreover, Σ also picks a particular chamber in the fiber.
• The choice of two triangles of Σ, {i, j, n} and {p, q, n} corresponds to selecting the two boundaries of the chamber corresponding to Σ and defined by the lines L ij and L pq .
• Performing triangle moves which are not associated with either {i, j, n} or {p, q, n} means that we cover chambers in RP 2 which are within lines L ij and L pq . Note that two lines in RP 2 partition the space into only two regions. We are choosing chambers in the region that already contains that associated with Σ.
Let us mention some of the consequences of the geometric perspective. Even though we started with a GCO, the picture reveals that the decoupling identity is more naturally associated with the pair of lines L ij and L pq and to the choice of "inside" and "outside".
As we will see in the examples, if L ij ∩ L pq is not one of the n − 1 special points, the decoupling identities are always irreducible as one can show that there exists a chamber for which algorithm II leads to them. If L ij ∩ L pq is one of the special points, the identity is not necessarily irreducible. In fact, there are n − 2 lines passing through any given special point. If a circle is drawn centered around the point, an identity is irreducible if the two lines intersect the circle at points that are adjacent.

Types of Irreducible Decoupling Identities and (3, 5) Examples
We mentioned that there are two types of irreducible identities distinguished by whether their complement in a decoupling set is also an irreducible identity or not. Let us call the former a partitioning identity and the latter a non-partitioning one. In section 6, we show that the non-partitioning ones are, in fact, the analog of the U (1) decoupling identities in k = 2 and that they can be thought of as "fundamental", in the sense that, conjecturally, all other identities are linear combinations of them.
There is a simple combinatorial characterization to distinguish them. An irreducible identity constructed using a GCO Σ and triangles {i, j, n} and {p, q, n} is partitioning if and only if {i, j} ∩ {p, q} = ∅.
Let us illustrate this concept with two (3,5) identities. In fig. 3, we start with a fibration of X(3, 5), thought of as the configuration space of five points on RP 2 , over X (3,4). The figure represents the base X(3, 4) and shows the lines where if point 5 were located, then it would be collinear with two other points. The regions bounded by lines are chambers and there are 12 in this case. Each chamber is associated with a GCO and Σ is chosen to be in chamber C 1 .
There are three triangles, i.e., boundaries that can be chosen to create the identity. These are {3, 4, 5}, {4, 5, 1}, and {5, 1, 2}. In fig. 3 we chose {3, 4, 5} and {5, 1, 2}. Since {3, 4} ∩ {1, 2} = ∅, we expect a partitioning identity. The algorithm for finding the corresponding identity boils down to selecting the set of chambers bounded by the two red lines. Note that there are only two regions divided by two lines in RP 2 . Here we consider the region which contains Σ to get the identity {C 1 , C 2 , C 3 , C 4 , C 5 , C 6 }. Since this is a partitioning identity, we also have that give rise to an irreducible identity. Here we list the integrands associated with the twelve chambers: labels of the chambers in the same region as Σ one finds a 4-term identity  Figure 5. A point in X (3,5). The five points on RP 2 are represented by the solid circles. Also shown are the 10 lines containing pairs of such points. The 10 lines intersect partitioning RP 2 into 31 regions (or chambers). Once point six is placed on a chamber, it cannot visit another chamber unless it becomes collinear with two other points, i.e. it must cross a line. Each chamber corresponds to a GCO and the corresponding types are shown. There are five type 0, fifteen type I, ten type II, and a single type III GCO in this decoupling set.

(3, 6) Irreducible Decoupling Identities
Let us start again by fibering X(3, 6) over X (3,5) with fiber in the position of point 6. This means that the base is given by the configuration space of five points. In fig. 5, we show a point in X (3,5). We also draw all lines connecting any pair of the five points as they are the boundaries of the connected chambers where point 6 can live. There are 31 chambers and we -25 -have indicated the type of the GCO associated with each of them. Note that there are five type 0 GCOs as expected from the fact that this is a decoupling set and exactly five (2,6) color factors get identified after removing point 6.
There are a total of 9 different types of irreducible identities that can be derived by selecting a pair of lines in fig. 5.
There is one 7-term, one 8-term, one 9-term, two 12-term, one 14-term, one 17-term, and two 19-term identities. GCOs. By the way, the associated GCOs for the chambers C 2 , C 19 , C 5 , C 8 are the type 0, I, II and III representatives shown in (3.7)-(3.10) respectively whose arrangements of lines are given in fig. 1.
In order to explicitly construct the identities we have to label the points so that a pair of lines can be selected. In fig. 6, we show the five points with labels and the four lines that contain point 1. The four lines split RP 2 into four regions. The chambers in each of the four regions define an irreducible identity. The explicit form of the GCOs and integrands associated with each chamber can be found in appendix D.2. Here we simply point out that there are one 8-term, one 9-term identity, and two 7-term identities which are related by a relabeling. The seven-term identity bounded by L 14 , L 15 is explicitly given by We have also included figures of the other kinds of identities in appendix D.3. For the reader's convenience, we also present the 9 kinds of GCO sets for irreducible identities in the ancillary file irreID_36_9.m. For example, in the ancillary file, the GCO sets for the seven-term identity in (4.11)  Here {type [1], perm[1, 5, 4, 3, 2, 6]} means a type I GCO using the seed given in (3.8) under a relabelling, (3.8)| {1,2,3,4,5,6}→{1,5,4,3,2,6} . We have also shown that the decoupling particle is 6 and the walls are {1, 4}, {1, 5}. The first GCO in irreID[•] is the original GCO in Algorithm II.
It is clear to see there is one type 0, four type I, and two type II GCOs in this identity.

