Cluster algebras of type $$D_4$$ D 4 , tropical planes, and the positive tropical Grassmannian

Abstract

We show that the number of combinatorial types of clusters of type D 4 modulo reflectionrotation is exactly equal to the number of combinatorial types of tropical planes in TP 5 . This follows from a result of Sturmfels and Speyer which classifies these tropical planes into seven combinatorial classes using a detailed study of the tropical Grassmannian Gr(3, 6). Speyer and Williams show that the positive part Gr + (3, 6) of this tropical Grassmannian is combinatorially equivalent to a small coarsening of the cluster fan of type D 4 . We provide a structural bijection between the rays of Gr + (3, 6) and the almost positive roots of type D 4 which makes this connection more precise. This bijection allows us to use the pseudotriangulations model of the cluster algebra of type D 4 to describe the equivalence of "positive" tropical planes in TP 5 , giving a combinatorial model which characterizes the combinatorial types of tropical planes using automorphisms of pseudotriangulations of the octogon.

Tropical Grassmannians, cluster algebras and scattering amplitudes

“…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.…”

“…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 ),…”

“…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.…”

The Geometry of Gaussoids

“…This variety has dimension 6 and degree 16 = 6!/(1 3 3 2 5). The following code in Macaulay2 [15] computes the 21 quadrics from the 35 quadrics that cut out the Grassmannian Gr (3,6) in P 19 :…”

“…They lie in tetrahedra EEEG of the tropical Grassmannian trop(Gr (3,6)). • 24 triangles of type ppC, like { p, p 12 , a 13|2 +a 23|1 +2p 12 +2p 123 }.…”