Tropical Grassmannians, cluster algebras and scattering amplitudes

We provide a cluster-algebraic approach to the computation of the recently introduced generalised biadjoint scalar amplitudes related to Grassmannians \Gr{k,n}. A finite cluster algebra provides a natural triangulation for the tropical Grassmannian whose volume computes the scattering amplitudes. Using this method one can construct the entire colour-ordered amplitude via mutations starting from a single term.


Introduction
Recently a very interesting connection between scattering amplitudes and tropical geometry has been uncovered [1,2]. The connection outlined so far is for tree-level biadjoint φ 3 amplitudes, which can be related to the series of tropical Grassmannians Gr(2, n) and a generalisation to higher Grassmannians Gr(k, n). Such amplitudes also have a formulation in terms of a set of scattering equations which generalise the usual scattering equations of [3][4][5] Tropical Grassmannians are defined as a space of solutions to a set of tropical hypersurface conditions which derive from the defining Plücker relations of the Grassmannian. An important ingredient in the relation to the generalised biadjoint scattering amplitudes is the notion of positivity which singles out a particular region in the tropical Grassmannian. We describe here the tropical formulation of the Grassmannian spaces and how to select the positive region. We will see that this coincides with the criteria recently used in [1,2] to determine the generalised φ 3 amplitudes for Gr (3,6) and Gr (3,7).
We will also develop the link further and describe a relation of the positive tropical Grassmannians to certain cluster algebras, as developed by Fomin and Zelevinsky [6,7]. These same cluster algebras have also arisen in the study of the singularities of loop amplitudes in planar N = 4 super Yang-Mills theory [8]. The cluster algebra picture provides extremely efficient calculational tools for determining the relevant positive solutions to the tropical hypersurface conditions, allowing for spaces of even quite large dimension to be simply constructed.
Once the positive region is obtained, the generalised biadjoint φ 3 amplitudes can be constructed as its volume in a direct generalisation of the picture described in [9]. Such a volume can be obtained additively via a triangulation of the region into simplexes. One such triangulation is provided by the (dual of the) associated cluster polytope. For the Gr(2, n) cases these polytopes are the A n−3 associahedra. In the Gr (3,6) case this corresponds to the D 4 polytope while in the Gr (3,7) case it is the E 6 polytope familiar from heptagon amplitude in planar N = 4 super Yang-Mills theory. For the Gr (3,8) case we can obtain a triangulation from the E 8 cluster polytope. This triangulation has the feature that it makes use of eight spurious vertices generated by the cluster algebra but not strictly needed to compute the volume. The above cases exhaust the list of finite Grassmannian cluster algebras.
A feature of the polytopes arising as positive tropical Grassmannians is that in general their facets are not all simplexes. This means that there is a redundancy in parametrising their volumes since they may be triangulated (or cut into simplexes) in multiple ways, each yielding a seemingly different but actually equivalent way of obtaining the volume. In physical language this means there are multiple ways of writing the amplitude which are in fact equivalent due to non-trivial identities between different contributions.
The non-simplicial nature of certain facets may also have a bearing on the analytic structure of loop amplitudes in planar N = 4 theory. In the case of Gr (3,7) it would be relevant for the heptagon amplitudes studied in [10,11] where it should be related to the recently discovered property of cluster adjacency [11,12] which forbids certain consecutive pairs of branch cuts in loop amplitudes and is related to the Steinmann relations. The non-simplicial facets can be thought of as a combination of simplexes, which corresponds in the cluster polytope to shrinking edges so that many clusters combine together.
We describe how the cases of Gr (3,6) and Gr(3, 7) fit into the above picture and we extend it to the case of Gr (3,8) whose positive tropical version corresponds to the E 8 cluster algebra. Since all these cluster algebras are finite, the triangulation procedure works in exactly the same way for all of them. Nevertheless the correspondence of between the cluster algebra and the fan for each case contains intricacies of different nature with valuable lessons and we elaborate on these in sections dedicated to different Grassmannians.
Before this we review the interpretation of the biadjoint φ amplitude as the volume of the dual to a kinematic realisation of the associahedron. We then illustrate all the main principles of the tropical Grassmannian, its positive part and the connection to cluster algebras in the example of Gr(2, 5).