(3, 7),(3, 8) and (3, 9) Irreducible Decoupling Identities
As n increases the number of different kinds of identities grows very fast. We have implemented algorithm II and made an exhaustive search for n = 7, 8, 9. They have 78, 1432, and 52 444 classes of GCO sets for irreducible decoupling identities respectively, among which 18, 234, and 6193 classes respectively are for non-partitioning identities. We store all of the classes of (3, 7) and (3,8) GCO sets in the ancillary file while we just put the 6193 classes of (3,9) GCO sets for non-partitioning identities there to save the size of the file. As will be argued in section 6, their combinations are supposed to express all other irreducible decoupling identities. So they should be enough.
In (3,9), there are GCOs sets for irreducible identities whose integrands contain some of the four free parameters. However, their identities hold no matter what the free parameters are.
Note that the irreducible decoupling identities express one integrand by others. If other integrands are already known, we get one integrand almost for free. One can express the unknown integrand as a linear combination of others with few undetermined signs first but these signs can be further fixed by matching the residues of this unknown integrand with its neighbors using (3.6). This also provides a way to get new integrands from known integrands. In particular, it produces the last integrand of (3, 9) with 21 triangles mentioned in section 3.3.3.
As we already know, there is a type of non-realizable pseudo-GCOs in (3,9). So there might be a problem when we apply Algorithm II as a combinatorial triangle flip of a GCO may lead to a non-realizable pseudo-GCO. In this case, we have to relax the requirement in Algorithm II a little bit and also allow us to put pseudo-GCOs in the set. This way, we may get a set of pseudo-GCOs including non-realizable ones even if we start with a GCO to generate this set. If we set the free parameter y 0 mentioned in (B.3) as 0 and set the "integrand" associated with the non-realizable pseudo-GCOs also as 0, we still get an irreducible decoupling identity in terms of integrands associated with the remaining GCOs in that set.
There are 558 kinds of such sets of pseudo-GCOs and we also put them in the ancillary file.

General (k, n) Irreducible Decoupling Identities
The algorithm presented in this section for generating irreducible (3, n) identities can, in principle, be extended to higher values of k. The practical obstacle is deciding which sets of simplices should not be used for simplex flips. The naive generalization would be to choose any set of (k − 1) simplices of the original GCO that contain the decoupling particle and use them to act as walls to bound a region in RP k−1 . This procedure always produces a set of GCOs whose associated integrands constitute an identity but it is not guaranteed to be irreducible due to possible lower k (but larger than 2) reducible identities on the boundary of the region. The higher dimensionality of the space makes it hard to develop a simple rule to generate the set of simplices needed to guarantee the irreducibility of the identities as it was done for k < 4. We leave its systematic study to future work.
Of course, when k = n − 2, one can develop the dual of Algorithm II and apply it to get identities.
It is clear to see the choice ofŜ i leads to an n−1 i -term irreducible identity and it is isomorphic to the one obtained by selectingŜ n−1−i for any i ∈ {1, · · · , n − 2}.
We have seen some examples in (4.9) and (4.10) which correspond to the selectionŜ 1 andŜ 2 respectively and here we generalize them to any n.
Eq. (4.14) looks very like the Kleiss-Kuijf (KK) relations 10 and the latter can indeed help us to understand (4.14).

Connecting Different Descriptions of GCOs
The attentive reader might be puzzled by the fact that CEGM integrals (2.3) are defined in the configuration space of n points in CP k−1 , X(k, n), while generalized color orderings (GCOs) are given in terms of arrangements of n hyperplanes in RP k−1 (see (3.1)).
It turns out that GCOs have several equivalent representations. In this section, we discuss three of them and their relations. The first is the one we have used so far, i.e., as arrangements of n hyperplanes in RP k−1 . The second one is as a configuration of n points in RP k−1 , i.e., the real model of the space over which the CEGM integral is defined. The third is as reorientation classes of realizable oriented matroids (or chirotopes) [3]. Due to the realizability constraint, we can think of chirotopes simply as a vector containing the signs of all Plucker variables of the n points in RP k−1 .

