The combinatorics of minuscule posets in mirror symmetry for homogeneous spaces

We will demonstrate the various ways minuscule posets appear in mirror symmetry statements for cominuscule homogeneous spaces in the joint work with Charles Wang (UMich). The mirror models of these spaces arise through Langlands duality, and hence use the corresponding minuscule representations. The weight spaces of these representations can be described using minuscule posets, so it is not surprising that the combinatorics of these posets govern the various objects appearing.