Amplitudes from volumes of dual associahedra
In [9] a connection between biadjoint scalar amplitudes and volumes was made. The main idea is to introduce a kinematic realisation of the associahedron. This is done as follows. Given an ordered set of light-like momenta p 1 , . . . , p n satisfying momentum conervation one introduces dual coordinates, with all indices treated modulo n. The 1 2 n(n − 3) square distances (x i − x j ) 2 = X ij can be related to Mandelstam invariants via Note that the momenta being null implies X i,i+1 = 0. The two-particle Mandelstam invariants s ij = (p i + p j ) 2 can be related to the dual variables via To define the kinematic associahedron we take all X ij positive and choose (n−3) coordinates, e.g. the X 1,i for i = 3, . . . , (n − 1). The remaining 1 2 (n − 2)(n − 3) independent variables need to be constrained in order to obtain a space of dimension (n − 3). To do this we impose 1 2 (n − 2)(n − 3) conditions which we take to be of the form for positive constants c ij . The coordinates X 1,i are then constrained to run only over a certain region: the kinematic associahedron. For the n = 5 example the conditions (2.4) become The coordinates (X 13 , X 14 ) then run over a region with the shape of a pentagon as shown in Fig. 1.
To obtain the dual of the kinematic associahedron it is helpful to embed it into projective space P n−3 . We introduce the auxiliary point Y = (1, X 13 , X 14 , . . . , X 1,n−1 ). The boundary conditions X ij = 0 of the kinematic associahedron become Y · W ij = 0 with W ij given by projective dual vectors determined by the conditions (2.4).
In the case n = 5 we have Y = (1, X 13 , X 14 ) and W 13 = (0, 1, 0) , These dual vectors define the dual to the Gr(2, 5) kinematic associahedron. Its volume may be computed by first triangulating it, e.g. by picking the reference point W * = (1, 0, 0) and adding the volume of the five triangles formed by W * and two adjacent dual vectors according to . (2.7) In this way we obtain the sum of five terms,

Tropical Grassmannians and amplitudes
The Grassmannian Gr(k, n) is the space of k-planes in n dimensions. The Grassmannian can therefore be parametrised by a k × n complex matrix with the k rows specifying a k plane. Since the plane is invariant under the action of GL(k) transformations one must mod out by the action of GL(k), leaving a space of dimension k(n − k). The Grassmannian may also be specified in terms of the minors of the matrix. The (k × k) minors p i 1 ,...,i k (Plücker coordinates) of any matrix obey homogeneous quadratic relations (Plücker relations) obtained by antisymmetrising (k + 1) indices, p i 1 ,...,ir,[i r+1 ,...i k p j 1 ,...,j r+1 ],j r+2 ,...,j k = 0 . (3.1) In the Gr(2, n) case the Plücker relations are given by the familiar n 4 three-term equations p ij p kl − p ik p jl + p il p jk = 0 , 1 ≤ i < j < k < l ≤ n . (3. 2) The Plücker relations define a subspace in the Plücker space parametrised by the n k Plücker coordinates p i 1 ,...,i k . Algebraically this space may be thought of as the ideal generated by the quadratic Plücker relations inside the ring of polynomials in the Plücker coordinates. After quotienting by a global rescaling of all Plücker coordinates the subspace satisfying the Plücker relations can be identified with the Grassmannian Gr(k, n) of dimension k(n − k).
The original Plücker relations are actually homogeneous in n independent rescalings p i 1 ,...,i k → t i 1 . . . t i k p i 1 ,...,i k with t i ∈ C * . If we quotient by all of these scalings instead of just the overall scaling we obtain a smaller space, which has dimension m = (k − 1)(n − k − 1) and corresponds to taking the columns of our original (k × n) to be elements of P k−1 instead of C k . There exists a tropical version of the above construction. In tropical geometry one takes the generating relations of the ideal and replaces multiplication with addition and addition with minimum. For example the generating quadratic polynomials of the Gr(2, n) Plücker relations (3.2) become the tropical polynomials which are piecewise linear maps on the space of n 2 variables w ij ∈ R. Piecewise linear maps have special surfaces between one region of linearity and another. Such surfaces are called tropical hypersurfaces and are attained when at least two of the terms of the tropical polynomial simultaneously attain the minimum. In other words the tropical polynomial (3.4) defines the following tropical hypersurfaces, w ij + w kl = w ik + w jl ≤ w il + w jk or w ij + w kl = w il + w jk ≤ w ik + w jl or w ik + w jl = w il + w jk ≤ w ij + w kl . (3.5) When we have many polynomial relations we must simultaneously satisfy the conditions arising from each polynomial relation. In the case of Gr(2, n) we must simultaneously satisfy the hypersurface relations coming from every Plücker relation, i.e. for every choice of {i, j, k, l} in (3.2).