Reading a GCO from a Configuration of Points
In section 4, we found that relating configurations of points and GCOs was very useful in geometrizing the decoupling identities. Here we explain the connection in detail and show how to read GCOs directly from the configuration of points. This will allow us to find a geometric understanding of the "basic" decoupling identities explained in the next section which are the natural analog of the U (1) decoupling identities.
Let us start with k = 3. A (3, n) GCO, which is encoded in the intersection pattern of lines {L 1 , L 2 . . . , L n } in RP 2 , can also be represented in the dual configuration given by Figure 7.
Left: A presentation of a configuration of points in RP 2 by setting z = 1. Right: A presentation of an arrangement of lines in RP 2 by setting b = 1. A point (x, y, z) ∈ RP 2 in one projective space is mapped to the line L = {(a, b, c) ∈ RP 2 : ax + by + cz = 0} in the other projective space and vice versa. More explicitly, the points P i , P t are mapped to the lines L i , L t in the arrangement of lines and the line P i,t is mapped to the point L i ∩ L t = (−tanθ t , 1, 0), which only depends on the angle θ t . As one can imagine, as θ t increases from 0 to π, the point L i ∩ L t moves along axis a from the origin to (−∞, 0) and then comes back from the other direction. For example, in the last graph, imagining that we start with the line crossing point P 1 and P 5 , rotate it around P 5 , it will cross P 3 ,P 2 and P 4 consecutively until it crosses point P 1 again. According to the argument in figure 7, in the dual graph, i.e., the arrangement of lines, there will be intersection points, L 5 ∩ L 1 , L 5 ∩ L 3 , L 5 ∩ L 2 , L 5 ∩ L 4 consecutively on the line L 5 , resulting in a standard color ordering (1324) there. So do other graphs in the bottom of this figure, resulting in a (3, 5) GCO ((2435), (1354), (1425), (1253), (1324)), which is dual to (13524). Compared with fig. 3, the present P 5 is put in the chamber C 5 there and the present resulting GCO is consistent with the Parke-Taylor factor associated with C 5 given in (4.7). Figure 9.
Left: A presentation of a configuration of points in RP k−1 by setting z = 1. Right: A presentation of an arrangement of hyperplanes in RP k−1 by setting b = 1. A point (x, y, z, x 4 , x 5 , · · · , x k ) ∈ RP k−1 in one projective space is mapped to the codim-1 hyperplane H = {(a, b, c, a 4 , a 5 , · · · , a k ) ∈ RP k−1 : ax + by + cz + k j=4 a j x j = 0} in the other projective space and vice versa. More explicitly, the unique codim-2 hyperplane denoted as P i1,i2,··· ,i k−2 that crosses points P i1 , P i2 , · · · , P i k−2 is mapped to the line H i1 ∩ H i1 · · · ∩ H i k−2 in the arrangement of hyperplanes and the codim-1 hyperplane P t,i1,i2,··· ,i k−2 is mapped to the point H t ∩ H i1 · · · ∩ H i k−2 . The coordinate system is chosen such that the codim-2 hyperplane P i1,i2,··· ,i k−2 is described by {(x, y, z, x 4 , x 5 , · · · , x k ) ∈ RP k−1 : x = 0 & y = 0} but different choices of coordinate system will not affect the intersection relations of hyperplanes in the dual space. Under the current coordinate system, we see the position of the point H t ∩ H i1 · · · ∩ H i k−2 only depends on the angle θ t and one can imagine that as θ t increases from 0 to π, the point H t ∩ H i1 · · · ∩ H i k−2 moves along axis a from the origin to (−∞, 0) and then comes back from the other direction.
points {P 1 , P 2 , . . . , P n } in RP 2 using the following procedure. For each point, P i , draw lines connecting it to other n − 1 points. Starting on any of the lines, circle around P i recording the labels of the lines as they are crossed. The resulting list is the (2, n − 1) color ordering in the i th entry of the GCO.
Let us denote the resulting list asσ (i) while the one on the line L i as σ (i) . To provẽ σ (i) = σ (i) , one just needs to make use of the explicit duality relation which maps a point (a, b, c) ∈ RP 2 to the line L = {(x, y, z) ∈ RP 2 : ax + by + cz = 0}. As shown in fig. 7, the point P i is mapped to L i and the line P i,t that crosses P i and P t is mapped to the point L i ∩L t whose position only depends on θ t . In a configuration of n points in X(3, n), P t could be any points except P i . As we read the lines P i,t clockwise or anti-clockwise, it gives the ordering σ (i) ; in the meantime, their dual points L i ∩ L t sit consecutively on the line L i . While the ordering of the latter is just σ (i) . Hence,σ (i) = σ (i) .
Reading the (2, n − 1) color ordering for every point P i with i ∈ [n], we get the whole (3, n) GCO for the configuration of points in RP 2 . See an example in fig. 8.
The generalization to any k ≥ 3 is straightforward.
Proposition 5.1. Given a configuration of n points in RP k−1 , for any k − 2 points {P i 1 , P i 2 , · · · , P i k−2 }, find the unique codim-2 hyperplane denoted as P i 1 ,··· ,i k−2 that crosses all of the k − 2 points. Start with any codim-1 hyper-plane that crosses the codim-2 one and rotate it around the codim-2 one, it will cross the remaining points consecutively, resulting in a list of n − k + 2 labels. As proved in fig. 9, the resulting list is the standard color ordering in the entry of the GCO with labels {i 1 , · · · , i k−2 } removed. This way, we get a (k, n) GCO directly from the configuration of n points in RP k−1 .

