math.AG — Algebraic Geometry
Laurent Phenomenon Algebras on Minuscule Grassmannians and their Mirrors
We construct Laurent phenomenon algebra structures on the homogeneous coordinate rings of the half-spin Grassmannians and the Freudenthal variety. Their frozen variables are the linear coordinates in a Coxeter orbit, and the resulting boundary divisors are anticanonical. Together with the construction of Daisey--Ducat for the Cayley plane, these examples admit explicit mirrors described by Newton polytopes. We identify the upper Laurent phenomenon algebras of the mirror seeds with the intrinsic mirror algebras of Keel--Yu at the specialization , including the theta functions associated with the individual boundary components. The half-spin construction is uniform in the rank. In the exceptional cases, contact calculations and Laurent divisibility determine the local expansions; the Freudenthal comparison also uses exact calculations of divisorial valuations. We obtain integral comparisons and valuative independence of the intrinsic theta basis in every reachable seed.