Note that for any solution {w ij }, any global positive rescaling of the w ij will also obey the conditions. Solutions therefore form rays emanating from the origin and can be represented by an n 2 -component vector, or more generally for Gr(k, n) an n k -component vector. Note also that if {w ij } are solutions of the above conditions then so are {w ij + a i + a j } for any set of n constants a i ∈ R. Such a shift symmetry is referred to as lineality. In the context of generalised biadjoint scattering amplitudes it corresponds to momentum conservation.
Quotienting the space of solutions of the tropical hypersurface conditions (3.5) by a single global shift with a i = a corresponds to the tropical version of the Grassmannian. Quotienting by all shifts corresponds to the tropical version of the space Conf n (P k−1 ). Here we are interested in the latter case where we quotient by all shifts. Despite this we will refer to the space obtained simply as the tropical Grassmannian and we use the notation Tr(k, n) to denote it.
The sign of the individual terms of the Plücker relations (3.2) is lost through tropicalisation. We can recover the information by identifying positive hypersurfaces regions as those whose defining terms in (3.2) have opposite signs [13]. The positive part of Tr(2, n) (denoted Tr + (2, n)) is closely related to the dual of the kinematic associahedron that we described above and hence can be identified with the canonically ordered amplitude of the bi-adjoint φ 3 theory. This fact is at the heart of the recent generalisation of the biadjoint amplitudes to general Tr(k, n) [1].
Such generalised biadjoint amplitudes can also be related to a generalisation of the scattering equations [1,3,4] to CP k−1 and through them to amplitudes of a generalised scalar bi-adjoint theory [5]. Focusing for simplicity to k = 3, we consider homogenous coordinates of n particles on CP 2 and form the 3 × n matrix We then define the potential function where [ijk] represent minors of m and s ijk are generalized Mandelstam variables that satisfy j =k s ijk = 0, ∀i. We can now write down the amplitude of a generalised scalar theory as where S 3 , i denotes derivative with respect to i and the generalized Parke-Taylor factors involve two orderings α and β and are given by . Let us now describe the Gr(2, 5) case. In this case we have ten Plücker coordinates p ij and the Plücker relations are given by (3.10) and four more relations obtained from cyclic rotation of the labels. These relations give rise to the tropical hypersurface conditions (3.5) for {i, j, k, l} given by {1, 2, 3, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {1, 3, 4, 5} and {2, 3, 4, 5}. Each of these five cases must be simultaneously satisfied.
We arrange the coordinates in the standard, lexicographical order, and define ray vectors as In this case we can also combine certain solutions. For example we find that any positive linear combination ae 12 + be 34 with a, b > 0 is also a solution. However no positive linear combination ae 12 + be 13 is a solution. We thus obtain a notion of connectivity of solutions: two solutions are connected if any positive linear combination of them is a solution. We say that there is an edge between such solutions. In the case of Gr(2, 5) we can never combine three or more solutions to obtain another solution. In higher dimensional examples one can obtain triangles of solutions and higher dimensional faces.
Performing permutations on the indices leads us to find 15 edges between the 10 vertices given by the e ij . The full set of solutions corresponding to the tropical Grassmannian Tr (2,5) can be depicted by the Petersen graph shown in

The positive tropical Grassmannian from webs
In [15] an alternative way of describing just the positive part Tr + (k, n) was given. In this approach one introduces a grid called a web diagram with labels {1, . . . , k} on the horizontal edge and labels {(k + 1), . . . , n} on the vertical edge. The squares of x 1 x 3 x 5 x 2 x 4 x 6 Figure 3: Example web diagram for Gr (3,7).