Construction of Chirotopes from GCOs
Denote the sign of a Plucker variable at a generic point in the Grassmannian G(k, n) by We shall assume that all Plucker coordinates are nonzero, in which case we obtain a point (χ) ∈ {±1} ( n k ) known as a (realizable) simplicial chirotope, or equivalently an oriented uniform matroid [3].
The positive Grassmannian is the subset of the real Grassmannian where all Plucker coordinates are positive, ∆ j 1 ,...,j k > 0. In [26,27], it is argued that the Parke-Taylor factor PT (k) (I) in Equation (2.10), multiplied by the canonical measure of the positive Grassmannian is its canonical form, while the Parke-Taylor factor itself is sometimes called the canonical function. See [21] for details about positive geometry and canonical forms 11 . Now, given any generic point in the real Grassmannian, after modding out by the torus (R ̸ =0 ) n we can fix an affine chart such that a certain collection of n Plucker coordinates are all positive. Then the hypersurfaces ∆ i 1 ,i 2 ,··· ,i k = 0 in the real Grassmannian induce a decomposition of the configuration space X(k, n) into chambers. These open chambers are characterized by so-called reorientation classes of simplicial chirotopes, that is, a vector of signs ±1 of length n k , modulo the scaling action of the torus (Z/2) n . Among all simplicial chirotopes, one has been particularly well-studied: the chirotope with all positive signs χ J = 1.
As explained in [11, eq 6.8], modding out by the torus T + = (R >0 ) n , one obtains the positive configuration space X + (k, n) = G + (k, n)/T + ; moreover, the PT function can also be used to compute the canonical form of G + (k, n)/T + . In the same way, it was proposed in [22] that the 372 (3,6) forms provide an analog for spaces that are supposed to be closely related to the chambers just mentioned where the Plucker variables are allowed to have other signs.
Let us now formulate a dictionary between GCOs and chirotopes 12 . More precisely, for each GCO we construct an equivalence class of chirotopes modulo the torus action 13 : we 11 While the general definition of canonical form is defined in [21], here we give some intuition. The simplest canonical form is the one for an interval x ∈ [0, 1] given by dx x(1−x) which has two unit residues at the boundaries of the interval. In general, the canonical form of a positive geometry is required to have a residue on each stratum that is again a canonical form, for the new semi-algebraic set. 12 Compare to for example [28] for a rank three construction. 13 According to the torus action, we can multiply the i th column of the M matrix shown in (2.2) by −1, select a representative chirotope by fixing a projective frame. The nontrivial direction is to map GCO to chirotope; we state the full solution. Fix a GCO Σ = (σ (12···k−2) , . . . , σ (n−k+3,...,n) ) as in (3.1), where σ (L) is a cyclic order on {1, . . . , n} \ L with L = {ℓ 1 , . . . , ℓ k−1 }. Here each element (χ J ) ∈ {±1} ( n k ) is antisymmetric in its indices. For example, when k = 3 we have χ abc = −χ bac = χ bca . We first fix a frame by putting χ I = 1 for all I = {i 1 , . . . , i k } ⊂ [1, k + 1] and χ 1,2,··· ,k−1,j = 1 for j ∈ [k + 2, n].
In general, given a pseudo GCO Σ, then χ Σ can still be found but it may be non-realizable, in the sense that there would not exist a point in the Grassmannian whose Plucker coordinates ∆ abc have sign χ abc .

(5.3)
which turn every χ ib...c of a chirotope to be its opposite. We say the resulting chirotope and the original one are in the equivalent class.
We comment that the type 0 and I GCOs (3.7) and (3.8) are related by a triangle flip via {1, 2, 6}, as shown in their arrangements of lines in fig. 1 or in the configurations of points in RP 2 in fig. 6 and here we see their chirotopes are related by flipping χ 126 .

Double Extensions and Fundamental Identities
According to Definition 4.1, the notion of decoupling for general k given in [2] is closely related to the projection operator defined in (3.4), π i : CO k,n → CO k,n−1 .
GCOs are grouped together according to the result from this projection. In other words, one defines equivalence classes The GCOs in each equivalence class produce identities that are in general reducible, i.e. the set of integrands associated with the GCOs satisfy more than one linear relation. There are two ways to think about this. Either the correct analog of U (1) decoupling for k ≥ 3 would necessarily yield reducible identities, or else there is a finer, deeper structure, which selects more elementary collections of GCOs whose integrands satisfy a unique relation.
Clearly, one would like to find a deeper structure that results in irreducible identities; therefore we need to refine Definition 4.1. In section 4.1, we discussed techniques for constructing irreducible identities for k = 3 and found two classes. In this section, we show that the non-partitioning identities can be obtained using such a finer notion of decoupling.
Our proposal for the refinement involves constraints imposed simultaneously on CO k,n and its dual CO n−k,n ; consequently, any identities or associated algebraic structures that may arise would be manifestly compatible with the duality on configuration spaces X(k, n) ↔ X(n − k, n). Actually, we are going to define a GCO set by using the intersection of one decoupling set in X(k, n) and the dual of the other in X(n − k, n) obtained by decoupling different particles.
On the contrary, we denote by π (j) the k-decreasing projection As above, again on the level of the configuration space, the k-decreasing projection in the j th direction acts by As explained in [2], the dual of a k-decreasing projection of a GCO is a k-preserving projection of the dual GCO.
Now we prepare to introduce double extensions and fundamental decoupling identities.
Moreover, we also claim that all other irreducible decoupling identities for k = 3 that can be obtained using the algorithm explained in section 4 are linear combinations of the fundamental ones.
We have verified Conjecture 6.2 for all decoupling identities among CEGM integrands for CO 3,6 and CO 3,7 .
The fundamental decoupling identities also have a beautiful geometric interpretation. Consider a (k, n−1) GCO expressed as an arrangement of n−1 hyperplanes with labels in the set [n] \ {i}. Now introduce another hyperplane, labeled i, in the arrangement in all possible ways but subject to the condition that H i ∩ H j defines a predetermined arrangement. This geometric picture will allow us the connect fundamental identities with the non-partitioning identities in section 4 and hence provide a proof of conjecture 6.2 for k = 3.

