Symbol alphabets from plabic graphs III: n = 9

Symbol alphabets of n-particle amplitudes in $$ \mathcal{N} $$ N = 4 super-Yang-Mills theory are known to contain certain cluster variables of G(4, n) as well as certain algebraic functions of cluster variables. In this paper we solve the C Z = 0 matrix equations associated to several cells of the totally non-negative Grassmannian, combining methods of arXiv:2012.15812 for rational letters and arXiv:2007.00646 for algebraic letters. We identify sets of parameterizations of the top cell of G+(5, 9) for which the solutions produce all of (and only) the cluster variable letters of the 2-loop nine-particle NMHV amplitude, and identify plabic graphs from which all of its algebraic letters originate.

Symbol alphabets from plabic graphs II: rational letters

Symbol alphabets of n-particle amplitudes in $$ \mathcal{N} $$ N = 4 super-Yang-Mills theory are known to contain certain cluster variables of G(4, n) as well as certain algebraic functions of cluster variables. The first paper arXiv:2007.00646 in this series focused on n = 8 algebraic letters. In this paper we show that it is possible to obtain all rational symbol letters (in fact all cluster variables) by solving matrix equations of the form C Z = 0 if one allows C to be an arbitrary cluster parameterization of the top cell of G+(n−4, n).

Symbol alphabets from plabic graphs

Symbol alphabets of n-particle amplitudes in $$ \mathcal{N} $$ N = 4 super-Yang-Mills theory are known to contain certain cluster variables of Gr(4, n) as well as certain algebraic functions of cluster variables. In this paper we suggest an algorithm for computing these symbol alphabets from plabic graphs by solving matrix equations of the form C ∙ Z = 0 to associate functions on Gr(m, n) to parameterizations of certain cells of Gr(k, n) indexed by plabic graphs. For m = 4 and n = 8 we show that this association precisely reproduces the 18 algebraic symbol letters of the two-loop NMHV eight-particle amplitude from four plabic graphs.

Links in the complex of weakly separated collections

International audience Plabic graphs are combinatorial objects used to study the totally nonnegative Grassmannian. Faces of plabic graphs are labeled by k-element sets of positive integers, and a collection of such k-element sets are the face labels of a plabic graph if that collection forms a maximal weakly separated collection. There are moves that one can apply to plabic graphs, and thus to maximal weakly separated collections, analogous to mutations of seeds in cluster algebras. In this short note, we show if two maximal weakly separated collections can be mutated from one to another, then one can do so while freezing the face labels they have in common. In particular, this provides a new, and we think simpler, proof of Postnikov's result that any two reduced plabic graphs with the same decorated permutations can be mutated to each other.

The twist for positroids

International audience There are two reasonable ways to put a cluster structure on a positroid variety. In one, the initial seed is a set of Plu ̈cker coordinates. In the other, the initial seed consists of certain monomials in the edge weights of a plabic graph. We will describe an automorphism of the positroid variety, the twist, which takes one to the other. For the big positroid cell, this was already done by Marsh and Scott; we generalize their results to all positroid varieties. This also provides an inversion of the boundary measurement map which is more general than Talaska's, in that it works for all reduced plabic graphs rather than just Le-diagrams. This is the analogue for positroid varieties of the twist map of Berenstein, Fomin and Zelevinsky for double Bruhat cells. Our construction involved the combinatorics of dimer configurations on bipartite planar graphs.

Full-Rank Valuations and Toric Initial Ideals

Let $V(I)$ be a polarized projective variety or a subvariety of a product of projective spaces, and let $A$ be its (multi-)homogeneous coordinate ring. To a full-rank valuation ${\mathfrak{v}}$ on $A$ we associate a weight vector $w_{\mathfrak{v}}$. Our main result is that the value semi-group of ${\mathfrak{v}}$ is generated by the images of the generators of $A$ if and only if the initial ideal of $I$ with respect to $w_{\mathfrak{v}}$ is prime. As application, we prove a conjecture by [ 7] connecting the Minkowski property of string polytopes to the tropical flag variety. For Rietsch-Williams’ valuation for Grassmannians, we identify a class of plabic graphs with non-integral associated Newton–Okounkov polytope (for ${\operatorname *{Gr}}_k(\mathbb C^n)$ with $n\ge 6$ and $k\ge 3$).