the grid are populated with variables x i . In Fig. 3 we illustrate the general procedure in the case of Tr (3,7). A Plücker coordinate is indexed by a set K of k distinct labels chosen from {1, . . . , n}. We denote the set {1, . . . , k} by [k]. We may then associate a Plücker coordinate p K to a set of paths on the web diagram as follows. Consider sets S of non-intersecting paths consistent with the arrows which go from We denote the set of all such sets as Path(K). For each path in a given set S we record the product of the variables in the squares above the path (if there are no squares above the path we record the value 1). For a set S of paths we take the product over all paths in the set which we denote by Prod S (x) (if the set is empty we record the value 1). Finally we sum over all possible choices of sets S of such non-intersecting paths, i.e. we sum over S ∈ Path(K), The procedure is best illustrated with an example: consider the Plücker coordinate p 357 in the case illustrated in Fig. 3. We need to consider sets of non-intersecting paths from {1, 2} to {5, 7}. We find the possible choices illustrated in Fig. 4. The final result for the Plücker coordinate is therefore, To consider the tropical Grassmannian we tropicalise the resulting polynomial, replacing multiplication with addition and addition with minimum to obtain w K . Following exactly the same logic for the simpler example of Gr(2, 5) we obtain 3 2 1 4 5 6 7 x 1 x 3 x 5 x 2 x 4 x 6 3 2 1 4 5 6 7 x 1 x 3 x 5 x 2 x 4 x 6 3 2 1 4 5 6 7 x 1 x 3 x 5 x 2 x 4 x 6 3 2 1 4 5 6 7 x 1 x 3 x 5 x 2 x 4 x 6 3 2 1 4 5 6 7 x 1 x 3 x 5 x 2 x 4 x 6 (as in [15]) The resulting tropical minors are piecewise linear functions in the space parametrised by (x 1 ,x 2 ). Each such function defines tropical hypersurfaces in exactly the same way as before. Taking the union over the tropical hypersurfaces gives rise to a fan with five domains of linearity separated by five rays as illustrated in Fig. 5a. We may label the five rays by where e 1 and e 2 are the two-component vectors, More generally, the tropical minors in Tr + (k, n) define a polyhedral fan in the (k − 1)(n − k − 1)-dimensional space ofx i variables with many domains of linearity separated by walls of codimension one. The walls intersect in surfaces of codimension two and so on all the way down to individual rays of dimension one defined by the multiple intersection of (at least) ((k − 1)(n − k − 1) − 1) walls. We illustrate the fan obtained in the case of Tr + (2, 6) in Fig. 5b.   where the equivalence corresponds to the linearity shift w ij → w ij + a i + a j with (a 1 , a 2 , a 3 , a 4 , a 5 ) = 1 2 (1, 1, −1, −1, −1). Doing the same for each of the five rays in (4.3) we indeed obtain the ten-component vectors {e 12 , e 45 , e 23 , e 15 , e 34 }, precisely the five positive rays in the list of ten solutions given in (3.14). The regions between the rays in Fig. 5a then correspond to the edges between the positive rays in Fig. 2.
We may also recover the rays (4.3) from {e 12 , e 45 , e 23 , e 15 , e 34 } by tropically evaluating the coordinates x 1 and x 2 which are given by The first component of the dual vector W 24 may be recovered by demanding for example Y · W 24 = y · ev(−e 1 ) = s 23 = X 24 , (4.8) where we recall Y = (1, X 13 , X 14 ) and y = (s 12 , . . . , s 45 ). Since the dual vectors are equivalent to the defining constraints of the kinematic associahedron, this gives us a way to recover the kinematic associahedron from the tropical minors. The expressions of the web variables x i in terms of Plücker coordinates in fact identifies them with the cluster X -variables of [6,7] for the initial cluster of the Gr(2, 5) cluster algebra. Indeed more generally the web variables are identified with the X -coordinates of the initial cluster for any Gr(k, n). As we now outline, we can use the algebraic machinery of the cluster algebra to generate all the ray vectors describing the positive tropical Grassmannian Tr + (k, n).

The tropical Grassmannian and cluster algebras
As mentioned above, we can identify cluster X -coordinates with web variables. As we shall see we can also identify the ray vectors with cluster A-coordinates. This allows us to generalise the notion of mutation to these rays such that we can generate all rays in the fan in a cluster algebraic way. Before we demonstrate this it is useful to revisit mutation for A-coordinates in Grassmannian cluster algebras.