Relation to Non-partitioning Irreducible Identities and k = 3 Proof
In section 4, we discussed two kinds of k = 3 irreducible decoupling identities, partitioning and non-partitioning. Here we prove that non-partitioning identities coincide with the fundamental identities.
Let us start by noting the following property. Proposition 6.3. All (3, n) GCOs that participate in a non-partitioning irreducible identity where label i was decoupled and which is generated by walls sharing label j all have the same (2, n − 1) color ordering in the j th position. In other words, their k-decreasing projections π (j) (Σ) coincide.
Sketch of Proof. Start with a (3, n−1) GCO Σ (i) in the set [n]\{i}. This defines a decoupling set in (3, n) by collecting all GCOs such that π i (Σ) = Σ (i) . Σ (i) also defines a configuration of n − 1 points in RP 2 . Now, according to the construction in section 4, a non-partitioning irreducible decoupling is found by selecting one of the points, say P j , drawing two lines intersecting at P j and containing two other points, such that the lines are adjacent. This means that the lines split RP 2 into two regions so that all the other n − 3 points are located in one of the regions. The identity is obtained by letting point P n wander around in the "empty" region and collecting all GCOs obtained in doing so. Clearly, any such GCOs will have the same (2, n − 1) color ordering in the j th position since such an ordering is obtained by reading the lines joining P j to each point, including P n , as we circle around P j , but all point are fixed, except P n (see section 5). However, P n is not allowed to escape the region bounded by two existing lines. This means that all GCOs participating in this identity have the same (2, n − 1) color ordering σ (j) in the j th entry.
A simple corollary is that all k = 3 non-partitioning identities are fundamental identities. We leave the proof to the reader. Since k = 3 non-partitioning identities were proven in appendix C, it follows that conjecture 6.2 holds for k = 3.

