Classification of codimension-one boundaries

Determine all codimension-one boundary components of the reflected Lagrangian amplituhedron for positive external data, proving that they are precisely the reduced symmetric boundaries defined by \(\mu_r=0\) and the half-contained interval Mandelstam boundaries defined by \(s_{[a,b]}=0\), with the maximal cells having the stated block-Lagrangian and fiber-product forms, respectively.

Background

The paper constructs the reflected Lagrangian amplituhedron as the image of Karpman’s nonnegative reflected Lagrangian Grassmannian under multiplication by positive external data. Its proposed primary boundary functionaries are the reduced symmetric quantities $\mu_r$, whose squares are the symmetric Mandelstam variables $s_{J_r}=\mu_r^2$, and the Mandelstam variables $s_{[a,b]}$ for intervals contained in one half of the reflected label set.

The paper describes candidate maximal domain cells for these boundaries. Symmetric boundaries are expected to come from products of two smaller reflected Lagrangian Grassmannians, whereas half-interval boundaries are expected to have a fiber-product structure involving an ordinary momentum-amplituhedron factor, a smaller reflected Lagrangian core, and the reflected color-dual ordinary factor. The conjecture also asserts that asymmetric axis-crossing Mandelstams do not yield additional codimension-one boundary supports.