M Theory Lesson 285

After several weeks of wondering where it was, I finally managed to track down a paper by Sergey Fomin and Nathan Reading, Root systems and generalized associahedra. Read chapter 3.

On page 38 they consider Grassmannian spaces, in a way not unlike that currently popular among twistor (ex-string) theorists. Consider the example Gr$(2,4)$. For any complex $2 \times 4$ matrix, we can define $2 \times 2$ submatrices of the form $M_{kl} = (z_{1k}, z_{1l}; z_{2k}, z_{2l})$. Letting $P_{kl} = \textrm{det} M_{kl}$ for all allowed $k$ and $l$, we have the relation

$P_{ik} P_{jl} = P_{ij} P_{kl} + P_{il} P_{jk}$

Fomin and Reading call this an exchange relation, because in the form

$xy = ac + bd$

it describes a relation between different chorded square pieces of a polygon, just like in the associahedra diagrams. Each exchange relation describes an edge in an associahedron. There are as many variables as one needs to label the sides of a square, and the diagonals, namely $2n + 3$, where $n = 1$ in the case of the basic square.

Aside: Of course, I tried googling exchange relation and BCFW, but there were, unfortunately, zero hits.