Discussions
In this work we completed the second element of the triality proposed in [2]. The triality refers to three different ways of computing color-dressed biadjoint scalar amplitudes. Color-dressed amplitudes are defined as a sum over generalized color orderings (GCOs) multiplied by partial amplitudes (see eq. (1.2)). In this first element of the triality we have the generalized Feynman diagram (GFD) technique described in detail in [2]. Each partial amplitude is computed as a sum over all GFDs which are locally planar with respect to the GCOs defining the partial amplitude. The second element in the triality is the computation of biadjoint amplitudes using CEGM integrals. In this work, we proposed CEGM integrands associated with a given GCO so that when integrated against the scattering equations they would produce the corresponding generalized biadjoint amplitude. This was explicitly checked for all (3,6) and (3, 7) partial amplitudes. The third element is construction in terms of a generalization of the positive tropical Grassmannian, associated with a given chirotope (and hence a given GCO), which we called chirotopal tropical Grassmannians in section 13 of [2]. In particular, in [29], the Global Schwinger integral over R (k−1)(n−k−1) was introduced as a compact formula for generalized partial amplitudes with GCOs of type 0 (i.e. k = 2 descendants). The main question here to complete the triality is the following: to what extent can this story be generalized to other GCOs?
The CEGM integrands we introduced in this work provide compatible systems of rational functions, associated with the connected components of the real configuration space X(k, n), as indexed by Generalized Color Orders. The numerator of each such integrand is uniquely determined (up to a sign) on X(3, n) for n ≤ 8, by identifying the one-dimensional residues on the common boundaries with each of the neighbors of the connected component. Starting at (3,9) and (4,8), some integrands are not completely determined in this way.
Could it be that additional constraints could arise by identifying higher dimensional residues? We do not a priori get a constraint from integrands of non-realizable pseudo-GCOs. Even so, we would not be able to eliminate all free parameters. It is not clear if taking iterated residues could fix the numerators of the integrands (this means that it is not clear whether our situation matches the positive geometry axioms of [21]). At this point, it seems that the natural way to fix the free parameters is by computing explicit generalized biadjoint amplitudes and requiring their values to match that obtained by summing over generalized Feynman diagrams.
There are numerous questions for future investigations into the CEGM integrals with integrands of general types. First of all, it is desirable to study whether we can read the kinematic poles [15,30] that appear in the doubly partial amplitudes m (k) (Σ,Σ) by just looking at the CEGM integrands, especially the Plucker variables in their denominators.
In CHY formulas, it is very easy to determine the poles of m (2) (α, β) by just considering the Plucker variables in the denominators of the standard Parke-Taylor factors (c.f. [31][32][33][34][35]). By making use of compatibility rules of poles, which is universal for any orderings, one can easily construct the whole amplitudes just from the poles [36,37], and in the case of generalized biadjoint amplitudes, [38] using the planar basis and matroid subdivisions. It would be interesting to explore how far away one can go for CEGM integrals in this direction. Other directions include studying the contribution of singular solutions [13] or the behavior of amplitudes [12,39] in soft and hard limits, factorization [40], smoothly splitting amplitudes and semi-locality [41], minimal kinematics to simplify the scattering equations and the resulting amplitudes [42,43], etc.
In this paper, we also studied irreducible decoupling identities in terms of CEGM integrands and it is highly desirable to find a systematic way to make use of them to generate algebraically linear independent sets of CEGM integrands, as the analog of the Kleiss-Kuijf basis in the stand k = 2 case [25]. Looking even further into the future, one would like to find the BCJ-like relations among CEGM integrands on the support of scattering equations, or BCJ-like basis [31,[44][45][46] of integrands. The latter may lay a foundation for possible BCJlike double copy relations [44,[47][48][49] among generalized amplitudes including the generalized biadjoint amplitudes.
It is well known that the CHY formula is closely related to the leading order of disk integrals of open string theory [50]. The latter was formally generalized to higher k formulas called Grassmannian stringy integrals [11], whose leading orders are closely related to the CEGM integrals.
There are many more avenues to explore. It is known that for globally planar integrands, what we call type 0, there is a connection to cluster algebras [10,14,[51][52][53][54][55][56]. It is also known that for X(3, n) with n < 9 there is an action of W (E n ), the Weyl group of E n , acting on the points mapping chambers to chambers [57,58]. It would be fascinating to make a connection with either of these topics. In fact, after the original posting of this article, the investigation was initiated in [59] for n=6, 7, where the connected components of the del Pezzo moduli space Y (3, n) were constructed. In 2009, Arkani-Hamed, Cachazo, Cheung, and Kaplan (ACCK) conjectured a formula that uses the higher k Parke-Tayler factors (2.10) as integrands and expresses tree-level amplitudes or loop discontinuities in N = 4 SYM in the sector with k negative helicity gluons in the planar limit as contour integrals [27]. The CEGM integrands can be thought of as a natural generalization of the the higher k Parke-Taylor factors and it would be interesting to see their possible applications to the ACCK formulas, especially the possible connections to non-planar on-shell diagrams and their Grassmannian formulations (see [60][61][62][63]).

Towards a Realization of Generalized Color Factors
In this work, we have treated the generalized color factors, c(Σ), as unknown. And their purpose so far has been that of a bookkeeping device for the partial amplitudes (1.2), When k = 2, color factors are given in terms of traces of generators of a Lie algebra. Decoupling identities are automatically obtained by a judicious choice of the generators and by asking the full color-dressed amplitude to vanish. The most pressing problem in this line of research is to find the explicit realization of generalized color factors, c(Σ), and use them to directly obtain decoupling identities.
In this work, we have derived identities among CEGM integrands and called them "decoupling" in analogy with the k = 2 version. Of course, for them to actually decouple something, it is necessary that the color-dressed amplitude vanishes when the color factors behave in a particular way.
As explained in [2], when we "decouple" one particle in a (3, 6) amplitude, say 1, all 372 GCOs are identified with one of twelve possibilities. Each set contains 31 GCOs that make up a decoupling set. We do not know the precise behavior of color factors, but if we assume that when two GCOs, say Σ I and Σ J , are identified, the corresponding color factors satisfy c(Σ I ) = ±c(Σ J ), then (7.1) would split into the 12 sets with 31 partial amplitudes in each. However, the 31 partial amplitudes can be made to cancel in smaller sets, i.e. the set is reducible.
The new element we learned in section 6 is that if we consider the double extension procedure one always gets irreducible identities.
This hints at the fact that for k > 2, one has to select a pair of particles i ̸ = j to define the decoupling. It is as if one were decoupling i with respect to j. In this case, the 31 GCO in a decoupling set are not all identities but they are separated into four sets. From the view point of the 372 GCOs, the {i, j} decoupling splits (7.1) (for (k = 3, n = 6)), into 48 sets. Each set contains either 7,8, or 9 GCOs. Moving on to (3,7), the 27240 GCOs are partitioned into 372 under any i th decoupling and into 1860 sets under any {i, j} decoupling.
We expect that requiring the generalized color factors to produce all identities, including the corresponding signs, would lead to enough constraints to provide a hint as to what algebraic structure is underlying their construction.