A Gr(k, n) cluster is identified by its m = (k − 1)(n − k − 1) unfrozen nodes and an m × m exchange matrix B which encodes the connectivity of the nodes within the cluster. Mutating at node k transforms B to B ′ given by where [x] + = max(x, 0). The mutated node also transforms, given by Generalising mutations to rays requires additional information, namely an additional matrix C (the coefficient matrix), its mutation given by 2 To each unfrozen A-coordinate we associate a ray vector g. We start by constructing the initial cluster such that the m unfrozen nodes are the m basis vectors for R m g l = e l , l = (1, . . . , m).

(5.4)
We then select a node k to mutate on, following the mutation rule where b 0 j corresponds to the jth column of B 0 , the exchange matrix for the initial cluster. We can then repeat this process as many times as required to generate a vector for each unfrozen A-coordinate. In the cases where the cluster algebra is of finite type (in this context the cases are Gr(2, n), Gr(3, 6), Gr(3, 7) and Gr (3,8)) we obtain a finite cluster polytope by performing all mutations where each vertex is associated to a cluster. Each face of the polytope is associated to an unfrozen A-coordinate a and also by the above procedure a vector g.
The advantage of having the relation of the positive tropical fan to the cluster algebra is that it gives us a very easy algebraic way to generate the relevant ray vectors to describe the fan. Once we have the fan we can embed it into the original Plücker space using the tropical minors and compute its volume to obtain the generalised scattering amplitude.
We illustrate the resulting polytope in the simplest case is given in Figure 6a. It has five clusters connected in the shape of a pentagon. This pentagon is the dual of the pentagon obtained from intersecting the the fan illustrated in Fig. 5a with the unit circle; its edges are labelled with ray vectors (4.3).
In fact for Gr(2, n) the polytope obtained by intersecting the positive tropical fan with the unit sphere is always the dual polytope of the Gr(2, n) associahedron or Stasheff polytope. For example in Fig. 6b we show the vectors associated to the faces of the A 3 associahedron. The dual polytope coincides with the intersection of the Gr(2, 6) positive tropical fan with the unit sphere given in Fig. 5b.
For the other finite cases the tropical positive fan gives polytopes that are closely  related to the duals of the cluster polytopes as we now describe.
6 Tr + (3, 6) Let us now consider the first case of the generalised biadjoint amplitudes which was addressed in [1]. In analogy to the Gr (2, n) cases of the previous section, the generalised amplitude for higher k and n can be interpreted as the volume of the computed by triangulating the relevant Tr + (3, 6) fan. Following [14] we start by considering by the Plücker relations of Gr (3,6), of which there are two kinds, three-term relations and four-term relations, x 1 x 3 x 2 x 4 Figure 7: The web diagram for Gr (3,6).
space R 20 separated by hypersurfaces defined as the set of points at which the two smallest arguments of the min functions are equal. Consider for instance, the first tropical polynomial in (6.2). It gives rise to a boundary between two cones if one of the following is satisfied: This polytope contains 65 vertices [14]. The part of the polytope that is relevant for a planar ordering is its positive part Tr + (3,6). In [1] the positive vertices were determined by requiring compatibility with a planar ordering for the scattering amplitude. Here we identify the positive rays by requiring that they satisfy the hypersurface conditions generated by monomials in the the Plücker coordinates with opposite signs as we described in Sect. 3. This leaves us with 16 rays out of 65, coinciding precisely with the set of [1]. They are e 123 and cyclic, f 1234 and cyclic and g 12,34,56 , g 23,45,61 , g 34,12,56 and g 45,23,61 .
The Gr + (3, 6) web diagram shown in Fig. 7 produces a matrix with following piecewise linear tropical minors [13,15], w 12i = w 134 = w 234 = 0 , w 135 = min(0,x 1 ), The regions of linearity of the tropical minors (6.6) define the fan for Tr + (3, 6) and its intersection with the unit sphere S 3 is a polytope with 16 vertices, 66 edges, 98 triangles and 48 three-dimensional facets. The tropical X -coordinates are given bỹ With the above relations (6.6) and (6.7) we can go back and forth between the representation of the 16 positive vertices in terms of the e ijk and in terms of a fourcomponent representation which we can also obtain from cluster mutations as we now describe.

