We have seen that every on-shell graph is associated with a (k×n) matrix C, where a reduced graph with nF faces gives us an (nF 1)-dimensional submanifold of the Grassmannian G(k,n). We have also seen that the invariant content of an on-shell diagram is given by the permutation that labels it. We will now link these two observations by showing that the submanifold in the Grassmannian associated with an on-shell graph is also characterized—for geometric reasons—by the same permutation which labels the graph.

Our discussion will be most transparent if we think of the Grassmannian in a complementary way to our presentation so far: instead of viewing the (k×n) matrix C horizontally, as a k-plane spanned by its rows, we want to now view C vertically—as a collection of n, k-dimensional columns. The GL(k)-invariant data to describe any configuration are ratios of minors: (a1 … ak)/(b1 … bk). Intuitively, a generic plane C would be one for which none of its minors vanish. Such a configuration would have k(n k) degrees of freedom. The vanishing of any minor of C implies some linear dependence among its columns. Allowing for all possible linear dependencies among the columns of C leads to the “matroid stratification” [102] of configurations, which is known to be arbitrarily complicated. (Indeed, it was proven in [103] that all algebraic varieties are part of this matroid stratification, so understanding this amounts to completely taming the entire category of algebraic varieties!) However, if we impose one small restriction on the set of admissible linear dependencies, we will find that a rich, simple, and very beautiful structure emerges.

The geometry and combinatorics of the positroid stratification

Notice that any configuration C associated with an on-shell, planar graph is endowed with a cyclic ordering for the columns ﹛c1, …, cn﹜. It is therefore natural to consider a stratification of G(k, n) that involves only linear dependencies among (cyclically) consecutive chains of columns. This is known as the positroid stratification [39, 40] (see also [34, 104])—the cells of which are called positroids—and will turn out to be precisely what is relevant to the physics of on-shell diagrams. In order to understand the connection most clearly, we will first discuss the stratification in some detail on its own, and show how these configurations are characterized by permutations.