Number of Linearly Independent Integrands
For k = 2 it is known that the space of integrands has dimension (n − 2)!. In other words, all (n − 1)!/2 can be expressed in terms of a basis of (n − 2)! integrands. This has a geometric interpretation as the dimension of the top cohomology group of X(2, n). This is not an accident as when each integrand is combined with the measure on CP 1 it becomes an element in the top cohomology. It is also known that U (1) decoupling identities are enough to reduce the space to the correct dimension. The generalization to k > 2 and any n is an open problem. However, it is natural to expect that the fundamental decoupling identities can help us find the algebraically linearly independent CEGM integrands.
Let us comment on what we know so far and the prospects for the future. For (3, 6), we have proven that fundamental decoupling identities are indeed enough to reduce all 372 CEGM integrands down to 126 linearly independent ones.
The situation for (3, 7) is more complicated and we do not have a definite answer at this point. At this point, we have been able to show that fundamental identities can be used to express the 27240 CEGM integrands in terms of only 7890. However, it is known that the dimension should be 7470 = 7890 − 420. One possible explanation for the 420 discrepancy is that identities from the dual (4, 7) space are needed to reduce the number. We leave this exploration for the future.
However, it is worth mentioning that there are computations using finite fields that can be used to compute information of X(2, n) and (3, n < 10). The basic idea is to determine the number of rank k uniform matroids over F q . Somewhat surprisingly, in the cases mentioned above, there is a quasi-polynomial expression for the numbers. In appendix A of [17], T. Lam uses such polynomials to compute the Euler characteristic of the configuration spaces by evaluating the quasi-polynomials at q = 1. As it turns out, evaluating the quasi-polynomials at q = 0 provides the dimensions we are after [64]. For X(3, 6) one indeed gets 126 while for (3, 7) one gets 7470.
It is fascinating that we encounter matroid theory in all its guises in this subject. From positroids (i.e., matroids over the reals with special conditions), chirotopes (oriented matroids), to matroids over finite fields.
3 . can imagine, if a connected broken line does not have any intersection with a straight line, it is forced to form a polygon even in the projective space. Anyway, the above claim does not apply to n-gons. However, it is easy to see that only type 0 (3, n) GCO has an n-gon whose ordering is consistent with the k = 2 color ordering from which the type 0 (3, n) GCO descends. Therefore, we are still able to find all polygons for any k = 3 GCOs combinatorially. Claim: Any (3, n) arrangement of lines has n(n − 1)/2 + 1 polygons.
This can be proved easily as there are 4 polygons in (3, 3) arrangement of lines and whenever we add a generic line to an (3, n − 1) arrangement of lines, we get n − 1 more polygons.

B Comments on (3,9) Pseudo-GCOs
In this appendix, we collect two interesting properties of (3,9) pseudo-GCOs that we expect to be present in all (3, n > 8).

B.1 (3, 9) Non-realizable Pseudo-GCOs
Combinatorially, the non-realizable pseudo-GCO (3.54) is indistinguishable from any other GCO and so one could try to compute its corresponding integrand by looking at the set of "combinatorial" triangles, i.e., look for all triples {i, j, k}, such that {j, k}, {i, k} and {i, j} (B.1) This would suggest that the corresponding integrand should have ten poles of the form 1/∆ abc , with {a, b, c}, in the list of triangles. For n = 9, this would be a basic integrand with p = 1 and hence completely determined by the torus action on it. A simple counting of labels reveals that the numerator should be ∆ 456 . Now we find a contradiction. The list (B.1) contains the triangle {4, 5, 6} and hence the denominator has a factor of ∆ 456 . But now we find that it cancels with the numerator. It is as if the triangle had disappeared.
This phenomenon has a beautiful geometric interpretation. While it is impossible to realize (3.54) as an arrangement of lines, as shown in fig. 11, which was first proposed in [65] (see also [66]), it is possible to construct an arrangement of pseudo-lines, i.e., lines that bend, that has the intersection structure dictated by (3.54). If one tries to "straighten" the pseudo-lines to get an arrangement of lines, as shown in fig. 12, one finds that all pseudotriangles with bent edges, including {4, 5, 6}, are forced to degenerate into a point and hence disappear! Due to the presence of the non-realizable pseudo-GCOs, some triangle flips of GCOs are geometrically forbidden even though their arrangements of lines indeed contain those triangles. See an example in fig. 13 whose GCO reads, via {1, 6, 8} is geometrically forbidden because if we force triangle {1, 6, 8} to shrink in fig. 13, then the arrangement of lines will be forced to degenerate into the singular case in fig. 12 where all other triangles are also gone. We cannot blow up the triangle {1, 6, 8} again in the opposite direction unless we allow the straight lines to bend a little bit, which leads to the arrangements of pseudo-lines in fig. 11. Now let us comment on what we can do in Algorithm I once a triangle flip of a target GCO leads to a pseudo-GCO. By making use of the other 11 triangle flips of the GCO (B.2) and the residue relations there, the integrand for that GCO is fixed up to a single parameter, If we assume that the associated integrand for the non-realizable pseudo-GCO is zero and require that the residue of (B.3) at ∆ 168 to vanish, i.e., that it has a spurious pole, 1/∆ 168 , we can fix the parameter, y 0 → 0. If instead, we could force the residue to match that of the non-realizable psedu-GCO. Recall we concluded that the (3,9) pseudo-GCO is basic, i.e., p = 1. So we can assign it a "naive integrand", 1/(∆ 147 ∆ 159 ∆ 168 ∆ 249 ∆ 258 ∆ 267 ∆ 348 ∆ 357 ∆ 369 ). Then we find y 0 → ±1 instead and (B.3) would have the same residue at ∆ 168 up to a sign with the naive integrand of the non-realizable pseudo-GCO.