Triangulating Tr + (3, 6) with clusters
Unlike in Gr(2, 5), the Tr + (3, 6) fan contains facets that are not simplicial. In particular, it contains 46 simplicial facets and two bipyramids defined by five vertices. This is a common feature of k > 2 (tropical) Grassmannians. To see this, first recall that (k − 1)(n − k + 1) rays define a facet of the fan if an arbitrary positive linear combination of them solves the positive versions of inequalities derived from the Plücker relations. In particular Tr + (3, 6) has 2 such facets with five vertices that form bipyramids. These non-simplicial bipyramids are arranged inside the fan Tr + (3, 6) as sketched in Figure 8. The fan Tr + (3, 6) is closely related to the dual of the Gr + (3, 6) associahedron in that the latter provides a natural triangulation of the former [13]. The vertices of the dual of the associahedron correspond to cluster A-coordinates. Two vertices are connected by an edge when the corresponding pair of A-coordinates appear together in a cluster, i.e. are cluster-adjacent in the sense of [11]. By definition, a pairwise connected quadruplet of vertices of the dual Gr + (3, 6) associahedron corresponds to a cluster, which in turn can be identified as a simplex triangulating Tr + (3, 6).
We begin with the initial cluster where we have also given their evaluations through the tropical minors (6.6) and the corresponding positive solutions given above. The fact that these five vertices form a single bipyramid rather than two tetrahedral facets can be seen from the linear relation, f 1234 + f 1256 + f 3456 = g 12,34,56 + g 34,12,56 . (6.10) Note that the cluster algebra provides a canonical way of determining a triangulation. In particular the bipyramid formed by the five rays described above is triangulated by two clusters whose vertices are given by {f 1234 , f 1256 , f 3456 , g 12,34,56 } and {f 1234 , f 1256 , f 3456 , g 34,12,56 }.
Equipped with the cluster triangulation, we can express the scattering amplitude as a sum over clusters: where as before y = (s 123 , . . . , s 456 ) is the lexicographically ordered vector of Mandelstam invariants, r a is the representation of the A coordinate a as a ray inx coordinates and ev means the evaluation using the tropical minors in (6.6).
Using this identification, we can write read off the two terms in the amplitude from the directly from the two clusters as 3 which was noted in [1] to correspond to a different triangulation of the bipyramid. However the cluster algebra prefers a particular one of these triangulations.
7 Tr(3, 7): the amplitude from E 6 clusters In this section we explicitly demonstrate how the triangulation of the fan associated to the positive tropical Grassmannian Tr + (3, 7) can be worked out from the Gr(3, 7) cluster algebra. As in the previous section, one can either compute F 3,7 from the web Web 3,7 or run the cluster-algebra machinery to obtain the generalised amplitude without even referring to Tr (3,7). Nevertheless let us first describe Tr + (3, 7) starting from Tr (3,7) and elaborate on a situation that is not encountered in Grassmannians of lower dimension.
The To compute positive Grassmannian Tr + (3, 7), we select out of the 721 rays above those which solve the positive versions of tropicalised Plücker relations. One finds that 49 of them satisfy such relations. This seems incompatible with the fact that the cluster algebra has 42 distinct unfrozen A-coordinates.
The resolution to this discrepancy is that seven positive rays of the type b 6 are linear combinations of three mutually-connected rays of type b 3 , any positive linear combination of which is a solution. In other words, b 6,123456 is in the middle of a triangular 2-face of Tr + (3, 7) and is not necessary to define a cone of the fan.
We again resort to the relevant cluster algebra E 6 to obtain a triangulation on which we evaluate the amplitude. The E 6 cluster algebra has 833 clusters that give the vertices of the associahedron. These 833 clusters make up the simplexes of the triangulation each of which contain six vertices.
If we employ the duality between Gr(3, 7) and Gr (4,7) and work in terms of the latter, we can relate the positive vertices above to the established notation for A-coordinates in the literature on N = 4 amplitudes [10,11]. The different types of rays classified in (7.1a)-(7.1f) nicely match the conventional cluster A coordinates: where the rest of the correspondence can be worked out by cyclic rotations of the second indices of the a ij and the arguments of the b i . With this correspondence, we find that the E 6 initial cluster a 24 a 37 a 13 a 17 a 32 a 27 (7.3) produces the following term in the amplitude 1 (y · b 1,1234567 )(y · b 3,1234567 )(y · b 2,1234567 )(y · b 2,4567123 )(y · b 3,6712345 )(y · b 1,5671234 ) , (7.4) with y = (s 123 , . . . , s 567 ). We then mutate these rays according to (5.5) iteratively until we cover all 833 clusters of the E 6 polytope. Recovering the corresponding kinematic invariants using (7.2), we can construct the Gr(3, 7) amplitude as volume of the positive tropical Grassmannian. An expression for this amplitude is provided in the ancillary file Gr37amp.m .