B.2 Another New Phenomenon for (3, 9) GCOs
In (3,9), there are GCOs connected by a triangle flip and share the same set of triangles. For example, the following two GCOs Their integrands have the same denominators but different numerators, The two GCOs are related by 2 ↔ 9, 3 ↔ 8, 4 ↔ 7 as are their integrands. Such GCOs are not necessarily of the same type. For example, the following two GCOs are also connected by flipping triangle {1, 5, 6},  where their numerators are related by 2 ↔ 7 but their denominators are not.

C Proof of k = 3 Irreducible Decoupling Identities
Here we prove Proposition 4.2 in the main text by making use of the residue theorem. Here all y variables are considered to be generic and fixed. Every minor ∆ abn becomes a linear function of z and without loss of generality, we assume that no two ∆ abn (z) share a root. This produces a rational function for every choice of v, f v (z) := (I(Σ 1 , z), I(Σ 2 , z), . . . , I(Σ N , z)) · v.
. The reason is the torus scaling (2.5). We also know that all possible poles of f v (z) are simple and of the form ∆ abn (z). We now have to prove that there exists a choice of v such that f v (z) has a zero residue at z * , and such that ∆ abn (z * ) = 0 for any {a, b}.
Finally, we are left to prove that the choice of v found above ensures that f v (z) has a zero residue at the solution to ∆ ijn (z) = 0 and to ∆ pqn (z) = 0.
In order to proceed it is necessary to introduce another degree of freedom by modifying our parametrization to include a second coordinate, w, M :=    1 0 0 1 y 15 y 16 · · · y 1n + z 0 1 0 1 y 25 y 26 · · · y 2n + w 0 0 1 1 1 1 · · · 1    . (C.5) Now, localizing on ∆ ijn (z, w) = 0 by solving for z = z(w) and treating each column in M (C.5) as a point in CP 2 implies that the points labeled i, j, n are on a projective line, i.e., a CP 1 . We can take w as the inhomogenous coordinate of the line. Our function f v (z(w), w) = g v (w) is now a rational function of the CP 1 defined by points i and j. Those elements in {I(Σ 1 ), I(Σ 2 ), . . . , I(Σ N )} that contain poles on the w, CP 1 , different from ∆ pqn (z(w), w) = 0, by construction always have a companion with the same residue but opposite in sign. This is because such poles correspond to triangle flips used in the algorithm. Therefore the only possible singularities of the function g v (w) is a simple pole at ∆ pqn (z(w), w) = 0. However, g v (w) = O(w −2 ) as w → ∞ and since ∆ pqn (z(w), w) is linear in w it must be that g v (w) = 0. Hence the residue of f v (z, w) at ∆ ijn (z, w) = 0 vanishes.
Repeating the same argument but localizing to the line ∆ pqn (z, w) = 0 we conclude that the function f v (z) is identically zero and hence we have a decoupling identity.
The last statement to prove is that the identity is irreducible. But this follows from the fact that on the CP 1 , the functions cancel their poles pairwise in analogy to a k = 2 decoupling identity, and therefore there are no proper subsets that can be made to vanish.

D Tables of GCOs and Identities
In this appendix, we present useful information needed to follow some of the arguments in the main text.

D.1 (3, 7) GCOs
Here we list the 11 representatives of (3,7) GCOs of different types given in Table 2  Here are the 31 explicit forms of GCOs, their types, and the integrands in fig. 6. In section 4, we introduced irreducible identities and a combinatorial way to find them. In the case (3,6), one can explicitly visualize the identities by selecting two of the ten lines in RP 2 constructed using the five points left after removing point 6. Each pair of lines intersects at a point and separates RP 2 into two regions.
When the intersection point is not one of the five special points, then the identities obtained by combining the GCOs on either one of the two regions give rise to an irreducible identity, i.e., these are partition identities. There are three different such configurations in (3,6). Two of them give rise to a pair of a 12-term and a 19-term identities. The other one leads to a pair of identities with 14 and 17 terms. These are shown in figure 15, 16, and 17.
When the intersection point is one of the five special points, then the identities obtained by combining the GCOs on either one of the two regions are not guaranteed to be irreducible. This is the case discussed in the main text but we repeat the description here for completeness. Partitioning identity obtained by considering lines L 14 and L 25 . The identity that contains chamber C 8 has 12 terms while the one that contains C 17 has 19 terms.  Partitioning identity obtained by considering lines L 12 and L 35 . The identity that contains chamber C 8 has 12 terms while the one that contains C 17 has 19 terms. Partitioning identity obtained by considering lines L 12 and L 45 . The identity that contains chamber C 8 has 14 terms while the one that contains C 17 has 17 terms.
Any special point belongs to four lines. The four lines split RP 2 into four regions, and each region gives rise to an identity. This is shown in fig. 6 where one can easily see that the identities contain 7, 7, 8 and 9 terms. Clearly, 7 + 7 + 8 + 9 = 31 since they partition the projective plane.