Gr(3, 8): redundant triangulations
In this section we will run the same construction in Gr (3,8) to provide a conjecture for the canonically-ordered part of the generalised biadjoint amplitude that one would obtain by solving the scattering equations for this Grassmannian.
We start with the initial cluster of Gr (3,8) where the b vertices are given below in (8.5) and as before y = (s 123 , . . . , s 678 ).
We then generate all 25,080 clusters using the mutation rules of [16] which we have adapted in equation (5.5). These clusters contain 128 distinct vectors in R 8 , identified with the 128 A coordinates of Gr (3,8). As usual, the Plücker coordinates of these vectors provides us the factors in the denominator of every term in the amplitude. We provide all 25080 terms in the ancillary file Gr38amp.m .
Let us comment further on the correspondence between the Tr + We can compare the Plücker coordinates of the vectors we obtain to the rays of another object called the Dressian Dr (3,8), studied in [18]. Dr (3,8) is a nonsimplicial fan that consists of 15470 rays which split into 12 symmetry classes of size (56, 70, 28, 420, 56, 1260, 420, 560, 1680, 840, 5040, 5040). These define facets in groups of sizes ranging from 8 to 12 . While all rays of Dr(k, n) are expected to be rays of Gr(k, n), the converse is not true. Indeed the Dressian Dr (3,8) does not capture the rays b 8 which give rise to "superfluous" triangulations.
The rays of Dr (3,8), positive and non-positive, are explicitly given as:  lie in the positive region in the sense that they satisfy the positive version of the inequalities (3.5). These vectors are in one-to-one correspondence with the 120 nonredundant rays generated by the cluster algebra. Note that the redundant vectors b e that we encountered in Gr (3,8) are of different nature to the b 6 of Gr (3,7). While both types of vectors are not rays of the relevant fan, unlike the b e , the b 6 are not generated by the cluster algebra.
9 Conclusions and outlook to Gr (4,8) In this paper we have utilised cluster algebra technology to construct tree-level biadjoint amplitudes on Gr(3, n) for n = 6, 7, 8. These amplitudes arise from scattering equations on the corresponding Grassmannians [1,2] and the relevance of cluster algebras for these amplitudes arises from the interpretation of these amplitudes as volumes of certain geometric objects. In the cases we studied in this paper these objects are polyhedra in (k − 1)(n − k − 1) − 1 dimensions, where k = 3.
Cluster algebras provide a natural triangulation of the polyhedra whose volumes correspond to the scattering amplitudes. Therefore we were able to employ mutation rules to "bootstrap" the amplitude starting from a single term only. In particular we provided a prescription for the volume of the simplex that corresponds to the initial cluster and obtained the volumes of the remaining simplexes through consecutive cluster mutations.
Each of the cases we considered has new features that provide important lessons. In n = 6 we saw that the clusters triangulate the bipyramids of Tr + (3, 6) into two simplexes. In n = 7 we identified that positive rays that define cones of Tr + (3, 7) are not rays of the Tr + (3, 7) fan and are also not detected by the cluster algebra. When we studied the n = 8 case, we found that the cluster algebra generates redundant triangulations of the Tr + (3,8) fan.
Having studied the fans corresponding to various Grassmannians, a natural direction to take is to attempt to construct the fan for Tr + (4,8), which corresponds to the positive part of Gr (4,8). The cluster algebra of the latter is expected to capture the rational symbol letters of 8-particle amplitudes in N = 4 but the fact that the Gr(4, 8) cluster algebra is infinite has been a forbidding obstacle in utilising cluster algebras in the computations of these amplitudes.
We find that restricting the mutations to clusters that contain only rays obeying the full number of intersection conditions for Tr + (4, 8) closes on a finite number of 169,192 clusters. The corresponding A coordinates in these clusters provides us with a finite, dihedrally-complete alphabet of 356 rational letters. In particular, this alphabet contains the rational letters reported in [19]. It would be interesting to check if these letters are in correspondence with the faces of the polytope found by Arkani-Hamed, Lam and Spradlin, as reported in [20].